<
Intro
duction
Advanced
Calculus
Introduction
So
fa
r
in
calculus
y
ou
have
develop
ed
the
to
ols
to
answ
er
the
follo
wing
questions
ab
out
a
function
of
one
va
riable:
1
Ho
w
quickly
do
es
the
value
of
the
function
change
as
the
input
changes?
2
Ho
w
do
w
e
estimate
the
value
of
the
function
nea
r
a
p
oint?
2
Introduction
3
What
a
re
the
maximum
and
minimum
values
of
the
function?
3
Introduction
4
What
is
the
a
rea
under
the
graph
of
the
function?
What
do
es
it
mean?
These
a
re
all
useful
to
ols,
but
w
e
can’t
apply
them
everywhere
that
w
e
w
ould
lik
e
to.
4
Introduction
Many
measurable
quantities
can
b
e
found
to
dep
end
on
the
value
of
multiple
inputs.
These
are
multiva
riable
functions
like
z
=
F
(
x
,
y
),
where
z
is
a
function
of
t
w
o
indep
endent
va
riables.
Examples
app
ea
r
in
all
the
sciences
1
Chemistry:
V
=
nrt
P
2
Physics:
F
=
GMm
r
2
3
Economics:
P
=
P
0
e
rt
Figure:
The
graph
of
a
t
w
o-va
riable
function
5
Introduction
W
e’ll
also
develop
to
ols
fo
r
integrating
functions
over
mo
re
exciting
objects,
fo
r
instance:
1
The
a
rea
ab
ove
a
curve
in
the
plane.
6
Introduction
2
A
vecto
r
field
acting
on
a
pa
rticle
traveling
through
it.
7
Introduction
3
A
fluid
flo
wing
through
a
surface.
8
Introduction
Goals
By
the
end
of
this
course,
w
e
should
have
the
to
ols
to:
cho
ose
a
purchase
that
maximizes
utilit
y
,
given
a
budget
constraint,
p
redict
the
p
otential
erro
r
in
a
chemistry
exp
eriment,
derive
the
surface
a
rea
of
a
sphere,
and
calculate
the
amount
of
energy
abso
rb
ed
b
y
a
sola
r
panel.
9
Section
1.3
Double
Integrals
in
P
ola
r
Co
o
rdinates
Goals:
1
Convert
integrals
from
Ca
rtesian
to
p
ola
r
co
o
rdinates.
2
Evaluate
integrals
in
p
ola
r
co
o
rdinates.
Question
1.3.1
What
Are
Pola
r
Co
ordinates?
Definition
The
p
ola
r
coordinates
of
a
point
are
denoted
(
r
,
θ
)
where
θ
(“theta”)
is
the
direction
to
the
point
from
the
origin
(measured
anticlo
ckwise
from
the
p
ositive
x
axis).
r
is
the
distance
to
the
p
oint
in
that
direction
(negative
r
means
travel
backw
a
rds).
Unlik
e
Ca
rtesian
co
o
rdinates,
a
p
oint
can
b
e
rep
resented
in
several
different
w
a
ys.
(1
,
0)
=
(1
,
2
π
)
=
(1
,
4
π
).
(1
,
0)
=
(
−
1
,
π
)
(0
,
α
)
=
(0
,
β
)
for
all
α,
β
.
11
Question
1.3.1
What
Are
Polar
Co
ordinates?
Exercise
Plot
and
lab
el
the
follo
wing
p
oints
and
sets
in
p
ola
r
co
o
rdinates
A
=
(2
,
π
3
)
B
=
(1
.
5
,
3
π
)
C
=
(
−
3
,
−
π
4
)
R
=
{
(
r
,
θ
)
:
0
≤
r
≤
2
}
S
=
{
(
r
,
θ
)
:
π
6
≤
θ
≤
π
4
,
r
≥
1
}
12
Question
1.3.1
What
Are
Polar
Co
ordinates?
Ca
rtesian
to
P
ola
r
p
(
r
,
θ
)
=
r
cos(
θ
)
i
+
r
sin(
θ
)
j
x
=
r
cos
θ
y
=
r
sin
θ
Notice:
x
2
+
y
2
=
r
2
r
=
p
x
2
+
y
2
θ
=
(
tan
−
1
y
x
x
>
0
tan
−
1
y
x
+
π
x
<
0
A
full
circle
is
0
≤
θ
≤
2
π
.
13
Question
1.3.2
What
Is
the
Jacobian
of
Pola
r
Co
ordinates?
Calculate
the
Jacobian
∂
(
x
,
y
)
∂
(
r
,
θ
)
such
that
dxdy
=
∂
(
x
,
y
)
∂
(
r
,
θ
)
drd
θ
.
14
Question
1.3.2
What
Is
the
Jacobian
of
Polar
Coordinates?
Main
Idea
The
Jacobian
of
p
ola
r
co
o
rdinates
is
r
.
Thus
dydx
=
rdrd
θ
15
Example
1.3.3
Integrating
Over
a
Disc
Let
D
be
the
disk:
x
2
+
y
2
≤
9.
Calculate
Z
Z
D
p
x
2
+
y
2
dA
.
16
Example
1.3.3
Integrating
Over
a
Disc
Let
D
be
the
disk:
x
2
+
y
2
≤
9.
Calculate
Z
Z
D
p
x
2
+
y
2
dA
.
Click to Load Applet
16
Example
1.3.4
Integrating
Over
a
W
edge
Let
D
=
{
(
x
,
y
)
:
x
≥
0
,
x
≤
y
,
x
2
+
y
2
≤
2
}
.
Sketch
D
and
calculate
Z
Z
D
x
2
dA
.
17
Example
1.3.4
../im
gicons/teacher.pdf
Integrating
Over
a
Wedge
T
rig
F
o
rmulas
Higher
p
o
w
ers
of
sine
and
cosine
a
rise
naturally
in
p
ola
r
integrals.
Y
ou’ll
b
e
resp
onsible
fo
r
applying
the
follo
wing
fo
rmulas.
F
o
rmulas
sin
2
θ
=
1
2
−
cos(2
θ
)
2
cos
2
θ
=
1
2
+
cos(2
θ
)
2
sin
3
θ
=
sin
θ
−
cos
2
θ
sin
θ
cos
3
θ
=
cos
θ
−
sin
2
θ
cos
θ
18
Example
1.3.4
Integrating
Over
a
Wedge
Exercise
F
o
r
each
of
the
integrals
b
elo
w,
sk
etch
the
domain
of
integration
then
convert
to
p
ola
r.
Y
ou
need
not
evaluate.
1
Z
Z
D
2
x
−
3
y
2
dydx
where
D
=
{
(
x
,
y
)
:
x
2
+
y
2
≤
16
,
−
y
≤
x
≤
y
}
2
Z
Z
D
x
2
ydydx
where
D
=
{
(
x
,
y
)
:
4
≤
x
2
+
y
2
≤
9
,
y
≤
0
}
3
Z
3
−
3
Z
√
9
−
y
2
0
x
2
+
y
2
dxdy
Which
of
y
our
integrals
can
b
e
solved
using
the
p
ro
duct
fo
rmula?
19
Example
1.3.5
A
Circle
Through
the
Origin
Let
D
be
the
domain
(
x
−
1)
2
+
y
2
≤
1.
Evaluate
Z
Z
D
x
2
+
y
2
dA
.
Click to Load Applet
20
Example
1.3.6
P
olar
Co
o
rdinates
in
T
riple
Integrals
Set
up
the
integral
fo
r
f
(
x
,
y
,
z
)
over
the
region
R
enclosed
b
etw
een
the
graphs
z
=
x
2
+
y
2
and
z
=
p
6
−
x
2
−
y
2
.
21
Example
1.3.6
P
olar
Co
o
rdinates
in
T
riple
Integrals
Set
up
the
integral
fo
r
f
(
x
,
y
,
z
)
over
the
region
R
enclosed
b
etw
een
the
graphs
z
=
x
2
+
y
2
and
z
=
p
6
−
x
2
−
y
2
.
Click to Load Applet
21
Example
1.3.6
Pola
r
Coordinates
in
T
riple
Integrals
Main
Idea
When
setting
up
a
triple
integral,
sometimes
the
domain
of
the
outer
t
w
o
va
riables
(usually
x
and
y
)
is
mo
re
conveniently
written
in
p
ola
r
co
o
rdinates.
Rema
rk
The
co
o
rdinate
system
(
r
,
θ
,
z
)
is
called
the
cylindrical
co
o
rdinate
system.
22
Section
1.8
T
riple
Integrals
in
Spherical
Co
o
rdinates
Goals:
1
W
rite
integrals
in
spherical
co
o
rdinates
Question
1.8.1
What
Are
Spherical
Co
o
rdinates?
Spherical
co
o
rdinates
a
re
a
three
dimensional
co
o
rdinate
system.
Here
ρ
(“rho”)
is
the
(three
dimensional)
distance
from
the
o
rigin.
ϕ
(“phi”)
is
the
angle
the
segment
from
the
o
rigin
mak
es
with
the
p
ositive
z
axis.
θ
is
the
angle
that
the
p
rojection
to
the
xy
-plane
mak
es
with
the
p
ositive
x
-axis.
Click to Load Applet
24
Question
1.8.1
What
Are
Spherical
Coordinates?
The
follo
wing
fo
rmulas
follo
w
from
trigonometry
.
Ca
rtesian
to
Spherical
x
=
ρ
cos
θ
sin
ϕ
y
=
ρ
sin
θ
sin
ϕ
z
=
ρ
cos
ϕ
Notice:
x
2
+
y
2
+
z
2
=
ρ
2
A
full
sphere
is
0
≤
θ
≤
2
π
0
≤
ϕ
≤
π
Click to Load Applet
25
Question
1.8.1
What
Are
Spherical
Co
ordinates?
Exercise
Describ
e
(o
r
dra
w?)
the
follo
wing
regions
in
spherical
co
ordinates.
1
R
=
{
(
ρ,
θ
,
ϕ
)
:
ϕ
=
π
2
}
2
R
=
{
(
ρ,
θ
,
ϕ
)
:
ρ
≤
5
}
3
R
=
{
(
ρ,
θ
,
ϕ
)
:
0
≤
θ
≤
π
4
}
4
R
=
{
(
ρ,
θ
,
ϕ
)
:
ϕ
≥
2
π
3
}
26
Question
1.8.1
What
Are
Spherical
Co
ordinates?
Theo
rem
The
Jacobian
fo
r
spherical
co
o
rdinates
is
ρ
2
sin
ϕ.
27
Example
1.8.2
The
V
olume
of
a
Sphere
Calculate
the
volume
of
a
sphere
of
radius
R
.
28
Example
1.8.3
Converting
to
Spherical
Co
o
rdinates
Convert
the
follo
wing
triple
integral
to
spherical
co
o
rdinates:
Z
3
0
Z
0
−
√
9
−
x
2
Z
√
9
−
x
2
−
y
2
0
yz
2
dzdydx
29
Example
1.8.3
Converting
to
Spherical
Co
o
rdinates
Convert
the
follo
wing
triple
integral
to
spherical
co
o
rdinates:
Z
3
0
Z
0
−
√
9
−
x
2
Z
√
9
−
x
2
−
y
2
0
yz
2
dzdydx
Click to Load Applet
29
Question
1.8.4
When
Do
We
Use
Spherical
Co
o
rdinates?
Spherical
co
o
rdinates
a
re
only
w
o
rth
using
if
the
domain
is
reasonably
w
ell
b
ehaved.
1
In
many
cases,
all
the
b
ounds
of
integration
a
re
constants.
2
The
b
ounds
of
ρ
involve
the
exp
ression
x
2
+
y
2
+
z
2
.
3
The
b
ounds
of
θ
a
re
given
by
inequalities
containing
only
x
and
y
.
Dra
w
these
in
the
plane.
4
The
b
ounds
of
ϕ
a
re
given
b
y
inequalities
concerning
z
.
5
In
some
mo
re
advanced
applications,
the
ρ
b
ounds
ma
y
b
e
a
function
of
ϕ
o
r
θ
,
meaning
ρ
should
b
e
the
inner
va
riable.
30
Question
1.8.4
When
Do
We
Use
Spherical
Co
ordinates?
Exercise
Set
up
the
integrals
of
g
(
x
,
y
,
z
)
over
the
following
regions
using
spherical
co
o
rdinates.
1
The
intersection
of
x
2
+
y
2
+
z
2
≤
4
and
z
≤
0.
2
The
intersection
of
the
sphere
x
2
+
y
2
+
z
2
≤
1
and
the
half-spaces
x
≥
0
and
y
≤
x
.
3
The
intersection
of
the
cone
z
≥
p
x
2
+
y
2
and
the
sphere
x
2
+
y
2
+
z
2
≤
9.
31
Section
12.1
Three-Dimensional
Co
ordinate
Systems
Goals:
1
Plot
p
oints
in
a
three-dimensional
co
o
rdinate
system.
2
Use
the
distance
fo
rmula.
3
Recognize
the
equation
of
a
sphere
and
find
its
radius
and
center.
4
Graph
an
implicit
function
with
a
free
va
riable.
Question
12.1.1
Ho
w
Do
Cartesian
Co
o
rdinates
Extend
to
Higher
Dimensions?
Recall
ho
w
w
e
constructed
the
Ca
rtesian
plane.
x
y
y
−
4
−
4
−
3
−
3
−
2
−
2
−
1
−
1
1
1
2
2
3
3
4
4
(2
,
0)
(2
,
3)
1
Assign
o
rigin
and
t
w
o
directions
(
x
,
y
).
2
y
is
90
degrees
anticlo
ckwise
from
x
.
3
Axes
consist
of
the
p
oints
d
isplaced
in
only
one
direction.
4
Co
o
rdinates
refer
to
displacement
from
the
o
rigin
in
each
direction.
5
Either
displacement
can
happ
en
first.
6
Each
p
oint
has
exactly
one
o
rdered
pair
that
refers
to
it.
33
Question
12.1.1
Ho
w
Do
Cartesian
Co
o
rdinates
Extend
to
Higher
Dimensions?
Recall
ho
w
w
e
constructed
the
Ca
rtesian
plane.
x
y
y
−
4
−
4
−
3
−
3
−
2
−
2
−
1
−
1
1
1
2
2
3
3
4
4
(2
,
0)
(2
,
3)
1
Assign
o
rigin
and
t
w
o
directions
(
x
,
y
).
2
y
is
90
degrees
anticlo
ckwise
from
x
.
3
Axes
consist
of
the
p
oints
d
isplaced
in
only
one
direction.
4
Co
o
rdinates
refer
to
displacement
from
the
o
rigin
in
each
direction.
5
Either
displacement
can
happ
en
first.
6
Each
p
oint
has
exactly
one
o
rdered
pair
that
refers
to
it.
33
Question
12.1.1
Ho
w
Do
Cartesian
Co
o
rdinates
Extend
to
Higher
Dimensions?
Recall
ho
w
w
e
constructed
the
Ca
rtesian
plane.
x
y
y
−
4
−
4
−
3
−
3
−
2
−
2
−
1
−
1
1
1
2
2
3
3
4
4
(2
,
0)
(2
,
3)
1
Assign
o
rigin
and
t
w
o
directions
(
x
,
y
).
2
y
is
90
degrees
anticlo
ckwise
from
x
.
3
Axes
consist
of
the
p
oints
displaced
in
only
one
direction.
4
Co
o
rdinates
refer
to
displacement
from
the
o
rigin
in
each
direction.
5
Either
displacement
can
happ
en
first.
6
Each
p
oint
has
exactly
one
o
rdered
pair
that
refers
to
it.
33
Question
12.1.1
Ho
w
Do
Cartesian
Co
o
rdinates
Extend
to
Higher
Dimensions?
Recall
ho
w
w
e
constructed
the
Ca
rtesian
plane.
x
y
y
−
4
−
4
−
3
−
3
−
2
−
2
−
1
−
1
1
1
2
2
3
3
4
4
(2
,
0)
(2
,
3)
1
Assign
o
rigin
and
t
w
o
directions
(
x
,
y
).
2
y
is
90
degrees
anticlo
ckwise
from
x
.
3
Axes
consist
of
the
p
oints
displaced
in
only
one
direction.
4
Co
o
rdinates
refer
to
displacement
from
the
o
rigin
in
each
direction.
5
Either
displacement
can
happ
en
first.
6
Each
p
oint
has
exactly
one
o
rdered
pair
that
refers
to
it.
33
Question
12.1.1
Ho
w
Do
Cartesian
Co
o
rdinates
Extend
to
Higher
Dimensions?
Recall
ho
w
w
e
constructed
the
Ca
rtesian
plane.
x
y
y
−
4
−
4
−
3
−
3
−
2
−
2
−
1
−
1
1
1
2
2
3
3
4
4
(2
,
0)
(2
,
3)
1
Assign
o
rigin
and
t
w
o
directions
(
x
,
y
).
2
y
is
90
degrees
anticlo
ckwise
from
x
.
3
Axes
consist
of
the
p
oints
displaced
in
only
one
direction.
4
Co
o
rdinates
refer
to
displacement
from
the
o
rigin
in
each
direction.
5
Either
displacement
can
happ
en
first.
6
Each
p
oint
has
exactly
one
o
rdered
pair
that
refers
to
it.
33
Question
12.1.1
Ho
w
Do
Cartesian
Co
o
rdinates
Extend
to
Higher
Dimensions?
Recall
ho
w
w
e
constructed
the
Ca
rtesian
plane.
x
y
y
−
4
−
4
−
3
−
3
−
2
−
2
−
1
−
1
1
1
2
2
3
3
4
4
(2
,
0)
(2
,
3)
1
Assign
o
rigin
and
t
w
o
directions
(
x
,
y
).
2
y
is
90
degrees
anticlo
ckwise
from
x
.
3
Axes
consist
of
the
p
oints
displaced
in
only
one
direction.
4
Co
o
rdinates
refer
to
displacement
from
the
o
rigin
in
each
direction.
5
Either
displacement
can
happ
en
first.
6
Each
p
oint
has
exactly
one
o
rdered
pair
that
refers
to
it.
33
Question
12.1.1
How
Do
Cartesian
Coordinates
Extend
to
Higher
Dimensions?
In
a
three-dimensional
Ca
rtesian
co
o
rdinate
system.
W
e
can
extrap
olate
from
t
w
o
dimensions.
Click to Load Applet
1
Assign
o
rigin
and
three
directions
(
x
,
y
,
z
).
2
Each
axis
mak
es
a
90
degree
angle
with
the
other
t
w
o.
3
The
z
direction
is
determined
b
y
the
right-hand
rule.
34
Question
12.1.2
Ho
w
Do
We
Establish
Which
Direction
Is
P
ositive
in
Each
Axis?
The
right
hand
rule
sa
ys
that
if
y
ou
mak
e
the
fingers
of
y
our
right
hand
follo
w
the
(counterclo
ckwise)
unit
circle
in
the
xy
-plane,
then
y
our
thumb
indicates
the
direction
of
the
p
ositive
z
-axis.
Click to Load Applet
Figure:
The
counterclo
ckwise
unit
circle
in
the
xy
-plane
35
Example
12.1.3
Dra
wing
a
Lo
cation
in
Three-Dimensional
Co
o
rdinates
The
p
oint
(2
,
3
,
5)
is
the
p
oint
displaced
from
the
o
rigin
b
y
2
in
the
x
direction
3
in
the
y
direction
5
in
the
z
direction.
Ho
w
do
w
e
dra
w
a
reasonable
diagram
of
where
this
p
oint
lies?
36
Example
12.1.3
Dra
wing
a
Lo
cation
in
Three-Dimensional
Co
o
rdinates
The
p
oint
(2
,
3
,
5)
is
the
p
oint
displaced
from
the
o
rigin
b
y
2
in
the
x
direction
3
in
the
y
direction
5
in
the
z
direction.
Ho
w
do
w
e
dra
w
a
reasonable
diagram
of
where
this
p
oint
lies?
Click to Load Applet
36
Example
12.1.3
Drawing
a
Location
in
Three-Dimensional
Co
ordinates
Ho
w
can
w
e
dra
w
a
reasonable
diagram
of
(
−
5
,
1
,
−
4)?
37
Example
12.1.3
Drawing
a
Location
in
Three-Dimensional
Co
ordinates
Ho
w
can
w
e
dra
w
a
reasonable
diagram
of
(
−
5
,
1
,
−
4)?
Click to Load Applet
37
Question
12.1.4
Ho
w
Do
W
e
Measure
Distance
in
Three-Space?
Theo
rem
The
distance
from
the
o
rigin
to
the
p
oint
(
x
,
y
,
z
)
is
given
by
the
Pythago
rean
Theo
rem
D
=
p
x
2
+
y
2
+
z
2
Click to Load Applet
38
Question
12.1.4
How
Do
We
Measure
Distance
in
Three-Space?
Theo
rem
The
distance
from
the
p
oint
(
x
1
,
y
1
,
z
1
)
to
the
p
oint
(
x
2
,
y
2
,
z
2
)
is
given
b
y
D
=
q
(
x
1
−
x
2
)
2
+
(
y
1
−
y
2
)
2
+
(
z
1
−
z
2
)
2
39
Question
12.1.5
What
Is
a
Graph?
Definition
The
graph
of
an
implicit
equation
is
the
set
of
p
oints
whose
co
o
rdinates
satisfy
that
equation.
In
other
w
o
rds,
the
t
w
o
sides
a
re
equal
when
w
e
plug
the
co
o
rdinates
in
fo
r
x
,
y
and
z
.
Example
The
graph
of
x
2
+
(
y
−
4)
2
+
(
z
+
1)
2
=
9
is
the
set
of
p
oints
that
a
re
distance
3
from
the
p
oint
(0
,
4
,
−
1)
Click to Load Applet
40
Example
12.1.6
Graphing
an
Equation
with
Tw
o
F
ree
V
a
riables
Sk
etch
the
graph
of
the
equation
y
=
3.
41
Example
12.1.6
Graphing
an
Equation
with
Two
Free
Va
riables
In
addition
to
co
o
rdinate
axes,
3-dimensional
space
has
3
co
o
rdinate
planes
.
1
The
graph
of
z
=
0
is
the
xy
-plane.
2
The
graph
of
x
=
0
is
the
yz
-plane.
3
The
graph
of
y
=
0
is
the
xz
-plane.
Click to Load Applet
Figure:
The
co
ordinate
planes
in
3-dimensional
space.
42
Example
12.1.7
Graphing
an
Equation
with
One
Free
V
a
riable
Sk
etch
the
graph
of
the
equation
z
=
x
2
−
3.
43
Question
12.1.8
What
Do
the
Graphs
of
Implicit
Equations
Lo
ok
Like
Generally?
Notice
that
the
graph
of
an
implicit
equation
in
the
plane
is
generally
one-dimensional
(a
curve),
whereas
the
graph
of
an
implicit
equation
in
three-space
is
generally
t
w
o-dimensional
(a
surface).
Figure:
The
curve
y
=
x
2
−
3
Click to Load Applet
Figure:
The
surface
z
=
x
2
−
3
44
Question
12.1.9
Ho
w
Do
W
e
Extrap
olate
to
Even
Higher
Dimensions?
W
e
can
use
a
co
o
rdinate
system
to
describ
e
a
space
with
mo
re
than
3
dimensions.
k
-dimensional
space
can
b
e
defined
as
the
set
of
p
oints
of
the
fo
rm
P
=
(
x
1
,
x
2
,
.
.
.
,
x
k
)
.
Theo
rem
The
distance
from
the
o
rigin
to
P
=
(
x
1
,
x
2
,
.
.
.
,
x
k
)
in
k
-space
is
q
x
2
1
+
x
2
2
+
·
·
·
+
x
2
k
There
is
no
right
hand
rule
fo
r
higher
dimensions,
b
ecause
w
e
can’t
dra
w
these
spaces
anyw
a
y
.
45
Section
12.1
Summa
ry
Questions
Q1
What
displacements
a
re
rep
resented
b
y
the
notation
(
a
,
b
,
c
)?
Q2
What
is
the
right
hand
rule
and
what
do
es
it
tell
y
ou
ab
out
a
three-dimensional
co
o
rdinate
system?
Q3
In
three-space,
what
is
the
y
-axis?
What
are
the
coordinates
of
a
general
p
oint
on
it?
Q4
In
three
space,
what
is
the
xz
-plane?
What
are
the
coordinates
of
a
general
p
oint
on
it?
What
is
its
equation?
Q5
Ho
w
do
w
e
use
a
free
va
riable
to
sk
etch
a
graph?
Q6
Ho
w
do
w
e
recognize
the
equation
of
a
sphere?
46
Section
12.1
Q11
Dra
w
diagrams
of
p
oints
with
the
follo
wing
co
o
rdinates.
a
(6
,
1
,
2)
b
(
−
3
,
0
,
0)
c
(2
,
−
1
,
4)
d
(0
,
3
,
5)
47
Section
12.1
Q11
Click to Load Applet
47
Section
12.1
Q50
The
graph
of
x
2
+
y
2
=
0
in
R
2
is
a
p
oint,
not
a
curve.
Use
this
idea
to
write
an
equation
fo
r
the
intersection
of
the
graphs
f
(
x
,
y
,
z
)
=
c
and
g
(
x
,
y
,
z
)
=
d
.
What
do
you
expect
the
dimension
of
this
intersection
to
b
e?
48
Section
12.2
V
ectors
Goals:
1
Distinguish
vecto
rs
from
scala
rs
(real
numb
ers)
and
p
oints.
2
Add
and
subtract
vecto
rs,
multiply
b
y
scala
rs.
3
Exp
ress
real
w
o
rld
vecto
rs
in
terms
of
their
comp
onents.
Question
12.2.1
What
is
a
V
ector?
Definition
A
vecto
r
in
2-space
consists
of
a
magnitude
(length)
and
a
direction.
Tw
o
vecto
rs
with
the
same
magnitude
and
the
same
direction
a
re
equal
.
Example
Here
a
re
four
vecto
rs
in
2-space
(the
plane)
rep
resented
b
y
a
rro
ws.
Tw
o
of
these
vecto
rs
a
re
equal.
50
Question
12.2.1
What
is
a
Vecto
r?
Here
a
re
some
vecto
rs
3
miles
south
The
fo
rce
that
a
magnetic
field
applies
to
a
cha
rged
pa
rticle
The
velo
cit
y
of
an
airplane
Here
a
re
some
non-vecto
rs
17
The
mass
of
an
automobile
3:15
PM
A
tlanta,
GA
51
Question
12.2.2
Ho
w
Do
W
e
Denote
Vecto
rs?
Endp
oint
Notation
The
vecto
r
v
from
point
A
to
p
oint
B
can
b
e
rep
resented
by
the
notation
−
→
AB
.
A
is
the
initial
point
and
B
is
the
terminal
p
oint
.
52
Question
12.2.2
How
Do
We
Denote
Vecto
rs?
Theo
rem
−
→
AB
=
−
→
CD
if
and
only
if
ABDC
is
a
parallelogram
(perhaps
a
squished
one).
53
Question
12.2.2
How
Do
We
Denote
Vecto
rs?
Co
o
rdinate
Notation
W
e
can
rep
resent
a
vecto
r
in
the
Ca
rtesian
plane
b
y
the
x
and
y
comp
onents
of
its
displacement.
If
A
=
(2
,
3)
and
B
=
(5
,
1),
then
−
→
AB
increases
x
b
y
5
−
2
=
3
and
y
by
1
−
3
=
−
2.
We
can
represent
−
→
AB
=
⟨
3
,
−
2
⟩
Click to Load Applet
Figure:
The
x
and
y
comp
onents
of
a
vecto
r
54
Question
12.2.2
How
Do
We
Denote
Vecto
rs?
Theo
rem
v
=
u
if
and
only
if
their
co
o
rdinate
rep
resentations
match
in
each
comp
onent.
W
e
can
also
measure
slop
e
using
the
co
o
rdinate
notation.
Fo
r
the
vector
v
=
⟨
a
,
b
⟩
:
b
rep
resents
the
displacement
in
the
y
-direction
(rise).
a
represents
the
displacement
in
the
x
-direction
(run).
The
slop
e
of
v
is
rise
run
=
b
a
.
55
Question
12.2.2
How
Do
We
Denote
Vecto
rs?
Every
p
oint
in
a
Ca
rtesian
co
o
rdinate
system
has
a
p
osition
vecto
r
,
which
gives
the
displacement
of
that
p
oint
from
the
o
rigin.
The
comp
onents
of
the
vecto
r
a
re
the
co
o
rdinates
of
the
p
oint.
Click to Load Applet
Figure:
There
is
only
one
p
oint
equal
to
(
−
5
,
1),
but
there
are
many
vecto
rs
equal
to
⟨−
5
,
1
⟩
.
56
Question
12.2.3
What
Arithmetic
Can
W
e
P
erfo
rm
with
V
ecto
rs?
V
ecto
r
Sums
The
sum
of
t
w
o
vecto
rs
v
+
u
is
calculated
b
y
p
ositioning
v
and
u
head
to
tail.
The
sum
is
the
vecto
r
from
the
initial
p
oint
of
one
to
the
terminal
p
oint
of
the
other.
In
coordinate
notation,
we
just
add
each
comp
onent
numerically
.
Click to Load Applet
⟨
1
,
3
⟩
+
⟨
3
,
−
1
⟩
⟨
4
,
2
⟩
57
Question
12.2.3
What
Arithmetic
Can
We
P
erform
with
Vecto
rs?
Scala
r
Multiples
Given
a
numb
er
(called
a
scala
r)
λ
and
a
vecto
r
v
we
can
p
ro
duce
the
scala
r
multiple
λ
v
,
which
is
the
vecto
r
in
the
same
direction
as
v
but
λ
times
as
long.
If
λ
is
negative
then
λ
v
extends
in
the
opp
osite
direction.
Either
wa
y
,
w
e
sa
y
λ
v
is
parallel
to
v
.
Click to Load Applet
In
coordinates
scalar
multiplication
is
distributed
to
each
component.
Fo
r
example:
2
.
5
⟨
6
,
4
⟩
=
⟨
15
,
10
⟩
58
Example
12.2.4
P
erforming
V
ecto
r
Arithmetic
Given
diagrams
of
t
w
o
vecto
rs
u
and
v
,
ho
w
would
w
e
calculate
1
2
u
+
v
?
What
if
w
e
a
re
instead
given
the
comp
onents
u
=
⟨
a
,
b
⟩
and
v
=
⟨
c
,
d
⟩
?
59
Question
12.2.5
What
Is
Standard
Basis
Notation?
W
e
can
rep
resent
any
vecto
r
in
the
plane
as
a
sum
of
scala
r
multiples
of
the
follo
wing
standa
rd
basis
vectors
.
Standa
rd
Basis
V
ecto
rs
The
emphstanda
rd
basis
vecto
rs
in
R
2
a
re
i
=
⟨
1
,
0
⟩
j
=
⟨
0
,
1
⟩
F
o
r
example,
the
vector
⟨
3
,
−
5
⟩
can
b
e
written
as
3
i
−
5
j
.
Y
ou
can
check
y
ourself
that
the
sum
on
the
right
gives
the
co
rrect
vecto
r.
60
Question
12.2.6
Ho
w
Do
W
e
Measure
the
Length
of
a
V
ecto
r?
Definition
The
length
o
r
magnitude
of
a
vecto
r
is
calculated
using
the
distance
fo
rmula
and
notated
|
v
|
.
If
v
=
a
i
+
b
j
,
then
|
v
|
=
p
a
2
+
b
2
61
Example
12.2.7
The
Length
of
a
V
ecto
r
If
v
=
⟨
3
,
−
5
⟩
calculate
|
v
|
62
Example
12.2.7
The
Length
of
a
Vecto
r
Definition
A
unit
vecto
r
is
a
vecto
r
of
length
1.
Given
a
vecto
r
v
the
scala
r
multiple
1
|
v
|
v
is
a
unit
vecto
r
in
the
same
direction
as
v
.
63
Question
12.2.8
Ho
w
Do
W
e
Measure
the
Direction
of
a
V
ecto
r?
Angles
a
re
a
go
o
d
w
a
y
of
compa
ring
directions.
In
general,
t
wo
vectors
will
not
intersect
to
fo
rm
an
angle,
so
w
e
use
the
follo
wing
definition:
Definition
The
angle
b
et
w
een
t
w
o
vecto
rs
is
the
angle
they
mak
e
when
they
a
re
placed
so
their
initial
p
oints
a
re
the
same.
If
they
mak
e
a
right
angle,
w
e
call
them
o
rthogonal
.
If
they
mak
e
an
angle
of
0
o
r
π
,
they
are
parallel.
64
Question
12.2.9
Ho
w
Do
W
e
Denote
Vecto
rs
in
Higher
Dimensions?
Higher
dimensional
vecto
rs
rep
resent
displacements
in
higher
dimensional
spaces.
W
e
can
call
a
vecto
r
in
n
-space
an
n
-vector.
W
e
can
still
denote
and
n
-vector
b
y
its
endp
oints.
We
can
also
denote
it
in
coordinate
notation,
but
w
e
need
mo
re
comp
onents.
Example
If
A
=
(2
,
4
,
1)
and
B
=
(5
,
−
1
,
3)
then
−
→
AB
=
⟨
3
,
−
5
,
2
⟩
.
65
Question
12.2.9
How
Do
We
Denote
Vecto
rs
in
Higher
Dimensions?
In
three
space,
w
e
add
another
standa
rd
basis
vecto
r
k
.
Standa
rd
basis
fo
r
3-vecto
rs
i
=
⟨
1
,
0
,
0
⟩
j
=
⟨
0
,
1
,
0
⟩
k
=
⟨
0
,
0
,
1
⟩
Example
⟨
3
,
−
5
,
2
⟩
=
3
i
−
5
j
+
2
k
Higher
dimensions
still
have
a
standa
rd
basis,
but
at
this
p
oint
the
naming
conventions
a
re
less
standa
rd.
{
e
1
,
e
2
,
e
3
,
.
.
.
,
e
n
}
is
common
fo
r
n
-vectors.
66
Question
12.2.9
How
Do
We
Denote
Vecto
rs
in
Higher
Dimensions?
Length
of
a
V
ector
The
length
of
an
n
-vector
derives
from
the
distance
fo
rmula
in
n
-space.
|⟨
a
1
,
a
2
,
a
3
,
.
.
.
,
a
n
⟩|
=
q
a
2
1
+
a
2
2
+
a
2
3
+
·
·
·
+
a
2
n
67
Question
12.2.9
How
Do
We
Denote
Vecto
rs
in
Higher
Dimensions?
Angles
Bet
ween
V
ecto
rs
Any
t
w
o
vecto
rs
with
the
same
initial
p
oint
lie
in
a
plane.
Their
angle
is
a
t
w
o-dimensional
measurement.
Ho
w
ever
there
is
no
go
o
d
w
a
y
to
measure
clo
ckwise
in
3
o
r
mo
re
dimensions.
The
angle
b
et
w
een
t
w
o
vecto
rs
is
never
negative,
no
r
mo
re
than
π
.
68
Question
12.2.9
How
Do
We
Denote
Vecto
rs
in
Higher
Dimensions?
Click to Load Applet
Figure:
Two
3-vecto
rs
with
a
common
initial
p
oint,
the
plane
that
contains
them,
and
the
angle
betw
een
them
69
Section
12.2
Summa
ry
Questions
Q1
Ho
w
is
a
vecto
r
simila
r
to
a
p
oint?
T
o
a
numb
er?
Q2
Ho
w
is
a
vecto
r
different
from
a
p
oint?
F
rom
a
numb
er?
Q3
Ho
w
can
y
ou
tell
if
t
w
o
vecto
rs
p
oint
in
the
same
direction?
Opp
osite
directions?
Q4
If
u
and
v
are
position
vectors
of
the
points
P
and
Q
,
how
a
re
u
and
v
related
to
−
→
PQ
?
70
Section
12.2
Q42
Let
u
and
v
b
e
non-parallel
vecto
rs
in
R
3
.
Ho
w
many
unit
vecto
rs
in
R
3
a
re
o
rthogonal
to
b
oth
u
and
v
?
71
Section
12.3
The
Dot
Pro
duct
Goals:
1
Calculate
the
dot
p
ro
duct
of
t
w
o
vecto
rs.
2
Determine
the
geometric
relationship
b
et
w
een
t
w
o
vecto
rs
based
on
their
dot
p
ro
duct.
3
Calculate
vecto
r
and
scala
r
p
rojections
of
one
vecto
r
onto
another.
Question
12.3.1
What
Is
the
Dot
Pro
duct?
Definition
The
dot
p
ro
duct
of
tw
o
vectors
is
a
number.
F
o
r
t
w
o
dimensional
vecto
rs
v
=
⟨
v
1
,
v
2
⟩
and
u
=
⟨
u
1
,
u
2
⟩
w
e
define
v
·
u
=
v
1
u
1
+
v
2
u
2
F
o
r
three
dimensional
vecto
rs
v
=
⟨
v
1
,
v
2
,
v
3
⟩
and
u
=
⟨
u
1
,
u
2
,
u
3
⟩
w
e
define
v
·
u
=
v
1
u
1
+
v
2
u
2
+
v
3
u
3
This
pattern
can
b
e
extended
to
any
dimension.
73
Example
12.3.2
Computing
a
Dot
Pro
duct
a
Calculate
⟨
2
,
3
,
−
1
⟩
·
⟨
4
,
1
,
5
⟩
b
Calculate
(
−
2
i
+
4
k
)
·
(
i
+
2
j
−
k
)
74
Example
12.3.2
../im
gicons/teacher.pdf
Computing
a
Dot
Product
Exercise
Let
u
=
⟨
2
,
3
⟩
,
v
=
⟨
4
,
−
1
⟩
and
w
=
⟨−
5
,
2
⟩
.
a
Compute
u
·
u
and
u
·
v
and
u
·
w
.
b
Compute
v
·
u
.
How
do
es
it
compare
to
u
·
v
?
c
Ho
w
is
u
·
u
related
to
|
u
|
?
d
Compute
3
u
and
3
v
then
take
their
dot
p
ro
duct.
How
is
it
related
to
u
·
v
?
e
Compute
v
+
w
then
compute
u
·
(
v
+
w
).
How
is
it
related
to
u
·
v
and
u
·
w
?
f
Why
do
y
ou
think
w
e
call
this
op
eration
a
“dot
p
ro
duct”
and
not
a
“dot
sum?”
g
If
y
ou
w
anted
to
p
rove
that
relationships
y
our
noticed
in
b
-
e
wo
rk
fo
r
all
p
ossible
vecto
rs,
ho
w
w
ould
y
ou
do
that?
75
Question
12.3.3
What
Are
the
Algebraic
Prop
erties
of
the
Dot
Product?
Theo
rem
The
follo
wing
algeb
raic
p
rop
erties
hold
fo
r
any
vecto
rs
u
,
v
and
w
and
scala
rs
m
and
n
.
Commutative
u
·
v
=
v
·
u
Distributive
u
·
(
v
+
w
)
=
u
·
v
+
u
·
w
Associative
m
u
·
n
v
=
mn
(
u
·
v
)
76
Question
12.3.4
What
Is
the
Geometric
Significance
of
the
Dot
Product?
Theo
rem
If
u
and
v
are
pa
rallel
then
u
·
v
=
(
|
u
||
v
|
if
u
and
v
have
the
same
direction
−|
u
||
v
|
if
u
and
v
have
opp
osite
directions
77
Question
12.3.4
What
Is
the
Geometric
Significance
of
the
Dot
Pro
duct?
Theo
rem
If
u
and
v
are
o
rthogonal
then
u
·
v
=
0
.
78
Question
12.3.4
What
Is
the
Geometric
Significance
of
the
Dot
Pro
duct?
Tw
o
vecto
rs
need
not
b
e
pa
rallel
o
r
o
rthogonal,
but
given
vecto
rs
u
and
v
we
can
alw
ays
write
v
=
v
proj
+
v
orth
.
Click to Load Applet
The
p
rop
erties
of
the
dot
p
ro
duct
tell
us
that
u
·
v
=
u
·
(
v
proj
+
v
orth
)
=
±
|
u
||
v
proj
|
+
0
Definition
The
numb
er
u
·
v
|
u
|
is
called
the
scala
r
p
rojection
of
v
onto
u
.
79
Question
12.3.4
What
Is
the
Geometric
Significance
of
the
Dot
Pro
duct?
Theo
rem
Let
u
and
v
have
the
same
initial
p
oint
and
meet
at
angle
θ
.
The
follo
wing
fo
rmula
holds
in
any
dimension:
u
·
v
=
|
u
||
v
|
cos
θ
Click to Load Applet
Recall
that
cos
θ
is
p
ositive
when
θ
<
π
/
2
negative
when
θ
>
π
/
2
zero
when
θ
=
π
/
2.
So
the
sign
of
u
·
v
tells
us
whether
θ
is
acute,
obtuse
or
right.
80
Example
12.3.5
Using
the
Cosine
F
ormula
What
is
the
angle
b
et
w
een
⟨
1
,
0
,
1
⟩
and
⟨
1
,
1
,
0
⟩
?
81
Example
12.3.5
Using
the
Cosine
F
ormula
What
is
the
angle
b
et
w
een
⟨
1
,
0
,
1
⟩
and
⟨
1
,
1
,
0
⟩
?
Click to Load Applet
81
Application
12.3.6
W
ork
In
physics,
w
e
sa
y
a
fo
rce
w
o
rks
on
an
object
if
it
moves
the
object
in
the
direction
of
the
fo
rce.
Given
a
fo
rce
F
and
a
displacement
s
,
the
fo
rmula
fo
r
w
o
rk
is:
W
=
Fs
82
Application
12.3.6
Wo
rk
In
higher
dimensions,
displacement
and
fo
rce
a
re
vecto
rs.
If
the
fo
rce
and
the
displacement
are
not
in
the
same
direction,
then
only
F
proj
contributes
to
w
o
rk.
W
=
F
proj
·
s
=
F
·
s
Click to Load Applet
83
Section
12.3
Summa
ry
Questions
Q1
What
algeb
raic
p
rop
erties
do
es
a
dot
p
ro
duct
sha
re
with
real
numb
er
multiplication?
Q2
What
is
the
significance
of
the
dot
p
ro
duct
of
t
w
o
pa
rallel
vecto
rs?
Q3
Ho
w
is
the
angle
b
et
w
een
t
w
o
vecto
rs
related
to
their
dot
p
ro
duct?
Q4
What
is
a
scala
r
p
rojection,
and
ho
w
do
y
ou
compute
it?
84
Section
12.3
Q16
If
|
u
|
=
6
and
|
v
|
=
10
what
a
re
the
greatest
and
least
p
ossible
values
of
u
·
v
?
85
Section
12.3
Q16
If
|
u
|
=
6
and
|
v
|
=
10
what
a
re
the
greatest
and
least
p
ossible
values
of
u
·
v
?
86
Section
12.3
Q22
Let
A
b
e
the
vertex
of
a
cub
e,
and
B
and
C
b
e
any
t
w
o
other
p
oints
on
the
cub
e.
Use
a
dot
p
ro
duct
to
explain
why
the
angle
b
etw
een
−
→
AB
and
−
→
A
C
cannot
b
e
larger
than
π
2
.
(Hint,
put
A
at
(0
,
0
,
0).)
87
Section
13.1
V
ector
F
unctions
and
Space
Curves
Goals:
1
Graph
certain
plane
curves.
2
Compute
limits
and
verify
the
continuit
y
of
vecto
r
functions.
Question
13.1.1
What
Is
a
V
ector-V
alued
Function?
Definition
A
general
vecto
r-valued
function
r
(
t
)
has
a
number
as
an
input
and
a
vecto
r
of
some
fixed
dimension
as
its
output.
If
the
outputs
are
tw
o-dimensional,
then
there
are
comp
onent
functions
f
(
t
)
and
g
(
t
)
such
that
r
(
t
)
=
⟨
f
(
t
)
,
g
(
t
)
⟩
o
r
r
(
t
)
=
f
(
t
)
i
+
g
(
t
)
j
.
The
domain
of
r
is
the
set
of
all
t
for
which
b
oth
comp
onent
functions
a
re
defined.
89
Question
13.1.1
What
Is
a
Vecto
r-Valued
Function?
Definition
The
graph
of
a
vecto
r-valued
function
r
(
t
)
is
the
set
of
points
whose
p
osition
vecto
r
is
r
(
t
)
fo
r
some
value
of
t
.
In
other
wo
rds,
they
are
the
p
oints
whose
co
o
rdinates
a
re
the
comp
onents
of
r
(
t
).
Rema
rk
Generally
the
graph
of
a
pa
rametric
equation
is
one-dimensional,
lik
e
a
line
o
r
a
curve.
W
e
call
these
graphs
plane
curves
o
r
space
curves
dep
ending
on
the
dimension
of
the
outputs
of
r
(
t
).
90
Question
13.1.1
What
Is
a
Vecto
r-Valued
Function?
Click to Load Applet
Figure:
The
graph
of
a
vecto
r
function
91
Question
13.1.1
What
Is
a
Vecto
r-Valued
Function?
Notation
W
e
can
alternately
exp
ress
a
vecto
r
function
as
a
set
of
pa
rametric
equations
.
F
o
r
instance
r
(
t
)
=
⟨
f
(
t
)
,
g
(
t
)
,
h
(
t
)
⟩
can
b
e
rewritten
as
x
=
f
(
t
)
y
=
g
(
t
)
z
=
h
(
t
)
The
va
riable
t
is
called
a
parameter
in
this
setting.
92
Question
13.1.2
What
is
the
V
ector
Equation
of
a
Line?
Here
is
a
w
a
y
to
describ
e
a
line
b
y
vecto
r
equation:
Equation
If
r
0
is
the
p
osition
vecto
r
of
an
kno
wn
point
,
and
v
is
a
direction
vecto
r
pa
rallel
to
the
line,
then
the
line
is
describ
ed
b
y
r
(
t
)
=
r
0
+
t
v
where
t
can
b
e
any
real
numb
er.
93
Question
13.1.2
What
is
the
Vecto
r
Equation
of
a
Line?
The
endp
oints
of
the
vecto
rs
r
(
t
)
trace
out
the
line
as
t
ranges
over
all
real
numb
ers.
Click to Load Applet
Figure:
A
line
and
the
vecto
rs
that
produce
its
vecto
r
function
94
Question
13.1.2
What
is
the
Vecto
r
Equation
of
a
Line?
In
addition
to
pa
rametric
notation,
lines
can
also
b
e
exp
ressed
as
symmetric
equations
Notation
The
follo
wing
a
re
equivalent
r
(
t
)
=
⟨
x
0
,
y
0
,
z
0
⟩
+
t
⟨
a
,
b
,
c
⟩
x
=
x
0
+
ta
y
=
y
0
+
tb
z
=
z
0
+
tc
x
−
x
0
a
=
y
−
y
0
b
=
z
−
z
0
c
95
Question
13.1.2
What
is
the
Vector
Equation
of
a
Line?
Exercise
a
Are
these
t
w
o
lines
pa
rallel?
Ho
w
can
y
ou
tell?
r
1
(
t
)
=
⟨
3
,
2
,
7
⟩
+
t
⟨
4
,
−
8
,
10
⟩
r
2
(
t
)
=
⟨
0
,
1
,
0
⟩
+
t
⟨−
6
,
12
,
−
15
⟩
b
Must
any
t
w
o
lines
in
three
space
either
b
e
pa
rallel
o
r
intersect?
Explain.
c
Quentin
claims
that
these
lines
do
not
intersect
r
3
(
t
)
=
⟨
0
,
6
,
0
⟩
+
t
⟨
2
,
−
1
,
4
⟩
r
4
(
t
)
=
⟨
0
,
0
,
8
⟩
+
t
⟨
3
,
0
,
4
⟩
He
a
rgues
that
the
equations
obtained
from
setting
the
co
o
rdinates
equal
do
not
have
a
solution.
2
t
=
3
t
6
−
t
=
0
4
t
=
8
+
4
t
What
do
y
ou
think
of
his
reasoning?
Do
the
lines
intersect?
96
Question
13.1.2
What
is
the
Vector
Equation
of
a
Line?
Click to Load Applet
Figure:
Two
intersecting
lines
in
three-space
97
Example
13.1.3
Other
Plane
Curves
to
Kno
w
Graph
the
plane
curves
asso
ciated
to
the
follo
wing
vecto
r
functions:
a
r
(
t
)
=
⟨
4
+
2
t
,
3
−
3
t
⟩
b
r
(
t
)
=
⟨
4
+
2
t
,
3
−
3
t
⟩
0
≤
t
≤
1
c
r
(
t
)
=
⟨
3
cos
t
,
3
sin
t
⟩
d
r
(
t
)
=
t
,
t
3
98
Example
13.1.3
Other
Plane
Curves
to
Know
c
r
(
t
)
=
⟨
3
cos
t
,
3
sin
t
⟩
Click to Load Applet
99
Example
13.1.3
Other
Plane
Curves
to
Know
d
r
(
t
)
=
t
,
t
3
Click to Load Applet
100
Example
13.1.3
Other
Plane
Curves
to
Know
Exercise
a
Sk
etch
the
plane
curve
of
r
(
t
)
=
(3
+
t
)
i
+
(5
−
4
t
)
j
0
≤
t
≤
1.
b
Sk
etch
the
plane
curve
of
r
(
t
)
=
⟨
2
cos(
t
)
,
2
sin(
t
)
⟩
0
≤
t
≤
2
π
.
c
Ho
w
w
ould
r
(
t
)
=
⟨
2
cos(
t
)
,
2
sin(
t
)
+
4
⟩
0
≤
t
≤
2
π
differ
from
b
?
Plot
some
p
oints
if
you
need
to.
d
Ho
w
w
ould
r
(
t
)
=
⟨
6
cos(
t
)
,
2
sin(
t
)
⟩
0
≤
t
≤
2
π
differ
from
b
?
Do
es
this
plane
curve
have
a
shap
e
y
ou
recognize?
e
What
graph
is
defined
b
y
r
(
t
)
=
(
t
3
−
4
t
)
i
+
t
j
?
101
Question
13.1.4
Ho
w
Do
W
e
Visualize
a
Space
Curve?
The
space
curve
defined
b
y
r
(
t
)
=
(1
−
cos(
t
)
−
sin(
t
))
i
+
cos(
t
)
j
+
sin(
t
)
k
is
b
est
understo
o
d
as
a
p
rojection.
Click to Load Applet
Figure:
A
unit
circle
in
the
yx
-plane
p
rojected
onto
x
=
1
−
y
−
z
102
Question
13.1.4
How
Do
We
Visualize
a
Space
Curve?
The
space
curve
defined
b
y
r
(
t
)
=
t
i
+
t
2
j
+
t
3
k
can
b
e
understoo
d
as
the
intersection
of
t
w
o
surfaces:
Click to Load Applet
Figure:
The
intersection
of
z
=
x
3
and
y
=
x
2
103
Question
13.1.4
How
Do
We
Visualize
a
Space
Curve?
The
space
curve
defined
b
y
r
(
t
)
=
cos(
t
)
i
+
sin(
t
)
j
+
t
4
k
can
b
e
understo
o
d
b
y
a
projectile
motion
a
rgument.
Click to Load Applet
Figure:
r
(
t
),
which
traces
the
unit
circle
above
the
xy
-plane
while
steadily
increasing
in
the
z
-direction
104
Question
13.1.5
What
Is
the
Limit
of
a
V
ector
F
unction?
Definition
If
r
(
t
)
=
⟨
f
(
t
)
,
g
(
t
)
,
h
(
t
)
⟩
then
lim
t
→
a
r
(
t
)
=
D
lim
t
→
a
f
(
t
)
,
lim
t
→
a
g
(
t
)
,
lim
t
→
a
h
(
t
)
E
Provided
the
limits
of
all
three
comp
onent
functions
exist.
105
Question
13.1.5
What
Is
the
Limit
of
a
Vecto
r
Function?
Definition
A
vecto
r
function
r
is
continuous
at
a
if
lim
t
→
a
r
(
t
)
=
r
(
a
)
.
This
is
the
case
if
and
only
if
the
comp
onent
functions
f
(
t
)
,
g
(
t
)
and
h
(
t
)
are
continuous
at
a
.
106
Example
13.1.6
T
esting
Continuit
y
Is
r
(
t
)
=
t
2
i
+
e
t
j
+
sin
t
t
k
continuous
at
t
=
0?
Justify
your
answ
er
using
the
definition
of
continuit
y
.
107
Section
13.2
Derivatives
of
a
V
ector
F
unctions
Goals:
1
Compute
derivatives
of
vecto
r
functions.
2
Interp
ret
derivatives
as
tangent
vecto
rs.
Question
13.2.1
What
Is
the
Derivative
of
a
V
ector
F
unction?
Definition
W
e
define
the
derivative
of
r
(
t
)
b
y
d
r
dt
=
r
′
(
t
)
=
lim
h
→
0
r
(
t
+
h
)
−
r
(
t
)
h
Notice
since
the
numerato
r
is
a
vecto
r
and
the
denominato
r
is
a
scala
r,
w
e
a
re
taking
the
limit
of
a
vecto
r
function.
109
Question
13.2.1
What
Is
the
Derivative
of
a
Vecto
r
Function?
If
r
(
t
)
=
f
(
t
)
i
+
g
(
t
)
j
then
what
is
r
′
(
t
)?
110
Question
13.2.1
What
Is
the
Derivative
of
a
Vecto
r
Function?
Theo
rem
If
r
(
t
)
=
f
(
t
)
i
+
g
(
t
)
j
then
r
′
(
t
)
=
f
′
(
t
)
i
+
g
′
(
t
)
j
,
Provided
these
derivatives
exist.
Simila
rly
,
if
r
(
t
)
=
f
(
t
)
i
+
g
(
t
)
j
+
h
(
t
)
k
then
r
′
(
t
)
=
f
′
(
t
)
i
+
g
′
(
t
)
j
+
h
′
(
t
)
k
,
Provided
these
derivatives
exist.
111
Question
13.2.1
What
Is
the
Derivative
of
a
Vecto
r
Function?
The
follo
wing
p
rop
erties
follo
w
from
applying
the
derivative
rules
y
ou
lea
rned
in
single-variable
calculus
to
each
component
of
a
vecto
r
function.
Theo
rem
F
o
r
any
differentiable
vecto
r
functions
u
(
t
)
,
v
(
t
),
differentiable
real-valued
function
f
(
t
)
and
constant
c
w
e
have
1
(
u
+
v
)
′
=
u
′
+
v
′
2
(
c
u
)
′
=
c
u
′
3
(
f
u
)
′
=
f
′
u
+
f
u
′
4
(
u
·
v
)
′
=
u
′
·
v
+
u
·
v
′
112
Question
13.2.2
What
Is
a
T
angent
V
ecto
r?
Definition
The
vector
r
′
(
t
0
)
is
called
a
tangent
vecto
r
to
the
curve
defined
by
r
(
t
).
If
r
(
t
0
)
defines
the
p
oint
P
,
then
we
call
r
′
(
t
0
)
the
tangent
vecto
r
at
P
.
By
replacing
t
0
with
a
va
riable
t
,
w
e
can
define
the
derivative
function
r
′
(
t
).
113
Question
13.2.2
What
Is
a
T
angent
Vector?
If
we
imagine
that
r
(
t
)
describ
es
the
p
osition
of
an
object
at
time
t
,
then
r
′
(
t
)
tells
us
the
velo
cit
y
(direction
and
magnitude)
of
the
object.
Click to Load Applet
Figure:
A
space
curve
and
its
tangent
vector
114
Question
13.2.2
What
Is
a
T
angent
Vector?
Here
a
re
t
w
o
closely
related
constructions
to
the
tangent
vecto
r.
Definition
The
unit
tangent
vecto
r
at
r
(
t
0
)
is
denoted
T
(
t
0
).
T
(
t
0
)
=
r
′
(
t
0
)
|
r
′
(
t
0
)
|
The
tangent
line
to
r
(
t
)
at
r
(
t
0
)
has
the
vecto
r
equation
L
(
s
)
=
r
(
t
0
)
+
s
r
′
(
t
0
)
115
Section
14.1
F
unctions
of
Several
Va
riables
Goals:
1
Convert
an
implicit
function
to
an
explicit
function.
2
Calculate
the
domain
of
a
multiva
riable
function.
3
Calculate
level
curves
and
cross
sections
.
Question
14.1.1
What
Is
a
F
unction
of
More
than
One
V
ariable?
Definition
A
function
of
t
w
o
va
riables
is
a
rule
that
assigns
a
numb
er
(the
output
)
to
each
o
rdered
pair
of
real
numb
ers
(
x
,
y
)
in
its
domain
.
The
output
is
denoted
f
(
x
,
y
).
Some
functions
can
b
e
defined
algeb
raically
.
If
f
(
x
,
y
)
=
p
36
−
4
x
2
−
y
2
then
f
(1
,
4)
=
p
36
−
4
·
1
2
−
4
2
=
4
.
117
Example
14.1.2
The
Domain
of
a
F
unction
Identify
the
domain
of
f
(
x
,
y
)
=
p
36
−
4
x
2
−
y
2
.
Figure:
The
domain
of
a
function
118
Example
14.1.2
The
Domain
of
a
F
unction
Identify
the
domain
of
f
(
x
,
y
)
=
p
36
−
4
x
2
−
y
2
.
Figure:
The
domain
of
a
function
118
Application
14.1.3
T
emp
erature
Maps
Many
useful
functions
cannot
b
e
defined
algeb
raically
.
There
is
a
function
T
(
x
,
y
)
which
gives
the
temp
erature
at
each
latitude
and
longitude
(
x
,
y
)
on
ea
rth.
T
(
−
71
.
06
,
42
.
36)
=
50
T
(
−
84
.
38
,
33
.
75)
=
59
T
(
−
83
.
74
,
42
.
28)
=
41
Figure:
A
temp
erature
map
119
Application
14.1.4
Digital
Images
A
digital
image
can
b
e
defined
b
y
a
b
rightness
function
B
(
x
,
y
).
y
x
687
1024
B
(339
,
773)
=
158
B
(340
,
773)
=
127
Figure:
An
image
represented
as
a
brightness
function
B
on
each
pixel
120
Question
14.1.5
What
Is
the
Graph
of
a
Tw
o-Va
riable
Function?
Definition
The
graph
of
a
function
f
(
x
,
y
)
is
the
set
of
all
p
oints
(
x
,
y
,
z
)
that
satisfy
z
=
f
(
x
,
y
)
.
The
height
z
ab
ove
a
p
oint
(
x
,
y
)
rep
resents
the
value
of
the
function
at
(
x
,
y
).
121
Question
14.1.5
What
Is
the
Graph
of
a
Two-V
ariable
Function?
In
this
figure,
f
(1
,
4)
is
equal
to
the
height
of
the
graph
ab
ove
(1
,
4
,
0).
Click to Load Applet
Figure:
The
graph
z
=
p
36
−
4
x
2
−
y
2
122
Question
14.1.6
Ho
w
Do
W
e
Visualize
a
Graph
in
Three-Space?
Definition
A
level
set
of
a
function
f
(
x
,
y
)
is
the
graph
of
the
equation
f
(
x
,
y
)
=
c
fo
r
some
constant
c
.
Fo
r
a
function
of
t
wo
variables
this
graph
lies
in
the
xy
-plane
and
is
called
a
level
curve
.
Example
Consider
the
function
f
(
x
,
y
)
=
p
36
−
4
x
2
−
y
2
.
The
level
curve
p
36
−
4
x
2
−
y
2
=
4
simplifies
to
4
x
2
+
y
2
=
20.
This
is
an
ellipse.
Other
level
curves
have
the
fo
rm
p
36
−
4
x
2
−
y
2
=
c
or
4
x
2
+
y
2
=
36
−
c
2
.
These
a
re
la
rger
o
r
smaller
ellipses.
123
Question
14.1.6
How
Do
We
Visualize
a
Graph
in
Three-Space?
Level
curves
tak
e
their
shap
e
from
the
intersection
of
z
=
f
(
x
,
y
)
and
z
=
c
.
Seeing
many
level
curves
at
once
can
help
us
visualize
the
shap
e
of
the
graph.
Click to Load Applet
Figure:
The
graph
z
=
f
(
x
,
y
)
,
the
planes
z
=
c
,
and
the
level
curves
124
Example
14.1.7
Dra
wing
Level
Curves
Where
a
re
the
level
curves
on
this
temp
erature
map?
Figure:
A
temp
erature
map
125
Example
14.1.8
Using
Level
Curves
to
Describ
e
a
Graph
What
features
can
w
e
discern
from
the
level
curves
of
this
top
ographical
map?
Figure:
A
top
ographical
map
126
Example
14.1.9
A
Cross
Section
Definition
The
intersection
of
a
plane
with
a
graph
is
a
cross
section
.
A
level
curve
is
a
t
yp
e
of
cross
section,
but
not
all
cross
sections
a
re
level
curves.
Find
the
cross
section
of
z
=
p
36
−
4
x
2
−
y
2
at
the
plane
y
=
1.
127
Example
14.1.9
A
Cross
Section
Click to Load Applet
Figure:
The
y
=
1
cross
section
of
z
=
p
36
−
4
x
2
−
y
2
128
Example
14.1.10
Converting
an
Implicit
Equation
to
a
F
unction
Definition
W
e
sometimes
call
an
equation
in
x
,
y
and
z
an
implicit
equation
.
Often
in
o
rder
to
graph
these,
w
e
convert
them
to
explicit
functions
of
the
fo
rm
z
=
f
(
x
,
y
)
W
rite
the
equation
of
a
pa
rab
oloid
x
2
−
y
+
z
2
=
0
as
one
or
mo
re
explicit
functions
so
it
can
b
e
graphed.
Then
find
the
level
curves.
129
Example
14.1.10
Converting
an
Implicit
Equation
to
a
Function
Click to Load Applet
Figure:
Level
curves
of
x
2
−
y
+
z
2
=
0
130
Question
14.1.11
Ho
w
Do
es
this
Apply
to
F
unctions
of
More
V
a
riables?
W
e
can
define
functions
of
three
va
riables
as
w
ell.
Denoting
them
f
(
x
,
y
,
z
).
F
o
r
even
more
variables,
w
e
use
x
1
through
x
n
.
The
definitions
of
this
section
can
b
e
extrap
olated
as
follo
ws.
V
a
riables
2
3
n
F
unction
f
(
x
,
y
)
f
(
x
,
y
,
z
)
f
(
x
1
,
.
.
.
,
x
n
)
Domain
subset
of
R
2
subset
of
R
3
subset
of
R
n
Graph
z
=
f
(
x
,
y
)
in
R
3
w
=
f
(
x
,
y
,
z
)
in
R
4
x
n
+1
=
f
(
x
1
,
.
.
.
,
x
n
)
in
R
n
+1
Level
Sets
level
curve
in
R
2
level
surface
in
R
3
level
set
in
R
n
131
Question
14.1.11
How
Does
this
Apply
to
Functions
of
More
Variables?
Observation
W
e
might
hop
e
to
solve
an
implicit
equation
of
n
variables
to
obtain
an
explicit
function
of
n
−
1
variables.
How
ever,
we
can
also
treat
it
as
a
level
set
of
an
explicit
function
of
n
variables
(whose
graph
lives
in
n
+
1
dimensional
space).
x
2
+
y
2
+
z
2
=
25
F
(
x
,
y
,
z
)
=
x
2
+
y
2
+
z
2
F
(
x
,
y
,
z
)
=
25
f
(
x
,
y
)
=
±
p
25
−
x
2
−
y
2
Both
viewp
oints
will
b
e
useful
in
the
future.
132
Section
14.1
Summa
ry
Questions
Q1
What
do
es
the
height
of
the
graph
z
=
f
(
x
,
y
)
rep
resent?
Q2
What
is
the
distinction
b
et
w
een
a
level
set
and
a
cross
section?
Q3
What
a
re
level
sets
in
R
2
and
R
3
called?
Q4
What
is
the
difference
b
et
w
een
an
implicit
equation
and
explicit
function?
133
Section
14.1
Q50
Consider
the
implicit
equation
zx
=
y
a
Rewrite
this
equation
as
an
explicit
function
z
=
f
(
x
,
y
).
b
What
is
the
domain
of
f
?
c
Solve
fo
r
and
sk
etch
a
few
level
sets
of
f
.
d
What
do
the
level
sets
tell
y
ou
ab
out
the
graph
z
=
f
(
x
,
y
)?
134
Section
14.1
Q50
Click to Load Applet
134
Section
14.2
Limits
and
Continuity
Goals:
1
Understand
the
definition
of
a
limit
of
a
multiva
riable
function.
2
Use
the
Squeeze
Theo
rem
3
Apply
the
definition
of
continuit
y
.
Question
14.2.1
What
Is
the
Limit
of
a
F
unction?
Definition
W
e
write
lim
(
x
,
y
)
→
(
a
,
b
)
f
(
x
,
y
)
=
L
if
w
e
can
mak
e
the
values
of
f
sta
y
a
rbitra
rily
close
to
L
b
y
restricting
to
a
sufficiently
small
neighb
o
rho
o
d
of
(
a
,
b
).
Proving
a
limit
exists
requires
a
fo
rmula
o
r
rule.
F
o
r
any
amount
of
closeness
required
(
ϵ
),
y
ou
must
b
e
able
to
p
ro
duce
a
radius
δ
a
round
(
a
,
b
)
sufficiently
small
to
keep
|
f
(
x
,
y
)
−
L
|
<
ϵ
.
136
Example
14.2.2
A
Limit
That
Do
es
Not
Exist
Sho
w
that
lim
(
x
,
y
)
→
(0
,
0)
x
2
−
y
2
x
2
+
y
2
do
es
not
exist.
Click to Load Applet
137
Example
14.2.3
Another
Limit
That
Do
es
Not
Exist
Sho
w
that
lim
(
x
,
y
)
→
(0
,
0)
xy
x
2
+
y
2
do
es
not
exist.
Click to Load Applet
138
Example
14.2.4
Y
et
Another
Limit
That
Do
es
Not
Exist
Sho
w
that
lim
(
x
,
y
)
→
(0
,
0)
xy
2
x
2
+
y
4
do
es
not
exist.
Click to Load Applet
139
Question
14.2.5
What
T
o
ols
Apply
to
Multi-Va
riable
Limits?
The
limit
la
ws
from
single-va
riable
limits
transfer
comfo
rtably
to
multi-va
riable
functions.
1
Sum/Difference
Rule
2
Constant
Multiple
Rule
3
Pro
duct/Quotient
Rule
The
Squeeze
Theo
rem
If
g
<
f
<
h
in
some
neighb
orhoo
d
of
(
a
,
b
)
and
lim
(
x
,
y
)
→
(
a
,
b
)
g
(
x
,
y
)
=
lim
(
x
,
y
)
→
(
a
,
b
)
h
(
x
,
y
)
=
L
,
then
lim
(
x
,
y
)
→
(
a
,
b
)
f
(
x
,
y
)
=
L
.
140
Question
14.2.6
What
Is
a
Continuous
F
unction?
Definition
W
e
sa
y
f
(
x
,
y
)
is
continuous
at
(
a
,
b
)
if
lim
(
x
,
y
)
→
(
a
,
b
)
f
(
x
,
y
)
=
f
(
a
,
b
)
.
Theo
rem
P
olynomials,
ro
ots,
trig
functions,
exp
onential
functions
and
loga
rithms
a
re
continuous
on
their
domains.
Sums,
differences,
p
ro
ducts,
quotients
and
comp
ositions
of
continuous
functions
a
re
continuous
on
their
domains.
In
each
of
our
examples,
the
function
w
as
a
quotient
of
p
olynomials,
but
(0
,
0)
was
not
in
the
domain.
141
Question
14.2.6
What
Is
a
Continuous
Function?
Rema
rk
Limits,
continuit
y
and
these
theo
rems
can
all
b
e
extrap
olated
to
functions
of
mo
re
va
riables.
142
Section
14.2
Summa
ry
Questions
Q1
Why
is
it
ha
rder
to
verify
a
limit
of
a
multiva
riable
function?
Q2
What
do
y
ou
need
to
check
in
o
rder
to
determine
whether
a
function
is
continuous?
143
Section
14.3
P
artial
Derivatives
Goals:
1
Calculate
pa
rtial
derivatives
.
2
Realize
when
not
to
calculate
pa
rtial
derivatives.
Question
14.3.1
What
Is
the
Rate
of
Change
of
a
M
ultiva
riable
Function?
Motivational
Example
The
fo
rce
due
to
gravit
y
b
et
w
een
t
w
o
objects
dep
ends
on
their
masses
and
on
the
distance
b
et
w
een
them.
Supp
ose
at
a
distance
of
8
,
000km
the
fo
rce
b
etw
een
t
w
o
pa
rticular
objects
is
100
newtons
and
at
a
distance
of
10
,
000km,
the
force
is
64
newtons.
Ho
w
much
do
w
e
exp
ect
the
fo
rce
b
et
w
een
these
objects
to
increase
o
r
decrease
p
er
kilometer
of
distance?
145
Question
14.3.1
What
Is
the
Rate
of
Change
of
a
Multivariable
Function?
Derivatives
of
a
single-va
riable
function
w
ere
a
w
a
y
of
measuring
the
change
in
a
function.
Recall
the
follo
wing
facts
ab
out
f
′
(
x
).
1
Average
rate
of
change
is
realized
as
the
slop
e
of
a
secant
line:
f
(
x
)
−
f
(
x
0
)
x
−
x
0
2
The
derivative
f
′
(
x
)
is
defined
as
a
limit
of
slop
es:
f
′
(
x
)
=
lim
h
→
0
f
(
x
+
h
)
−
f
(
x
)
h
3
The
derivative
is
the
instantaneous
rate
of
change
of
f
at
x
.
4
The
derivative
f
′
(
x
0
)
is
realized
geometrically
as
the
slop
e
of
the
tangent
line
to
y
=
f
(
x
)
at
x
0
.
5
The
equation
of
that
tangent
line
can
be
written
in
p
oint-slop
e
form:
y
−
y
0
=
f
′
(
x
0
)(
x
−
x
0
)
146
Question
14.3.1
What
Is
the
Rate
of
Change
of
a
Multivariable
Function?
A
pa
rtial
derivative
measures
the
rate
of
change
of
a
multiva
riable
function
as
one
va
riable
changes,
but
the
others
remain
constant.
Definition
The
pa
rtial
derivatives
of
a
t
wo-va
riable
function
f
(
x
,
y
)
are
the
functions
f
x
(
x
,
y
)
=
lim
h
→
0
f
(
x
+
h
,
y
)
−
f
(
x
,
y
)
h
and
f
y
(
x
,
y
)
=
lim
h
→
0
f
(
x
,
y
+
h
)
−
f
(
x
,
y
)
h
.
147
Question
14.3.1
What
Is
the
Rate
of
Change
of
a
Multivariable
Function?
Notation
The
pa
rtial
derivative
of
a
function
can
b
e
denoted
a
va
riet
y
of
w
a
ys.
Here
a
re
some
equivalent
notations
f
x
∂
f
∂
x
∂
z
∂
x
∂
∂
x
f
D
x
f
148
Example
14.3.2
Computing
a
Pa
rtial
Derivative
Find
∂
∂
y
(
y
2
−
x
2
+
3
x
sin
y
).
Main
Idea
T
o
compute
a
pa
rtial
derivative
f
y
,
p
erfo
rm
single-va
riable
differentiation.
T
reat
y
as
the
indep
endent
variable
and
x
as
a
constant.
149
Synthesis
14.3.3
Interp
reting
Derivatives
from
Level
Sets
Belo
w
a
re
the
level
curves
f
(
x
,
y
)
=
c
fo
r
some
values
of
c
.
Can
we
tell
whether
f
x
(
−
4
,
1
.
25)
and
f
y
(
−
4
,
1
.
25)
are
positive
or
negative?
Figure:
Some
level
curves
of
f
(
x
,
y
)
150
Question
14.3.4
What
Is
the
Geometric
Significance
of
a
Pa
rtial
Derivative?
The
pa
rtial
derivative
f
x
(
x
0
,
y
0
)
is
realized
geometrically
as
the
slop
e
of
the
line
tangent
to
z
=
f
(
x
,
y
)
at
(
x
0
,
y
0
,
z
0
)
and
traveling
in
the
x
direction.
Since
y
is
held
constant,
this
tangent
line
lives
in
y
=
y
0
.
Click to Load Applet
Figure:
The
tangent
line
to
z
=
f
(
x
,
y
)
in
the
x
direction
151
Example
14.3.5
Derivative
Rules
and
P
artial
Derivatives
Find
f
x
fo
r
the
follo
wing
functions
f
(
x
,
y
):
a
f
=
√
xy
(on
the
domain
x
>
0
,
y
>
0)
b
f
=
y
x
c
f
=
√
x
+
y
d
f
=
sin
(
xy
)
152
Question
14.3.6
What
If
We
Have
Mo
re
than
Tw
o
V
a
riables?
W
e
can
also
calculate
pa
rtial
derivatives
of
functions
of
mo
re
va
riables.
All
va
riables
but
one
a
re
held
to
b
e
constants.
F
o
r
example
if
f
(
x
,
y
,
z
)
=
x
2
−
xy
+
cos(
yz
)
−
5
z
3
then
w
e
can
calculate
∂
f
∂
y
:
153
Example
14.3.7
A
F
unction
of
Three
V
ariables
F
o
r
an
ideal
gas,
w
e
have
the
la
w
P
=
nRT
V
,
where
P
is
p
ressure,
n
is
the
numb
er
of
moles
of
gas
molecules,
T
is
the
temperature,
and
V
is
the
volume.
a
Calculate
∂
P
∂
V
.
b
Calculate
∂
P
∂
T
.
c
(Science
Question)
Supp
ose
w
e’re
heating
a
sealed
gas
contained
in
a
glass
container.
Do
es
∂
P
∂
T
tell
us
ho
w
quickly
the
p
ressure
is
increasing
p
er
degree
of
temp
erature
increase?
154
Question
14.3.8
Ho
w
Do
Higher
Order
Derivatives
W
o
rk?
T
aking
a
pa
rtial
derivative
of
a
pa
rtial
derivative
gives
us
a
higher
o
rder
pa
rtial
derivative.
W
e
use
the
follo
wing
notation.
Notation
(
f
x
)
x
=
f
xx
=
∂
2
f
∂
x
2
W
e
need
not
use
the
same
va
riable
each
time
Notation
(
f
x
)
y
=
f
xy
=
∂
∂
y
∂
∂
x
f
=
∂
2
f
∂
y
∂
x
155
Example
14.3.9
A
Higher
Order
P
artial
Derivative
If
f
(
x
,
y
)
=
sin(3
x
+
x
2
y
)
calculate
f
xy
.
156
Question
14.3.10
Do
es
Differentiation
Order
Matter?
No.
Sp
ecifically
,
the
following
is
due
to
Clairaut:
Theo
rem
If
f
is
defined
on
a
neighb
o
rho
od
of
(
a
,
b
)
and
the
functions
f
xy
and
f
yx
a
re
b
oth
continuous
on
that
neighb
o
rho
o
d,
then
f
xy
(
a
,
b
)
=
f
yx
(
a
,
b
).
This
readily
generalizes
to
la
rger
numb
ers
of
va
riables,
and
higher
o
rder
derivatives.
F
o
r
example
f
xyyz
=
f
zyxy
.
157
Section
14.3
Summa
ry
Questions
Q1
What
is
the
role
of
each
va
riable
when
w
e
compute
a
pa
rtial
derivative?
Q2
What
do
es
the
pa
rtial
derivative
f
y
(
a
,
b
)
mean
geometrically?
Q3
Can
y
ou
think
of
an
example
where
the
pa
rtial
derivative
do
es
not
accurately
mo
del
the
change
in
a
function?
Q4
What
is
Clairaut’s
Theo
rem?
158
Section
14.3
Q10
In
the
diagram
from
this
example,
use
a
p
oint
on
the
c
=
30
level
set
to
app
ro
ximate
f
y
(4
,
−
1
.
25).
Figure:
Some
level
curves
of
f
(
x
,
y
)
159
Section
14.3
Q20
Supp
ose
Jinteki
Co
rp
o
ration
mak
es
widgets
which
is
sells
fo
r
$100
each.
It
commands
a
small
enough
p
o
rtion
of
the
ma
rk
et
that
its
p
ro
duction
level
do
es
not
affect
the
demand
(p
rice)
fo
r
its
p
ro
ducts.
If
W
is
the
numb
er
of
widgets
p
ro
duced
and
C
is
their
op
erating
cost,
Jinteki’s
p
rofit
is
mo
deled
b
y
P
=
100
W
−
C
.
Since
∂
P
∂
W
=
100
do
es
this
mean
that
increasing
production
can
b
e
exp
ected
to
increase
profit
at
a
rate
of
$100
p
er
widget?
160
Section
14.3
Q28
Ho
w
many
third
pa
rtial
derivatives
do
es
a
t
w
o-va
riable
function
have?
Assuming
these
derivatives
a
re
continuous,
which
of
them
a
re
equal
acco
rding
to
Clairaut’s
theo
rem?
161
Section
12.5
No
rmal
Equations
of
Planes
Goals:
1
Give
equations
of
planes
in
b
oth
vecto
r
and
no
rmal
fo
rms.
2
Use
no
rmal
vecto
rs
to
measure
the
distance
to
a
plane.
Question
12.5.1
What
Is
the
Slop
e-Intercept
Equation
of
a
Plane?
Unlik
e
a
line,
a
non-vertical
plane
has
t
wo
slopes.
One
measures
rise
over
run
in
the
x
-direction,
the
other
in
the
y
-direction.
Click to Load Applet
Figure:
A
plane
with
slopes
in
the
x
and
y
directions.
163
Question
12.5.1
What
Is
the
Slope-Intercept
Equation
of
a
Plane?
Equation
A
plane
with
z
intercept
(0
,
0
,
b
)
and
slop
es
m
x
and
m
y
in
the
x
and
y
directions
has
equation
z
=
m
x
x
+
m
y
y
+
b
.
164
Example
12.5.2
W
riting
the
Equation
of
a
Plane
W
rite
the
equation
of
a
plane
with
intercepts
(4
,
0
,
0),
(0
,
6
,
0)
and
(0
,
0
,
8).
165
Example
12.5.2
Writing
the
Equation
of
a
Plane
Main
Idea
Given
three
p
oints
in
a
plane
A
=
(
x
1
,
y
1
,
z
1
),
B
=
(
x
2
,
y
2
,
z
2
)
and
C
=
(
x
3
,
y
3
,
z
3
)
1
If
t
w
o
p
oints
sha
re
an
x
-co
ordinate,
w
e
can
directly
compute
m
y
and
vice
versa.
2
F
ailing
that,
w
e
can
set
up
a
system
of
equations
and
solve
fo
r
m
x
,
m
y
and
b
.
166
Question
12.5.3
What
is
a
No
rmal
V
ecto
r
to
a
Plane?
In
algeb
ra,
y
ou
lea
rned
the
no
rmal
equation
of
a
line:
e.g.
2
x
+
3
y
−
12
=
0.
Why
is
it
called
this?
167
Question
12.5.3
What
is
a
No
rmal
V
ecto
r
to
a
Plane?
In
algeb
ra,
y
ou
lea
rned
the
no
rmal
equation
of
a
line:
e.g.
2
x
+
3
y
−
12
=
0.
Why
is
it
called
this?
167
Question
12.5.3
What
is
a
Normal
V
ector
to
a
Plane?
A
no
rmal
vecto
r
to
a
plane
is
orthogonal
to
every
vecto
r
in
the
plane.
Theo
rem
In
three-dimensional
space,
every
plane
has
no
rmal
vecto
rs.
They
a
re
all
pa
rallel
to
each
other.
Click to Load Applet
Figure:
A
plane
,
its
no
rmal
vector
n
,
and
a
vecto
r
−
→
PQ
in
the
plane
168
Question
12.5.3
What
is
a
Normal
V
ector
to
a
Plane?
Theo
rem
If
r
0
=
⟨
x
0
,
y
0
,
z
0
⟩
describ
es
an
kno
wn
p
oint
on
a
plane,
and
n
=
⟨
a
,
b
,
c
⟩
is
a
no
rmal
vecto
r.
Then
the
no
rmal
equation
of
the
plane
is
(
r
−
r
0
)
·
n
=
0
o
r
a
(
x
−
x
0
)
+
b
(
y
−
y
0
)
+
c
(
z
−
z
0
)
=
0
img/normalequation.pn
g
Notice
that
since
x
0
,
y
0
and
z
0
a
re
constants,
w
e
can
distribute
and
collect
them
into
a
single
term:
d
.
ax
+
b
y
+
cz
−
ax
0
−
by
0
−
cz
0
=
0
ax
+
b
y
+
cz
+
d
=
0
169
Question
12.5.3
What
is
a
Normal
V
ector
to
a
Plane?
This
reasoning
w
o
rks
in
any
dimension
to
define
a
set
of
p
oints
whose
displacement
from
a
kno
wn
p
oint
is
o
rthogonal
to
some
no
rmal
vecto
r.
Example
a
(
x
−
x
0
)
+
b
(
y
−
y
0
)
=
0
defines
a
line.
a
(
x
−
x
0
)
+
b
(
y
−
y
0
)
+
c
(
z
−
z
0
)
=
0
defines
a
plane.
a
1
(
x
1
−
c
1
)
+
a
2
(
x
2
−
c
2
)
+
·
·
·
+
a
n
(
x
n
−
c
n
)
=
0
defines
a
hyp
erplane
.
170
Example
12.5.4
Computing
a
Normal
V
ecto
r
Find
the
no
rmal
equation
of
the
plane
with
intercepts
(4
,
0
,
0),
(0
,
3
,
0)
and
(0
,
0
,
8).
Compute
a
normal
vecto
r.
171
Synthesis
12.5.5
Using
the
Normal
V
ecto
r
to
Compute
Distance
Consider
the
line
2
x
+
3
y
−
12
=
0.
This
is
the
line
with
normal
vector
n
=
⟨
2
,
3
⟩
and
kno
wn
p
oint
P
=
(3
,
2).
172
Synthesis
12.5.5
Using
the
Normal
Vector
to
Compute
Distance
Example
Let
P
1
=
(7
,
2)
and
P
2
=
(4
,
0).
1
Dra
w
the
vecto
rs
−
−
→
PP
1
and
−
−
→
PP
2
.
2
If
y
ou
didn’t
have
a
picture,
ho
w
could
y
ou
use
the
values
of
n
·
−
−
→
PP
1
and
n
·
−
−
→
PP
2
to
determine
which
side
of
the
line
P
1
and
P
2
lie
on?
173
Synthesis
12.5.5
Using
the
Normal
Vector
to
Compute
Distance
Theo
rem
Given
a
line,
plane,
o
r
hyperplane
with
no
rmal
equation
L
(
x
1
,
.
.
.
,
x
k
)
=
0
and
co
rresp
onding
no
rmal
vecto
r
n
,
the
signed
distance
from
the
hyp
erplane
to
the
p
oint
Q
=
(
q
1
,
.
.
.
,
q
k
)
is
L
(
q
1
,
.
.
.
,
q
k
)
|
n
|
.
174
Example
12.5.6
The
Distance
from
a
Plane
Compute
the
geometric
distance
from
the
o
rigin
to
the
plane
6
x
+
8
y
+
3
z
−
24
=
0.
175
Application
12.5.7
Supp
o
rt
V
ecto
r
Machines
One
t
yp
e
of
machine
lea
rning
involves
training
a
computer
to
distinguish
b
et
w
een
t
w
o
states.
F
o
r
example,
a
computer
might
b
e
trained
to
distinguish
b
et
w
een
a
cancerous
tumo
r
and
a
b
enign
one.
T
o
do
this
the
computer
is
given
a
la
rge
set
of
cases.
Each
case
is
measured
b
y
numerical
data,
such
as:
The
size
of
the
tumo
r
The
lo
cation
of
the
tumo
r
The
age
of
the
patient
Results
of
blo
o
d
tests
The
b
rightness
of
each
pixel
in
a
CT
scan
o
r
MRI
Each
data
t
yp
e
is
a
dimension,
and
each
case
is
a
p
oint
in
a
(p
robably
very
high)
dimensional
space.
176
Application
12.5.7
Support
Vector
Machines
Click to Load Applet
177
Section
12.5
Summa
ry
Questions
Q1
What
info
rmation
do
y
ou
need
in
o
rder
to
write
the
no
rmal
equation
of
a
plane?
Q2
Ho
w
a
re
the
no
rmal
vecto
rs
of
a
plane
related
to
each
other?
Q3
What
is
the
significance
of
the
co
efficients
in
the
no
rmal
equation
of
a
plane?
Q4
Ho
w
do
w
e
compute
the
signed
distance
from
a
p
oint
to
a
plane?
178
Section
12.5
Q14
Supp
ose
w
e
kno
w
the
planes
12
x
+
18
y
+
6
z
−
15
=
0
and
ax
+
b
y
+
4
z
+
d
=
0
a
re
pa
rallel.
What
can
y
ou
sa
y
ab
out
the
values
of
a
,
b
and
d
?
179
Section
12.5
Q30
Tw
o
planes
a
re
p
erp
endicula
r
if
their
no
rmal
vecto
rs
a
re
o
rthogonal.
a
Are
4
x
−
7
y
+
z
−
3
=
0
and
5
x
+
y
+
13
z
+
25
=
0
p
erpendicular?
b
If
t
w
o
planes
a
re
p
erp
endicula
r,
is
every
vecto
r
in
the
first
plane
o
rthogonal
to
every
vecto
r
in
the
second
plane?
180
Section
14.4
Linea
r
App
ro
ximations
Goals:
1
Calculate
the
equation
of
a
tangent
plane
.
2
Rewrite
the
tangent
plane
formula
as
a
linea
rization
o
r
differential
.
3
Use
linea
rizations
to
estimate
values
of
a
function.
4
Use
a
differential
to
estimate
the
erro
r
in
a
calculation.
Question
14.4.1
What
Is
a
T
angent
Plane?
Definition
A
tangent
plane
at
a
point
P
=
(
x
0
,
y
0
,
z
0
)
on
a
surface
is
a
plane
containing
the
tangent
lines
to
the
surface
through
P
.
Click to Load Applet
Figure:
The
tangent
plane
to
z
=
f
(
x
,
y
)
at
a
p
oint
182
Question
14.4.1
What
Is
a
T
angent
Plane?
Equation
If
the
graph
z
=
f
(
x
,
y
)
has
a
tangent
plane
at
(
x
0
,
y
0
),
then
it
has
the
equation:
z
−
z
0
=
f
x
(
x
0
,
y
0
)(
x
−
x
0
)
+
f
y
(
x
0
,
y
0
)(
y
−
y
0
)
.
Rema
rks
1
This
is
the
p
oint-slop
e
fo
rm
of
the
equation
of
a
plane.
f
x
(
x
0
,
y
0
)
and
f
y
(
x
0
,
y
0
)
a
re
the
slop
es.
2
x
0
and
y
0
a
re
numb
ers,
so
f
x
(
x
0
,
y
0
)
and
f
y
(
x
0
,
y
0
)
are
numb
ers.
The
va
riables
in
this
equation
a
re
x
,
y
and
z
.
183
Question
14.4.1
What
Is
a
T
angent
Plane?
The
cross
sections
of
the
tangent
plane
give
the
equation
of
the
tangent
lines
w
e
lea
rned
in
single
va
riable
calculus.
y
=
y
0
x
=
x
0
z
−
z
0
=
f
x
(
x
0
,
y
0
)(
x
−
x
0
)
+
0
z
−
z
0
=
0
+
f
y
(
x
0
,
y
0
)(
y
−
y
0
)
184
Example
14.4.2
W
riting
the
Equation
of
a
T
angent
Plane
Give
an
equation
of
the
tangent
plane
to
f
(
x
,
y
)
=
√
xe
y
at
(4
,
0)
185
Question
14.4.3
Ho
w
Do
W
e
Rewrite
a
T
angent
Plane
as
a
Function?
Definition
If
w
e
write
z
as
a
function
L
(
x
,
y
),
w
e
obtain
the
linea
rization
of
f
at
(
x
0
,
y
0
).
L
(
x
,
y
)
=
f
(
x
0
,
y
0
)
+
f
x
(
x
0
,
y
0
)(
x
−
x
0
)
+
f
y
(
x
0
,
y
0
)(
y
−
y
0
)
If
the
graph
z
=
f
(
x
,
y
)
has
a
tangent
plane,
then
L
(
x
,
y
)
app
roximates
the
values
of
f
nea
r
(
x
0
,
y
0
).
Notice
f
(
x
0
,
y
0
)
just
calculates
the
value
of
z
0
.
This
formula
is
equivalent
to
the
tangent
plane
equation
after
w
e
solve
fo
r
z
b
y
adding
z
0
to
b
oth
sides.
186
Example
14.4.4
App
roximating
a
F
unction
Use
a
linea
rization
to
app
ro
ximate
the
value
of
√
4
.
02
e
0
.
05
.
187
Question
14.4.5
Ho
w
Do
es
Differential
Notation
W
ork
in
More
V
a
riables?
The
differential
dz
measures
the
change
in
the
linea
rization
of
f
(
x
,
y
)
given
pa
rticula
r
changes
in
the
inputs:
dx
and
dy
.
It
is
a
useful
sho
rthand
when
one
is
estimating
the
erro
r
in
an
initial
computation.
Definition
F
o
r
z
=
f
(
x
,
y
),
the
differential
o
r
total
differential
dz
is
a
function
of
a
p
oint
(
x
0
,
y
0
)
and
t
w
o
indep
endent
va
riables
dx
and
dy
.
dz
=
f
x
(
x
0
,
y
0
)
dx
+
f
y
(
x
0
,
y
0
)
dy
=
∂
z
∂
x
dx
+
∂
z
∂
y
dy
Rema
rk
The
differential
fo
rmula
is
just
the
tangent
plane
fo
rmula
with
dz
=
z
−
z
0
dx
=
x
−
x
0
dy
=
y
−
y
0
.
188
Question
14.4.5
How
Does
Differential
Notation
Wo
rk
in
More
Variables?
An
old
trigonometry
application
is
to
measure
the
height
of
a
p
ole
b
y
standing
at
some
distance.
We
then
measure
the
angle
θ
of
incline
to
the
top,
as
w
ell
as
the
distance
b
to
the
base.
The
height
is
h
=
b
tan
θ
.
a
If
the
distance
to
the
base
is
13
m
and
the
angle
of
incline
is
π
6
,
what
is
the
height
of
the
p
ole?
b
Human
measurement
is
never
p
erfect.
If
our
measurement
of
b
is
off
b
y
at
most
0
.
1
m
and
our
measurement
of
θ
is
off
b
y
at
most
π
120
,
use
a
differential
to
app
ro
ximate
the
maximum
p
ossible
erro
r
in
our
h
.
189
Section
14.4
Summa
ry
Questions
Q1
What
do
y
ou
need
to
compute
in
o
rder
to
write
the
equation
of
a
tangent
plane
to
z
=
f
(
x
,
y
)
at
(
x
0
,
y
0
,
z
0
)?
Q2
F
o
r
what
kinds
of
functions
a
re
linea
r
app
ro
ximations
useful?
Q3
Ho
w
a
re
the
tangent
plane
and
the
linea
rization
related?
Q4
Ho
w
is
the
differential
defined
fo
r
a
t
w
o
va
riable
function?
What
do
es
each
va
riable
in
the
fo
rmula
mean?
190
Section
14.4
Q10
Let
g
(
x
,
y
)
=
3
x
2
+4
x
−
2
e
(
y
3
)
.
W
rite
the
equation
of
the
tangent
plane
to
z
=
g
(
x
,
y
)
at
(0
,
1).
191
Section
14.4
Q16
Sho
w
ho
w
to
use
an
app
rop
riate
linea
rization
to
app
ro
ximate
1
5
.
12
sin
31
π
30
.
a
What
function
f
(
x
,
y
)
w
ould
y
ou
linea
rize
to
mak
e
this
app
ro
ximation?
b
What
(
x
0
,
y
0
)
w
ould
y
ou
use
to
write
y
our
linea
rization?
c
What
x
and
y
w
ould
y
ou
plug
into
L
(
x
,
y
)
to
app
ro
ximate
1
5
.
12
sin
31
π
30
?
192
Section
14.4
Q21
Bo
ris
is
measuring
the
a
rea
of
a
rectangula
r
field,
so
he
can
decide
ho
w
much
grass
seed
to
buy
.
Acco
rding
to
his
measurements,
the
field
is
30
m
b
y
50
m
,
giving
an
area
of
1500
m
2
.
If
w
e
accept
that
each
of
his
measurements
has
an
erro
r
no
la
rger
than
0
.
2
m
,
use
a
differential
to
app
ro
ximate
the
maximum
erro
r
in
his
a
rea
computation.
193
Section
14.4
Q22
Supp
ose
I
decide
to
invest
$10
,
000
exp
ecting
a
6%
annual
rate
of
return
fo
r
12
y
ea
rs,
after
which
I’ll
use
it
to
purchase
a
house.
The
fo
rmula
fo
r
comp
ound
interest
P
=
P
0
e
rt
indicates
that
when
I
w
ant
to
buy
a
house,
I
will
have
P
=
10
,
000
e
0
.
72
.
I
accept
that
my
exp
ected
rate
of
return
might
have
an
erro
r
of
up
to
dr
=
2%.
Also,
I
may
decide
to
buy
a
house
up
to
dt
=
3
yea
rs
b
efo
re
or
after
I
exp
ected.
a
W
rite
the
fo
rmula
fo
r
the
differential
dP
at
(
r
0
,
t
0
)
=
(0
.
06
,
12).
b
Given
my
assumptions,
what
is
the
maximum
estimated
erro
r
dP
in
my
initial
calculation?
c
What
is
the
actual
maximum
erro
r
in
P
?
194
Section
14.4
Q24
Let
f
(
x
,
y
)
b
e
a
function.
What
differential
and
what
inputs
into
that
differential
w
ould
y
ou
use
to
app
ro
ximate
f
(5
.
5
,
3
.
2)
−
f
(4
.
7
,
3
.
8).
195
Section
14.5
The
Chain
Rule
Goals:
1
Use
the
chain
rule
to
compute
derivatives
of
compositions
of
functions.
2
P
erfo
rm
implicit
differentiation
using
the
chain
rule.
Section
14.5
The
Chain
Rule
Motivational
Example
Supp
ose
Jinteki
Co
rp
o
ration
mak
es
widgets
which
is
sells
fo
r
$100
each.
It
commands
a
small
enough
p
o
rtion
of
the
ma
rk
et
that
its
p
ro
duction
level
do
es
not
affect
the
demand
(p
rice)
fo
r
its
p
ro
ducts.
If
W
is
the
numb
er
of
widgets
p
ro
duced
and
C
is
their
op
erating
cost,
Jinteki’s
p
rofit
is
mo
deled
b
y
P
=
100
W
−
C
The
partial
derivative
∂
P
∂
W
=
100
do
es
not
correctly
calculate
the
effect
of
increasing
p
ro
duction
on
p
rofit.
How
can
w
e
calculate
this
co
rrectly?
197
Question
14.5.1
Ho
w
Do
We
Compute
the
Derivative
of
a
Comp
osition
of
Functions?
Given
a
function
f
(
x
,
y
)
where
x
=
x
(
t
)
and
y
=
y
(
t
),
we
can
ask
how
f
changes
as
t
changes.
We
can
visualize
this
change
b
y
dra
wing
the
graph
z
=
f
(
x
,
y
)
over
the
path
given
b
y
the
parametric
equations
x
(
t
)
and
y
(
t
).
Click to Load Applet
Figure:
The
comp
osition
f
(
x
(
t
)
,
y
(
t
)),
represented
by
the
height
of
z
=
f
(
x
,
y
)
over
the
path
(
x
(
t
)
,
y
(
t
))
198
Question
14.5.1
How
Do
We
Compute
the
Derivative
of
a
Comp
osition
of
Functions?
Theo
rem
(The
Chain
Rule)
Consider
a
differentiable
function
f
(
x
,
y
).
If
we
define
x
=
x
(
t
)
and
y
=
y
(
t
),
b
oth
differential
functions,
we
have
df
dt
=
∂
f
∂
x
dx
dt
+
∂
f
∂
y
dy
dt
199
Question
14.5.1
How
Do
We
Compute
the
Derivative
of
a
Comp
osition
of
Functions?
Rema
rks
f
(
x
(
t
)
,
y
(
t
))
is
a
function
(only)
of
t
.
Because
of
this,
df
dt
is
an
o
rdina
ry
derivative,
not
a
pa
rtial
derivative.
df
dt
is
not
the
slop
e
of
the
comp
osition
graph.
slop
e
=
rise
in
z
run
in
xy
-plane
df
dt
=
rise
in
z
change
in
t
The
chain
rule
is
easy
to
rememb
er
b
ecause
of
its
simila
rit
y
to
the
differential:
dz
=
∂
z
∂
x
dx
+
∂
z
∂
y
dy
.
The
p
ro
of
is
mo
re
complicated
than
just
sticking
a
dt
under
each
term.
200
Example
14.5.2
Using
the
Chain
Rule
If
P
=
R
−
C
and
w
e
have
R
=
100
w
and
C
=
3000
+
70
w
−
0
.
1
w
2
,
calculate
dP
dw
.
201
Question
14.5.3
What
If
We
Have
Mo
re
V
ariables?
The
chain
rule
w
o
rks
just
as
w
ell
if
x
and
y
a
re
functions
of
more
than
one
va
riable.
In
this
case
it
computes
pa
rtial
derivatives.
Theo
rem
If
f
(
x
,
y
),
x
(
s
,
t
)
and
y
(
s
,
t
),
are
all
differentiable,
then
∂
f
∂
s
=
∂
f
∂
x
∂
x
∂
s
+
∂
f
∂
y
∂
y
∂
s
202
Question
14.5.3
What
If
We
Have
More
Variables?
W
e
can
also
mo
dify
it
fo
r
functions
of
mo
re
than
t
w
o
va
riables.
Theo
rem
Given
f
(
x
,
y
,
z
),
x
(
t
),
y
(
t
)
and
z
(
t
),
all
differentiable,
w
e
have
df
dt
=
∂
f
∂
x
dx
dt
+
∂
f
∂
y
dy
dt
+
∂
f
∂
z
dz
dt
203
Example
14.5.4
A
Comp
osition
with
Mo
re
V
a
riables
Recall
that
fo
r
an
ideal
gas
P
(
n
,
T
,
V
)
=
nRT
V
.
R
is
a
constant.
n
is
the
numb
er
of
molecules
of
gas.
T
is
the
temp
erature
in
Celsius.
V
is
the
volume
in
meters.
Supp
ose
w
e
w
ant
to
understand
the
rate
at
which
the
p
ressure
changes
as
an
air-tight
glass
container
of
gas
is
heated.
a
Apply
the
chain
rule
to
get
an
exp
ression
fo
r
dP
dT
.
b
What
is
dn
dT
?
c
What
is
dT
dT
?
d
Supp
ose
that
dV
dT
=
(5
.
9
×
10
−
6
)
V
.
Calculate
and
simplify
the
exp
ression
y
ou
got
fo
r
dP
dT
.
204
Example
14.5.4
A
Comp
osition
with
Mo
re
V
a
riables
Recall
that
fo
r
an
ideal
gas
P
(
n
,
T
,
V
)
=
nRT
V
.
R
is
a
constant.
n
is
the
numb
er
of
molecules
of
gas.
T
is
the
temp
erature
in
Celsius.
V
is
the
volume
in
meters.
Supp
ose
w
e
w
ant
to
understand
the
rate
at
which
the
p
ressure
changes
as
an
air-tight
glass
container
of
gas
is
heated.
a
Apply
the
chain
rule
to
get
an
exp
ression
fo
r
dP
dT
.
204
Example
14.5.4
A
Comp
osition
with
Mo
re
V
a
riables
Recall
that
fo
r
an
ideal
gas
P
(
n
,
T
,
V
)
=
nRT
V
.
R
is
a
constant.
n
is
the
numb
er
of
molecules
of
gas.
T
is
the
temp
erature
in
Celsius.
V
is
the
volume
in
meters.
Supp
ose
w
e
w
ant
to
understand
the
rate
at
which
the
p
ressure
changes
as
an
air-tight
glass
container
of
gas
is
heated.
b
What
is
dn
dT
?
204
Example
14.5.4
A
Comp
osition
with
Mo
re
V
a
riables
Recall
that
fo
r
an
ideal
gas
P
(
n
,
T
,
V
)
=
nRT
V
.
R
is
a
constant.
n
is
the
numb
er
of
molecules
of
gas.
T
is
the
temp
erature
in
Celsius.
V
is
the
volume
in
meters.
Supp
ose
w
e
w
ant
to
understand
the
rate
at
which
the
p
ressure
changes
as
an
air-tight
glass
container
of
gas
is
heated.
c
What
is
dT
dT
?
204
Example
14.5.4
A
Comp
osition
with
Mo
re
V
a
riables
Recall
that
fo
r
an
ideal
gas
P
(
n
,
T
,
V
)
=
nRT
V
.
R
is
a
constant.
n
is
the
numb
er
of
molecules
of
gas.
T
is
the
temp
erature
in
Celsius.
V
is
the
volume
in
meters.
Supp
ose
w
e
w
ant
to
understand
the
rate
at
which
the
p
ressure
changes
as
an
air-tight
glass
container
of
gas
is
heated.
d
Supp
ose
that
dV
dT
=
(5
.
9
×
10
−
6
)
V
.
Calculate
and
simplify
the
exp
ression
y
ou
got
fo
r
dP
dT
.
204
Example
14.5.5
A
Comp
osition
with
Limited
Information
Supp
ose
g
(
p
,
q
,
r
)
=
re
p
2
q
.
Given
that
p
,
q
,
r
are
all
differentiable
functions
of
x
with
the
values
in
the
following
table,
compute
dg
dx
when
x
=
2.
x
0
1
2
3
p
(
x
)
3
1
5
10
p
′
(
x
)
−
3
2
3
4
q
(
x
)
6
2
−
2
3
q
′
(
x
)
−
1
−
5
2
3
r
(
x
)
10
11
7
3
r
′
(
x
)
1
0
−
1
−
3
205
Application
14.5.6
Implicit
Differentiation
Recall
that
an
implicit
equation
on
n
variables
defines
a
level
set
of
a
n
-variable
function.
Consider
the
graph
x
3
+
y
2
−
4
xy
=
0.
How
can
w
e
use
this
to
calculate
dy
dx
at
the
p
oint
(3
,
3)?
206
Application
14.5.6
Implicit
Differentiation
Figure:
The
graph
of
F
(
x
,
y
)
=
x
3
+
y
2
−
4
xy
=
0,
its
tangent
line
at
(3
,
3),
and
the
gradient
of
F
Main
Ideas
dy
dx
is
the
slop
e
of
the
tangent
line
to
F
(
x
,
y
)
=
c
.
The
chain
rule
allo
ws
us
to
derive
dy
dx
=
−
F
x
F
y
−
F
x
F
y
is
the
negative
recip
ro
cal
of
F
y
F
x
,
which
is
the
slop
e
of
∇
F
.
207
Application
14.5.7
Indirect
Profit
Functions
Supp
ose
a
firm
chooses
how
much
quantit
y
q
to
produce,
but
their
p
rofit
Π(
q
,
α
)
depends
on
some
pa
rameter
α
outside
their
control
(maybe
a
tax
o
r
a
measure
of
regulato
ry
burden).
The
firm,
once
it
kno
ws
the
value
of
α
,
will
cho
ose
the
q
that
maximizes
p
rofit.
Ho
w
will
their
p
rofit
change
as
α
changes?
208
Application
14.5.7
Indirect
Profit
Functions
Click to Load Applet
Figure:
Two
graphs
of
z
=
Π(
q
,
α
),
one
where
q
changes
to
b
e
the
optimal
choice
for
each
α
and
one
where
q
is
fixed
at
q
0
,
the
optimal
choice
fo
r
α
0
209
Section
14.5
Summa
ry
Questions
Q1
Ho
w
can
w
e
visualize
f
(
x
,
y
),
when
x
and
y
a
re
functions
of
t
?
Q2
Explain
why
df
dt
cannot
b
e
interp
reted
as
a
slop
e
of
f
over
the
xy
-plane.
Q3
What
is
the
difference
b
et
w
een
dz
dx
and
∂
z
∂
x
?
Ho
w
is
the
first
one
computed?
Q4
Ho
w
do
y
ou
use
the
chain
rule
to
differentiate
implicit
functions?
210
Section
14.5
Q12
Liam
sa
ys
“Suppose
f
is
a
function
of
x
and
y
.
If
x
and
y
a
re
increasing,
then
f
is
increasing.”
We
all
kno
w
Liam
is
inco
rrect.
How
could
we
use
the
chain
rule
to
refute
him?
211
Section
14.5
Q14
Let
x
=
t
2
and
y
=
sin
t
.
Let
f
(
x
,
y
)
=
xy
.
a
Compute
df
dt
using
the
multiva
riable
chain
rule.
b
Compute
df
dt
b
y
substituting
and
using
single-va
riable
differentiation.
c
What
ea
rlier
rule
of
differentiation
can
w
e
recover
b
y
applying
the
chain
rule
to
f
(
x
,
y
)
=
xy
?
212
Section
14.5
Q26
Another
p
rinciple
in
physics
is
the
conservation
of
energy
.
Kenetic
energy
is
given
b
y
E
=
1
2
mv
2
,
where
m
is
the
mass
and
v
is
the
linear
speed
of
the
object.
Supp
ose
that
w
e
have
a
ro
ck
drifiting
through
space.
Supp
ose
it
impacts
stationa
ry
ro
cks
and
the
combined
mass
sticks
together
(without
releasing
any
energy
as
heat,
light
or
sound).
Thus
the
mass
of
the
total
travelling
object
increases,
while
the
total
energy
sta
ys
the
same.
Derive
an
exp
ression
fo
r
ho
w
sp
eed
changes
p
er
unit
of
increase
in
mass.
213
Section
14.5
Q27
Supp
ose
that
x
is
a
function
of
t
and
that
when
t
=
9,
we
have
x
=
7
and
dx
dt
=
−
3.
Define
f
(
x
,
t
)
=
√
x
+
t
.
a
Compute
the
pa
rtial
derivate
∂
f
∂
t
(7
,
9)
.
b
Compute
the
total
derivative
df
dt
(7
,
9)
.
c
In
a
few
sentences,
explain
what
these
t
w
o
quantities
compute
and
why
they
a
re
different
from
each
other.
214
Section
14.5
Q30
Supp
ose
the
p
osition
of
a
pa
rticle
at
time
t
is
given
by
x
(
t
)
=
t
2
y
(
t
)
=
3
−
t
z
(
t
)
=
√
t
A
t
t
=
4,
ho
w
quickly
is
pa
rticle
travelling
a
w
a
y
from
the
plane
x
+
2
y
−
2
z
=
10?
215
Section
14.5
Q31
Here
is
a
diagram
of
the
level
curves
of
h
(
x
,
y
)
for
certain
values
of
c
.
a
Is
h
y
(2
,
1)
p
ositive
or
negative?
Explain
in
a
sentence
o
r
tw
o.
b
Add
a
vecto
r
to
the
diagram
that
indicates
the
direction
of
greatest
increase
of
h
at
(
−
2
,
0).
c
Supp
ose
x
=
4
−
5
t
and
y
=
3
t
2
.
Determine,
with
the
aid
of
a
relevant
calculation,
whether
dh
dt
is
p
ositive
o
r
negative
at
t
=
1.
216
Section
14.6
The
Gradient
Vecto
r
Goals:
1
Calculate
the
gradient
vecto
r
of
a
function.
2
Relate
the
gradient
vecto
r
to
the
shap
e
of
a
graph
and
its
level
curves.
3
Compute
directional
derivatives
.
Question
14.6.1
Ho
w
Do
We
Compute
Rates
of
Change
in
Another
Direction?
The
pa
rtial
derivatives
of
f
(
x
,
y
)
give
the
instantaneous
rate
of
change
in
the
x
and
y
directions.
This
is
realized
geometrically
as
the
slop
e
of
the
tangent
line.
What
if
w
e
w
ant
to
travel
in
a
different
direction?
Click to Load Applet
Figure:
The
tangent
line
to
z
=
f
(
x
,
y
)
in
the
x
direction
218
Question
14.6.1
How
Do
We
Compute
Rates
of
Change
in
Another
Direction?
Definition
Let
f
(
x
,
y
)
b
e
a
function
and
u
b
e
a
unit
vecto
r
in
R
2
.
The
directional
derivative
,
denoted
D
u
f
,
is
the
instantaneous
rate
of
change
of
f
as
w
e
move
in
the
u
direction.
This
is
also
the
slop
e
of
the
tangent
line
to
z
=
f
(
x
,
y
)
in
the
direction
of
u
.
Click to Load Applet
Figure:
The
tangent
line
to
f
(
x
,
y
)
in
the
direction
of
u
219
Question
14.6.1
How
Do
We
Compute
Rates
of
Change
in
Another
Direction?
Recall
that
w
e
compute
D
x
f
b
y
compa
ring
the
values
of
f
at
(
x
,
y
)
to
the
value
at
(
x
+
h
,
y
),
a
displacement
of
h
in
the
x
-direction.
D
x
f
(
x
,
y
)
=
lim
h
→
0
f
(
x
+
h
,
y
)
−
f
(
x
,
y
)
h
T
o
compute
D
u
f
fo
r
u
=
a
i
+
b
j
,
we
compa
re
the
value
of
f
at
(
x
,
y
)
to
the
value
at
(
x
+
ta
,
y
+
tb
),
a
displacement
of
t
in
the
u
-direction.
Limit
F
ormula
D
u
f
(
x
,
y
)
=
lim
t
→
0
f
(
x
+
ta
,
y
+
tb
)
−
f
(
x
,
y
)
t
220
Question
14.6.1
How
Do
We
Compute
Rates
of
Change
in
Another
Direction?
Questions:
1
What
direction
p
ro
duces
the
greatest
directional
derivative?
The
smallest?
2
Ho
w
a
re
these
directions
related
to
the
geometry
(sp
ecifically
the
level
curves)
of
the
graph?
3
Ho
w
these
directions
related
to
the
pa
rtial
derivatives?
221
Question
14.6.1
How
Do
We
Compute
Rates
of
Change
in
Another
Direction?
Click to Load Applet
Figure:
A
cross
section
of
z
=
f
(
x
,
y
)
and
a
tangent
line
in
the
direction
of
u
222
Question
14.6.2
What
Is
the
Gradient
V
ecto
r?
Definition
The
gradient
vecto
r
of
f
at
(
x
,
y
)
is
∇
f
(
x
,
y
)
=
⟨
f
x
(
x
,
y
)
,
f
y
(
x
,
y
)
⟩
Rema
rks:
1
The
gradient
vecto
r
is
a
function
of
(
x
,
y
).
Different
p
oints
have
different
gradients.
2
u
max
,
which
maximizes
D
u
f
,
p
oints
in
the
same
direction
as
∇
f
.
3
u
0
,
which
is
tangent
to
the
level
curves,
is
o
rthogonal
to
∇
f
.
223
Question
14.6.3
Ho
w
Do
W
e
Compute
a
Directional
Derivative?
The
tangent
lines
live
in
the
tangent
plane.
W
e
can
compute
their
slop
e
b
y
rise
over
run.
Let
u
b
e
a
unit
vecto
r
from
(
x
0
,
y
0
)
to
(
x
1
,
y
1
).
Let
the
asso
ciated
z
values
in
the
tangent
plane
b
e
z
0
and
z
1
resp
ectively
.
D
u
f
(
x
0
,
y
0
)
=
rise
run
=
z
1
−
z
0
|
u
|
=
f
x
(
x
0
,
y
0
)(
x
1
−
x
0
)
+
f
y
(
x
0
,
y
0
)(
y
1
−
y
0
)
=
∇
f
(
x
0
,
y
0
)
·
u
.
Click to Load Applet
224
Question
14.6.3
How
Do
We
Compute
a
Directional
Derivative?
F
unctions
of
Mo
re
V
ariables
W
e
can
also
define
directional
derivatives
of
higher
va
riable
functions
with
analogous
results.
f
(
x
1
,
.
.
.
,
x
n
)
is
a
differentiable
function.
u
is
a
unit
vecto
r
in
R
n
.
D
u
f
denotes
the
directional
derivative
in
the
direction
of
u
.
∇
f
=
⟨
f
x
1
,
.
.
.
,
f
x
n
⟩
is
an
n
-dimensional
vector
function
on
R
n
.
D
u
f
=
∇
f
·
u
225
Synthesis
14.6.4
Directional
Derivative
and
the
Cosine
Fo
rmula
No
w
that
w
e
have
a
fo
rmula
fo
r
directional
derivatives,
w
e
can
verify
our
observations
from
ea
rlier.
Supp
ose
f
(
x
,
y
)
is
a
differentiable
function
and
w
e
can
cho
ose
any
unit
vecto
r
u
.
a
W
rite
D
u
f
(
x
,
y
)
in
terms
of
the
length
of
a
vecto
r
and
an
angle.
b
In
what
direction
u
will
f
increase
fastest?
c
What
will
b
e
the
value
of
D
u
f
(
x
,
y
)
in
that
direction?
d
In
what
direction
u
will
D
u
f
(
x
,
y
)
=
0?
226
Synthesis
14.6.4
Directional
Derivative
and
the
Cosine
Formula
Click to Load Applet
Figure:
The
angle
b
etw
een
the
gradient
of
f
and
a
unit
vector
Main
Ideas
The
cosine
fo
rmula
fo
r
the
dot
p
ro
duct
lets
us
relate
the
directional
derivative
to
an
angle.
f
increases
fastest
in
the
direction
of
∇
f
(
x
,
y
).
D
u
f
(
x
,
y
)
=
0
when
∇
f
(
x
,
y
)
and
u
a
re
o
rthogonal.
227
Example
14.6.5
A
Directional
Derivative
Let
f
(
x
,
y
)
=
p
9
−
x
2
−
y
2
and
let
u
=
⟨
0
.
6
,
−
0
.
8
⟩
.
a
What
a
re
the
level
curves
of
f
?
b
What
direction
do
es
∇
f
(1
,
2)
p
oint?
c
Without
calculating,
is
D
u
f
(1
,
2)
p
ositive
or
negative?
d
Calculate
∇
f
(1
,
2)
and
D
u
f
(1
,
2).
228
Example
14.6.6
Dra
wing
the
Gradient
Let
h
(
x
,
y
)
give
the
altitude
at
longitude
x
and
latitude
y
.
Assuming
h
is
differentiable,
dra
w
the
direction
of
∇
h
(
x
,
y
)
at
each
of
the
p
oints
lab
eled
b
elo
w.
Which
gradient
is
the
longest?
A
B
C
Figure:
A
top
ographical
map
229
Application
14.6.7
Edge
Detection
The
length
of
the
gradient
of
a
b
rightness
function
detects
the
edges
in
a
picture,
where
the
b
rightness
is
changing
quickly
.
∂
B
∂
x
(336
,
785)
≈
185
−
187
1
∂
B
∂
y
(336
,
785)
≈
179
−
187
1
∇
B
(336
,
785)
≈
(
−
2
,
−
8)
∂
B
∂
x
(340
,
784)
≈
97
−
139
1
∂
B
∂
y
(340
,
784)
≈
72
−
139
1
∇
B
(340
,
784)
≈
(
−
42
,
−
67)
∇
B
∇
B
Figure:
A
long
gradient
vecto
r
indicates
a
swift
change
in
brightness.
Its
direction
suggests
the
shap
e
of
the
edges.
230
Application
14.6.7
Edge
Detection
The
length
of
the
gradient
of
a
b
rightness
function
detects
the
edges
in
a
picture,
where
the
b
rightness
is
changing
quickly
.
∂
B
∂
x
(336
,
785)
≈
185
−
187
1
∂
B
∂
y
(336
,
785)
≈
179
−
187
1
∇
B
(336
,
785)
≈
(
−
2
,
−
8)
∂
B
∂
x
(340
,
784)
≈
97
−
139
1
∂
B
∂
y
(340
,
784)
≈
72
−
139
1
∇
B
(340
,
784)
≈
(
−
42
,
−
67)
∇
B
∇
B
Figure:
A
long
gradient
vecto
r
indicates
a
swift
change
in
brightness.
Its
direction
suggests
the
shap
e
of
the
edges.
230
Application
14.6.7
Edge
Detection
The
length
of
the
gradient
of
a
b
rightness
function
detects
the
edges
in
a
picture,
where
the
b
rightness
is
changing
quickly
.
∂
B
∂
x
(336
,
785)
≈
185
−
187
1
∂
B
∂
y
(336
,
785)
≈
179
−
187
1
∇
B
(336
,
785)
≈
(
−
2
,
−
8)
∂
B
∂
x
(340
,
784)
≈
97
−
139
1
∂
B
∂
y
(340
,
784)
≈
72
−
139
1
∇
B
(340
,
784)
≈
(
−
42
,
−
67)
∇
B
∇
B
Figure:
A
long
gradient
vecto
r
indicates
a
swift
change
in
brightness.
Its
direction
suggests
the
shap
e
of
the
edges.
230
Application
14.6.7
Edge
Detection
The
length
of
the
gradient
of
a
b
rightness
function
detects
the
edges
in
a
picture,
where
the
b
rightness
is
changing
quickly
.
∂
B
∂
x
(336
,
785)
≈
185
−
187
1
∂
B
∂
y
(336
,
785)
≈
179
−
187
1
∇
B
(336
,
785)
≈
(
−
2
,
−
8)
∂
B
∂
x
(340
,
784)
≈
97
−
139
1
∂
B
∂
y
(340
,
784)
≈
72
−
139
1
∇
B
(340
,
784)
≈
(
−
42
,
−
67)
∇
B
∇
B
Figure:
A
long
gradient
vecto
r
indicates
a
swift
change
in
brightness.
Its
direction
suggests
the
shap
e
of
the
edges.
230
Application
14.6.8
T
angent
Planes
to
a
Level
Surface
Use
a
gradient
vecto
r
to
find
the
equation
of
the
tangent
plane
to
the
graph
x
2
+
y
2
+
z
2
=
14
at
the
p
oint
(2
,
1
,
−
3).
231
Application
14.6.8
T
angent
Planes
to
a
Level
Surface
Main
Idea
The
graph
of
an
implicit
equation
can
b
e
written
as
a
level
set
of
a
function.
The
gradient
of
that
function
is
a
no
rmal
vecto
r
to
the
level
set
and
also
to
its
tangent
line/plane/hyp
erplane.
Click to Load Applet
Figure:
The
level
surface
x
2
+
y
2
+
z
2
=
14
,
its
tangent
plane
and
∇
F
.
232
Section
14.6
Summa
ry
Questions
Q1
What
do
es
the
direction
of
the
gradient
vecto
r
tell
y
ou?
Q2
What
do
es
the
directional
derivative
mean
geometrically?
Q3
Ho
w
do
y
ou
compute
a
directional
derivative?
Q4
Ho
w
is
the
gradient
vecto
r
related
to
a
level
set?
233
Section
14.6
Q12
Supp
ose
the
linea
rization
of
f
(
x
,
y
)
at
(
−
3
,
9)
has
the
equation
L
(
x
,
y
)
=
4
+
2(
x
+
3)
−
1
3
(
y
−
9)
.
What
is
the
slop
e
of
L
from
(
−
3
,
9)
to
(5
,
3)?
234
Section
14.6
Q14
If
D
u
f
(
x
,
y
)
<
0,
what
can
y
ou
say
about
the
directions
of
∇
f
(
x
,
y
)
and
u
?
235
Section
14.6
Q16
Explain
why
it
mak
es
sense
that
if
D
u
f
(
a
,
b
,
c
)
=
0,
then
u
is
tangent
to
the
level
surface
of
f
through
(
a
,
b
,
c
).
236
Section
14.6
Q26
The
b
rightness
function
on
the
Mona
Lisa
image
ranges
from
0
to
255.
If
w
e
use
adjacent
p
oints
to
app
o
rixmate
the
gradient
as
in
the
example,
what
is
the
longest
gradient
vecto
r
w
e
could
theo
retically
p
ro
duce?
237
Section
14.6
Q28
Let
P
b
e
a
point
on
the
circle
x
2
+
y
2
=
r
2
.
Sho
w
that
the
p
osition
vecto
r
of
P
is
no
rmal
to
the
circle
at
P
.
238
Section
14.6
Q36
Supp
ose
that
f
(
x
,
y
,
z
)
is
a
differentiable
function,
and
f
(3
,
5
,
−
2)
=
13.
Supp
ose
further
that
the
vecto
rs
⟨
3
,
1
,
0
⟩
and
⟨
0
,
2
,
5
⟩
b
oth
lie
in
the
tangent
plane
to
the
surface
f
(
x
,
y
,
z
)
=
13
at
(3
,
5
,
−
2).
If
the
maximum
value
of
D
u
f
(3
,
5
,
−
2)
is
20,
find
all
p
ossible
values
of
∇
f
(3
,
5
,
−
2).
239
Section
14.7
Maximum
and
Minimum
V
alues
Goals:
1
Find
critical
points
of
a
function.
2
T
est
critical
p
oints
to
find
lo
cal
maximums
and
minimums.
3
Use
the
Extreme
V
alue
Theo
rem
to
find
the
global
maximum
and
global
minimum
of
a
function
over
a
closed
set.
Question
14.7.1
What
Are
Lo
cal
Extremes?
The
lo
cal
extremes
of
a
function
a
re
the
lo
cal
minimums
and
maximums.
Definition
Given
an
n
-variable
function
f
(
x
1
,
x
2
,
.
.
.
,
x
n
)
w
e
sa
y
that
a
p
oint
P
in
n
-space
is
1
a
lo
cal
maximum
if
f
(
P
)
≥
f
(
Q
)
fo
r
all
Q
in
some
neighborhoo
d
a
round
P
.
2
a
lo
cal
minimum
if
f
(
P
)
≤
f
(
Q
)
fo
r
all
Q
in
some
neighborhoo
d
a
round
P
.
241
Question
14.7.2
Where
Do
Lo
cal
Extremes
Lie?
In
the
case
of
a
t
w
o-va
riable
function,
w
e
can
visualize
this
as
follo
ws:
If
f
x
(
P
)
=
0,
then
we
could
travel
in
the
x
direction
to
increase
or
decrease
f
.
If
f
y
(
P
)
=
0,
then
we
could
travel
in
the
y
direction
to
increase
or
decrease
f
.
Thus
at
a
lo
cal
maximum
o
r
lo
cal
minimum,
the
tangent
plane
must
b
e
ho
rizontal.
Click to Load Applet
Figure:
T
angent
lines
must
have
slop
e
0
at
a
lo
cal
max.
242
Question
14.7.2
Where
Do
Lo
cal
Extremes
Lie?
Definition
W
e
sa
y
P
is
a
critical
point
of
f
if
either
1
∇
f
(
P
)
=
0
o
r
2
∇
f
(
P
)
do
es
not
exist
(b
ecause
one
of
the
partial
derivatives
do
es
not
exist).
Theo
rem
The
lo
cal
maximums
and
minimums
of
a
function
can
only
o
ccur
at
critical
p
oints.
243
Example
14.7.3
Finding
Critical
Points
The
function
z
=
2
x
2
+
4
x
+
y
2
−
6
y
+
13
has
a
minimum
value.
Find
it.
244
Question
14.7.4
Ho
w
Do
We
Identify
Two-V
ariable
Local
Maximums
and
Minimums?
A
critical
p
oint
could
b
e
a
lo
cal
maximum.
In
this
case
f
curves
do
wnw
a
rd
in
every
direction.
Click to Load Applet
Figure:
A
lo
cal
maximum
at
(0
,
0)
245
Question
14.7.4
How
Do
We
Identify
Two-V
ariable
Local
Maximums
and
Minimums?
A
critical
p
oint
could
b
e
a
lo
cal
minimum.
In
this
case
f
curves
upw
ard
in
every
direction.
Click to Load Applet
Figure:
A
lo
cal
minimum
at
(0
,
0)
246
Question
14.7.4
How
Do
We
Identify
Two-V
ariable
Local
Maximums
and
Minimums?
A
critical
p
oint
could
b
e
neither.
f
curves
upw
ard
in
some
directions
but
do
wnw
a
rd
in
others.
This
configuration
is
called
a
saddle
p
oint
.
Click to Load Applet
Figure:
A
saddle
p
oint
at
(0
,
0)
247
Question
14.7.4
How
Do
We
Identify
Two-V
ariable
Local
Maximums
and
Minimums?
Theo
rem
(The
Second
Derivatives
T
est)
Supp
ose
f
is
differentiable
at
(
P
)
and
f
x
(
P
)
=
f
y
(
P
)
=
0.
Then
we
can
compute
D
=
f
xx
(
P
)
f
yy
(
P
)
−
[
f
xy
(
P
)]
2
1
If
D
>
0
and
f
xx
(
P
)
>
0
then
P
is
a
lo
cal
minimum.
2
If
D
>
0
and
f
xx
(
P
)
<
0
then
P
is
a
lo
cal
maximum.
3
If
D
<
0
then
P
is
a
saddle
p
oint.
Unfo
rtunately
,
if
D
=
0,
this
test
gives
no
info
rmation.
248
Question
14.7.4
How
Do
We
Identify
Two-V
ariable
Local
Maximums
and
Minimums?
Definition
The
quantit
y
D
in
the
second
derivatives
test
is
actually
the
determinant
of
a
matrix
called
the
Hessian
of
f
.
f
xx
(
P
)
f
yy
(
P
)
−
[
f
xy
(
P
)]
2
=
det
f
xx
(
P
)
f
xy
(
P
)
f
yx
(
P
)
f
yy
(
P
)
|
{z
}
Hf
(
P
)
Hf
follows
a
logical
pattern
and
can
b
e
a
useful
mnemonic
for
the
second
derivatives
test.
249
Example
14.7.5
Classifying
a
Critical
P
oint
Let
f
(
x
,
y
)
=
cos(2
x
+
y
)
+
xy
a
V
erify
that
∇
f
(0
,
0)
=
⟨
0
,
0
⟩
.
b
Is
(0
,
0)
a
lo
cal
minimum,
a
lo
cal
maximum,
or
neither?
250
Example
14.7.5
Classifying
a
Critical
Point
Click to Load Applet
Figure:
The
graph
z
=
cos(2
x
+
y
)
+
xy
with
a
local
maximum
at
(0
,
0)
251
Question
14.7.6
Ho
w
Do
W
e
Find
Global
Extremes?
Theo
rem
(The
Extreme
V
alue
Theo
rem)
A
continuous
function
f
on
a
closed
and
b
ounded
domain
D
has
a
global
maximum
and
a
global
minimum
somewhere
in
D
.
Definition
Let
D
be
a
subset
of
n
-space.
D
is
closed
if
it
contains
all
of
the
points
on
its
b
oundary
.
D
is
bounded
if
there
is
some
upp
er
limit
to
how
fa
r
its
p
oints
get
from
the
o
rigin
(o
r
any
other
fixed
p
oint).
If
there
a
re
p
oints
of
D
a
rbitra
rily
fa
r
from
the
o
rigin,
then
D
is
unbounded
.
252
Question
14.7.6
How
Do
We
Find
Global
Extremes?
F
o
r
one-variable
functions.
The
EVT
requires
that
the
domain
be
a
union
of
finite,
closed
intervals
(and
ma
yb
e
finitely
many
isolated
p
oints).
Figure:
A
union
of
finite,
closed
intervals
253
Question
14.7.6
How
Do
We
Find
Global
Extremes?
Figure:
x
2
+
y
2
≤
9
is
closed.
Figure:
x
2
+
y
2
<
9
is
not
closed.
254
Question
14.7.6
How
Do
We
Find
Global
Extremes?
Figure:
−
2
≤
x
≤
2
and
−
3
<
y
<
3
is
not
closed.
Figure:
−
2
≤
x
≤
2
and
−
3
≤
y
≤
3
and
(
x
,
y
)
=
(1
,
2)
is
not
closed.
255
Question
14.7.6
How
Do
We
Find
Global
Extremes?
Figure:
−
2
≤
x
≤
2
and
−
3
≤
y
≤
3
is
b
ounded.
Figure:
−
2
≤
x
≤
2
is
unb
ounded.
256
Example
14.7.7
Finding
a
Global
Maximum
Consider
the
function
f
(
x
,
y
)
=
x
2
+
2
y
2
−
x
2
y
on
the
domain
D
=
{
(
x
,
y
)
:
x
2
+
y
2
≤
16
,
x
≤
0
}
a
Do
es
f
have
a
maximum
value
on
D
?
How
do
w
e
kno
w?
b
Find
the
critical
p
oints
of
f
.
c
Must
one
of
the
critical
p
oints
b
e
the
maximum?
d
Find
the
maximum
of
f
.
257
Example
14.7.7
Finding
a
Global
Maximum
Consider
the
function
f
(
x
,
y
)
=
x
2
+
2
y
2
−
x
2
y
on
the
domain
D
=
{
(
x
,
y
)
|
{z
}
points
in
R
2
:
x
2
+
y
2
≤
16
,
x
≤
0
|
{z
}
conditions
}
a
Do
es
f
have
a
maximum
value
on
D
?
How
do
w
e
kno
w?
b
Find
the
critical
p
oints
of
f
.
c
Must
one
of
the
critical
p
oints
b
e
the
maximum?
d
Find
the
maximum
of
f
.
257
Example
14.7.7
Finding
a
Global
Maximum
Consider
the
function
f
(
x
,
y
)
=
x
2
+
2
y
2
−
x
2
y
on
the
domain
D
=
{
(
x
,
y
)
|
{z
}
points
in
R
2
:
x
2
+
y
2
≤
16
,
x
≤
0
|
{z
}
conditions
}
a
Do
es
f
have
a
maximum
value
on
D
?
How
do
w
e
kno
w?
257
Example
14.7.7
Finding
a
Global
Maximum
Consider
the
function
f
(
x
,
y
)
=
x
2
+
2
y
2
−
x
2
y
on
the
domain
D
=
{
(
x
,
y
)
|
{z
}
points
in
R
2
:
x
2
+
y
2
≤
16
,
x
≤
0
|
{z
}
conditions
}
a
Do
es
f
have
a
maximum
value
on
D
?
How
do
w
e
kno
w?
257
Example
14.7.7
Finding
a
Global
Maximum
Consider
the
function
f
(
x
,
y
)
=
x
2
+
2
y
2
−
x
2
y
on
the
domain
D
=
{
(
x
,
y
)
|
{z
}
points
in
R
2
:
x
2
+
y
2
≤
16
,
x
≤
0
|
{z
}
conditions
}
a
Find
the
critical
p
oints
of
f
.
257
Example
14.7.7
Finding
a
Global
Maximum
Consider
the
function
f
(
x
,
y
)
=
x
2
+
2
y
2
−
x
2
y
on
the
domain
D
=
{
(
x
,
y
)
|
{z
}
points
in
R
2
:
x
2
+
y
2
≤
16
,
x
≤
0
|
{z
}
conditions
}
a
Must
one
of
the
critical
p
oints
b
e
the
maximum?
257
Example
14.7.7
Finding
a
Global
Maximum
Consider
the
function
f
(
x
,
y
)
=
x
2
+
2
y
2
−
x
2
y
on
the
domain
D
=
{
(
x
,
y
)
|
{z
}
points
in
R
2
:
x
2
+
y
2
≤
16
,
x
≤
0
|
{z
}
conditions
}
a
Find
the
maximum
of
f
.
257
Example
14.7.7
Finding
a
Global
Maximum
Click to Load Applet
257
Example
14.7.7
Finding
a
Global
Maximum
Main
Ideas
If
the
Extreme
V
alue
Theo
rem
applies,
then
all
we
need
to
do
is
find
the
critical
p
oints
and
evaluate
f
at
each.
One
is
guaranteed
to
be
the
maximum,
and
one
is
gua
ranteed
to
b
e
the
minimum.
∇
f
=
0
will
detect
critical
p
oints
on
the
interio
r,
but
not
on
the
b
ounda
ry
.
W
e
can
rewrite
the
function
on
a
b
ounda
ry
comp
onent
using
substitution.
Set
the
derivative
equal
to
0
to
find
critical
p
oints.
Derivatives
will
not
detect
maximums
at
the
endp
oints
of
a
b
oundary
curve.
These
must
b
e
included
in
y
our
set
of
critical
p
oints.
258
Section
14.7
Summa
ry
Questions
Q1
Where
must
the
lo
cal
maximums
and
minimums
of
a
function
o
ccur?
Why
do
es
this
make
sense?
Q2
What
do
es
the
second
derivatives
test
tell
us?
Q3
What
hyp
otheses
do
es
the
Extreme
V
alue
Theo
rem
require?
What
do
es
it
tell
us?
Q4
Assuming
a
maximum
and
minimum
exist,
where
must
y
ou
lo
ok
in
a
domain
to
b
e
sure
y
ou
find
them?
259
Section
14.7
Q6
Is
a
global
maximum
also
a
lo
cal
maximum?
Explain.
260
Section
14.7
Q12
Supp
ose
f
(
x
)
is
a
function
of
x
with
critical
p
oints
x
=
a
and
x
=
b
.
Supp
ose
g
(
y
)
is
a
function
of
y
with
critical
p
oints
y
=
c
and
y
=
d
.
What
a
re
the
critical
p
oints
of
h
(
x
,
y
)
=
f
(
x
)
+
g
(
y
)?
261
Section
14.7
Q16
F
o
r
what
values
of
a
do
es
f
(
x
,
y
)
=
x
2
+
y
2
+
axy
have
a
lo
cal
minimum
at
the
o
rigin?
262
Section
14.7
Q32
Let
f
(
x
,
y
)
b
e
a
differentiable
function
and
let
D
=
{
(
x
,
y
)
:
y
≥
x
2
−
4
,
x
≥
0
,
y
≤
5
}
.
a
Sk
etch
the
domain
D
.
b
Do
es
the
Extreme
V
alue
Theo
rem
gua
rantee
that
f
has
an
absolute
minimum
on
D
?
Explain.
c
List
all
the
places
y
ou
w
ould
need
to
check
in
o
rder
to
lo
cate
the
minimum.
263
Section
14.8
Lagrange
Multipliers
Goals:
1
Find
minimum
and
maximum
values
of
a
function
subject
to
a
constraint
.
2
If
necessa
ry
,
use
Lagrange
multipliers
.
Question
14.8.1
What
Is
a
Constraint?
Sometimes
w
e
aren’t
interested
in
the
maximum
value
of
f
(
x
,
y
)
over
the
whole
domain,
w
e
w
ant
to
restrict
to
only
those
p
oints
that
satisfy
a
certain
constraint
equation.
The
maximum
on
the
constraint
is
unlik
ely
to
b
e
the
same
as
the
unconstrained
maximum
(where
∇
f
=
0).
Can
we
still
use
∇
f
to
find
the
maximum
on
the
constraint?
Click to Load Applet
Figure:
Maximizing
f
such
that
x
+
y
=
1
265
Question
14.8.2
Ho
w
Do
W
e
Solve
a
Constrained
Optimization?
The
metho
d
of
Lagrange
Multipliers
mak
es
use
of
the
following
theo
rem.
Theo
rem
Supp
ose
an
objective
function
f
(
x
,
y
)
and
a
constraint
function
g
(
x
,
y
)
a
re
differentiable.
The
lo
cal
extremes
of
f
(
x
,
y
)
given
the
constraint
g
(
x
,
y
)
=
c
o
ccur
where
∇
f
=
λ
∇
g
fo
r
some
numb
er
λ
,
o
r
else
where
∇
g
=
0.
The
numb
er
λ
is
called
a
Lagrange
Multiplier
.
This
theo
rem
generalizes
to
functions
of
mo
re
va
riables.
266
Question
14.8.2
How
Do
We
Solve
a
Constrained
Optimization?
Click to Load Applet
Figure:
Where
∇
f
is
not
pa
rallel
to
∇
g
,
we
can
travel
along
g
(
x
,
y
)
=
c
and
increase
the
value
of
f
.
This
is
b
ecause
D
u
f
>
0
fo
r
some
u
along
the
constraint.
267
Example
14.8.3
The
Maximum
on
a
Curve
Find
the
p
oint(s)
on
the
ellipse
4
x
2
+
y
2
=
4
on
which
the
function
f
(
x
,
y
)
=
xy
is
maximized.
268
Example
14.8.3
The
Maximum
on
a
Curve
Figure:
The
four
p
oints
that
satisfy
∇
f
=
λ
∇
g
and
g
(
x
,
y
)
=
c
.
Main
Idea
The
level
set
of
a
continuous
(constraint)
function
is
alw
a
ys
closed.
If
it
is
also
b
ounded
and
the
objective
function
is
differentiable,
then
one
of
the
p
oints
p
ro
duced
b
y
Lagrange
multipliers
will
b
e
the
global
maximum
and
one
will
b
e
the
global
minimum
of
the
constrained
optimization.
269
Example
14.8.4
The
Maximum
on
a
Surface
Find
the
maximum
value
of
the
function
f
(
x
,
y
,
z
)
=
x
4
y
4
z
on
the
sphere
x
2
+
y
2
+
z
2
=
36.
Click to Load Applet
Figure:
The
gradient
vector
and
level
surface
of
a
constraint
function
and
the
gradient
vector
of
the
objective
function
270
Synthesis
14.8.5
Using
the
Extreme
Value
Theorem
and
Lagrange
Multipliers
Ho
w
can
Lagrange
multipliers
help
us
find
the
maximum
of
f
(
x
,
y
)
=
x
2
+
2
y
2
−
x
2
y
on
the
domain
D
=
{
(
x
,
y
)
:
x
2
+
y
2
≤
16
,
x
≤
0
}
?
271
Synthesis
14.8.5
Using
the
Extreme
Value
Theo
rem
and
Lagrange
Multipliers
Main
Idea
T
o
find
the
absolute
minimum
and
maximum
of
a
differentiable
function
f
(
x
,
y
)
over
a
closed
and
b
ounded
domain
D
:
1
Compute
∇
f
and
find
the
critical
p
oints
inside
D
.
2
Identify
the
b
ounda
ry
comp
onents.
Find
the
critical
p
oints
on
each
using
substitution
o
r
Lagrange
multipliers
.
3
Identify
the
endp
oints
(intersections)
of
the
b
ounda
ry
comp
onents.
4
Evaluate
f
(
x
,
y
)
at
all
of
the
ab
ove.
The
minimum
is
the
low
est
numb
er,
the
maximum
is
the
highest.
272
Question
14.8.7
Can
This
Lagrange
Apply
to
Mo
re
Than
One
Constraint?
If
w
e
have
t
w
o
constraints
in
three-space,
g
(
x
,
y
,
z
)
=
c
and
h
(
x
,
y
,
z
)
=
d
,
then
their
intersection
is
generally
a
curve.
Click to Load Applet
Figure:
The
intersection
of
the
constraints
g
(
x
,
y
,
z
)
=
c
and
h
(
x
,
y
,
z
)
=
d
273
Question
14.8.7
Can
This
Lagrange
Apply
to
More
Than
One
Constraint?
Acco
rding
to
our
ea
rlier
a
rgument
ab
out
directional
derivatives,
at
a
maximum
P
on
the
constraint,
∇
f
(
P
)
must
b
e
normal
to
the
constraint.
There
a
re
mo
re
w
a
ys
fo
r
this
to
happ
en
with
t
w
o
constraint
equations.
1
∇
f
(
P
)
could
b
e
parallel
to
∇
g
(
P
).
2
∇
f
(
P
)
could
b
e
parallel
to
∇
h
(
P
).
3
∇
f
(
P
)
could
b
e
the
vector
sum
of
a
vector
pa
rallel
to
∇
g
(
P
)
and
a
vecto
r
pa
rallel
to
∇
h
(
P
).
274
Question
14.8.7
Can
This
Lagrange
Apply
to
More
Than
One
Constraint?
Theo
rem
If
f
(
x
,
y
,
z
)
is
a
differentiable
function
and
g
(
x
,
y
,
z
)
=
c
and
h
(
x
,
y
,
z
)
=
d
a
re
t
w
o
constraints.
If
P
is
a
maximum
of
f
(
x
,
y
,
z
)
among
the
p
oints
that
satisfy
these
constraints
then
either
∇
f
(
P
)
=
λ
∇
g
(
P
)
+
µ
∇
h
(
P
)
fo
r
some
scala
rs
λ
and
µ
,
o
r
∇
g
(
P
)
and
∇
h
(
P
)
a
re
pa
rallel.
This
system
of
equations
is
usually
difficult
to
solve
b
y
hand.
275
Question
14.8.7
Can
This
Lagrange
Apply
to
More
Than
One
Constraint?
Rema
rk
Y
ou
can
check
the
reasonableness
of
this
metho
d
b
y
noting
that
it
gives
us
a
system
of
5
va
riables,
x
,
y
,
z
,
λ
,
µ
,
and
five
equations:
f
x
(
x
,
y
,
z
)
=
λ
g
x
(
x
,
y
,
z
)
+
µ
h
x
(
x
,
y
,
z
)
g
(
x
,
y
,
z
)
=
c
f
y
(
x
,
y
,
z
)
=
λ
g
y
(
x
,
y
,
z
)
+
µ
h
y
(
x
,
y
,
z
)
h
(
x
,
y
,
z
)
=
d
f
z
(
x
,
y
,
z
)
=
λ
g
z
(
x
,
y
,
z
)
+
µ
h
z
(
x
,
y
,
z
)
W
e
therefo
re
generally
exp
ect
this
system
to
have
a
finite
numb
er
of
solutions,
though
there
a
re
plent
y
of
counterexamples
to
this
exp
ectation.
276
Section
14.8
Summa
ry
Questions
Q1
What
is
a
constraint?
Q2
What
equations
do
y
ou
write
when
y
ou
apply
the
metho
d
of
Lagrange
multipliers?
Q3
Is
the
set
of
p
oints
that
satisfies
a
constraint
closed
and
b
ounded?
Explain.
Q4
Ho
w
do
es
a
constraint
a
rise
when
finding
the
maximum
over
a
closed
and
b
ounded
domain?
277
Section
14.8
Q8
Supp
ose
the
curve
b
elo
w
is
the
graph
of
g
(
x
,
y
)
=
k
.
Use
metho
ds
from
calculus
to
find
and
ma
rk
the
app
ro
ximate
lo
cation
of
the
p
oint
that
maximizes
the
function
f
(
x
,
y
)
=
3
y
−
x
subject
to
the
constraint
g
(
x
,
y
)
=
k
.
Justify
your
reasoning
in
a
few
sentences.
278
Section
14.8
Q10
Sho
w
that
(3
,
3)
is
not
a
lo
cal
maximum
of
f
(
x
,
y
)
=
2
x
2
−
4
xy
+
y
2
−
8
x
on
the
graph
x
3
+
y
3
=
6
xy
.
279
Section
14.8
Q18
Consider
the
follo
wing
t
w
o
questions:
Find
the
maximum
value
of
f
(
x
,
y
)
that
satisfies
x
2
+
y
2
≤
9.
Find
the
maximum
value
of
f
(
x
,
y
)
that
satisfies
x
2
+
y
2
=
9.
a
Ho
w
a
re
the
questions
different?
b
Which
question
tak
es
less
w
o
rk
to
solve?
Explain
ho
w
y
ou
kno
w.
c
Do
solutions
exist
to
b
oth
questions?
What
additional
info
rmation
w
ould
gua
rantee
that
they
do?
280
Section
14.8
Q20
Consider
the
function
f
(
x
,
y
)
=
x
2
+
6
xy
+
9
y
2
+
5.
Find
the
maximum
and
minimum
values
of
f
on
the
domain
D
=
{
(
x
,
y
)
:
y
≥
x
,
x
≥
0
,
x
2
+
y
2
≤
10
}
281
Section
15.1
Double
Integrals
Goals:
1
App
ro
ximate
the
volume
under
a
graph
b
y
adding
p
risms.
2
Calculate
the
volume
under
a
graph
using
a
double
integral.
Question
15.1.1
Ho
w
Do
We
App
ro
ximate
the
Volume
Under
z
=
f
(
x
,
y
)?
W
e
app
ro
ximated
the
a
rea
under
the
graph
y
=
f
(
x
)
by
rectangles.
Smaller
rectangles
give
a
b
etter
app
ro
ximation,
and
w
e
defined
the
limit
of
these
app
ro
ximations
to
b
e
the
definite
integral
.
Z
b
a
f
(
x
)
dx
=
lim
∆
x
→
0
n
X
i
=1
f
(
x
∗
i
)∆
x
Click to Load Applet
Figure:
The
area
under
y
=
f
(
x
)
appro
ximated
b
y
rectangles
283
Question
15.1.1
How
Do
We
Appro
ximate
the
Volume
Under
z
=
f
(
x
,
y
)?
A
simila
r
metho
d
app
ro
ximates
the
signed
volume
under
the
graph
z
=
f
(
x
,
y
)
(where
volume
below
the
xy
-plane
counts
as
negative).
We
divide
the
domain
Click to Load Applet
0
≤
x
≤
4
0
≤
y
≤
2
into
sub
rectangles
of
a
rea
A
.
W
e
dra
w
a
p
rism
over
each
rectangle
whose
height
is
the
value
of
the
function
over
some
test
p
oint
(
x
∗
i
,
y
∗
i
).
V
olume
≈
n
X
i
=1
f
(
x
∗
i
,
y
∗
i
)
A
.
284
Question
15.1.1
How
Do
We
Appro
ximate
the
Volume
Under
z
=
f
(
x
,
y
)?
If
our
domain
is
not
a
rectangle,
w
e
ma
y
not
b
e
able
to
divide
it
into
sub
rectangles.
Luckily
,
the
fo
rmula
fo
r
volume
of
a
p
rism
w
o
rks
fo
r
any
shap
e
base.
We
can
still
compute
V
olume
≈
n
X
i
=1
f
(
x
∗
i
,
y
∗
i
)
A
i
.
Click to Load Applet
Figure:
A
domain
sub
divided
into
irregula
r
subregions
285
Question
15.1.1
How
Do
We
Appro
ximate
the
Volume
Under
z
=
f
(
x
,
y
)?
F
o
r
a
reasonably
w
ell-b
ehaved
function
f
(
x
,
y
),
the
actual
volume
can
b
e
computed
b
y
taking
a
limit
of
these
app
ro
ximations.
W
e
call
this
limit
the
double-integral.
Definition
Let
D
be
a
domain
in
R
2
.
F
o
r
a
given
division
of
D
into
n
subregions
denote
A
i
,
the
a
rea
of
the
i
th
region.
(
x
∗
i
,
y
∗
i
),
any
p
oint
in
the
i
th
region
|
A
|
is
the
diameter
of
the
la
rgest
region.
W
e
define
the
double
integral
of
f
(
x
,
y
)
to
be
a
limit
over
all
p
ossible
divisions
of
D
.
Z
Z
D
f
(
x
,
y
)
dA
=
lim
|
A
|→
0
n
X
i
=1
f
(
x
∗
i
,
y
∗
i
)
A
i
286
Example
15.1.2
App
roximating
a
Double
Integral
Consider
Z
Z
D
x
2
ydA
,
where
D
is
the
region
sho
wn
here.
Appro
ximate
the
integral
using
the
division
of
D
sho
wn,
and
evaluating
f
(
x
,
y
)
at
the
midp
oint
of
each
rectangle.
x
y
1
2
1
287
Question
15.1.3
Ho
w
Do
W
e
Evaluate
Double
Integrals?
W
e
already
kno
w
another
w
a
y
of
computing
a
volume.
W
e
can
compute
the
a
rea
of
the
cross
sections
p
erp
endicula
r
to
the
x
-axis.
Let
the
function
A
(
x
)
denote
this
area
at
each
x
.
Then
V
olume
=
Z
b
a
A
(
x
)
dx
A
(
x
)
is
itself
the
area
under
a
curve.
In
a
particula
r
cross
section,
x
is
constant,
and
f
(
x
,
y
)
is
a
function
of
y
.
The
a
rea
b
elo
w
this
graph
is
the
integral
A
(
x
)
=
Z
d
c
f
(
x
,
y
)
dy
W
e
can
put
these
together
to
obtain
an
iterated
integral
,
an
integral
whose
integrand
is
itself
an
integral.
288
Question
15.1.3
How
Do
We
Evaluate
Double
Integrals?
Click to Load Applet
Figure:
Cross
sections
of
the
region
b
elow
the
graph:
z
=
f
(
x
,
y
)
289
Question
15.1.3
How
Do
We
Evaluate
Double
Integrals?
Theo
rem
(Fubini’s
Theorem)
F
o
r
any
domain
D
w
e
have
Z
Z
D
f
(
x
,
y
)
dA
=
Z
b
a
Z
d
c
f
(
x
,
y
)
dy
dx
where
a
and
b
a
re
the
x
b
ounds
of
D
,
and
c
and
d
are
the
y
b
ounds
of
the
cross
section
at
each
x
.
Alternately
,
we
can
write
Z
Z
D
f
(
x
,
y
)
dA
=
Z
d
c
Z
b
a
f
(
x
,
y
)
dx
dy
where
c
and
d
a
re
the
y
b
ounds
of
D
,
and
a
and
b
a
re
the
x
b
ounds
of
the
cross
section
at
each
y
.
290
Example
15.1.4
Using
F
ubini’s
Theo
rem
Compute
Z
Z
D
x
2
y
dA
,
where
D
is
the
region
shown
here:
x
y
1
2
1
291
Question
15.1.5
Can
We
Break
a
Double
Integral
into
a
Product
of
Single
Integrals?
In
general,
w
e
can’t
exp
ect
to
facto
r
out
the
inner
integral
of
R
R
D
f
(
x
,
y
)
dydx
(using
the
constant
multiple
rule).
The
y
-bounds
may
dep
end
on
x
,
and
the
y
terms
may
not
factor
out
of
the
integrand.
Ho
w
ever,
fo
r
certain
functions
and
domains,
this
facto
ring
is
p
ossible.
Theo
rem
Z
b
a
Z
d
c
f
(
x
)
g
(
y
)
dydx
=
Z
b
a
f
(
x
)
dx
Z
d
c
g
(
y
)
dy
W
e
w
on’t
b
e
able
to
use
this
theo
rem
all
the
time.
It
has
t
w
o
imp
o
rtant
requirements:
1
The
b
ounds
of
integration
(
a
,
b
,
c
,
d
)
are
constants.
We’ll
see
integrals
so
on
where
this
is
not
the
case.
2
The
integrand
can
b
e
facto
red
into
a
function
of
x
times
a
function
of
y
.
Most
tw
o-variable
functions
cannot.
292
Example
15.1.6
Integrating
a
Pro
duct
Use
a
p
ro
duct
decomp
osition
to
compute
R
R
D
x
2
ydA
,
where
D
is
the
region
sho
wn
here:
x
y
1
2
1
293
Application
15.1.7
Rates
(p
er
Area)
Single
integrals
can
compute
total
change
given
a
rate
of
change.
meters
traveled
p
er
second
−
→
total
meters
traveled.
GDP
gro
wth
p
er
y
ea
r
−
→
total
GDP
gro
wth.
mass
of
a
chemical
p
ro
duced
p
er
second
−
→
total
mass
p
ro
duced.
294
Application
15.1.7
Rates
(p
er
Area)
Integrating
rainfall
p
er
squa
re
kilometer
gives
the
total
rain
that
fell
in
a
w
atershed.
Figure:
A
rainfall
density
map
295
Application
15.1.7
Rates
(p
er
Area)
Integrating
w
atts
p
er
squa
re
meter
on
a
sola
r
a
rra
y
gives
the
total
energy
generated.
Figure:
Solar
panels
By
Jud
McCranie
-
Own
work,
CC
BY-SA
4.0
https://commons.wikimedia.org/w/index.php?curid=70132767
296
Application
15.1.8
Probabilit
y
If
w
e
generate
a
data
set
in
which
w
e
have
measured
t
w
o
va
riables,
then
the
p
robabilit
y
that
a
random
data
p
oint
lies
in
a
given
region
is
the
double
integral
of
a
joint
densit
y
function
over
that
area.
Figure:
A
highly
correlated
set
of
observations
and
an
uncorrelated
joint
densit
y
function
297
Section
15.1
Summa
ry
Questions
Q1
What
shap
e
do
w
e
use
to
app
ro
ximate
volume
under
a
surface?
Q2
What
fo
rmula
do
w
e
use
to
compute
the
exact
volume
under
a
graph
z
=
f
(
x
,
y
)?
Q3
What
do
es
F
ubini’s
Theo
rem
tell
us?
Q4
What
conditions
do
you
need
in
order
to
write
a
double
integral
as
a
p
ro
duct
of
single
integrals?
298
Section
15.1
Q10
Let
T
b
e
the
triangle
with
vertices
(0
,
0),
(1
,
0)
and
(0
,
2).
Sho
w
ho
w
to
app
ro
ximate
Z
Z
T
e
x
+
y
dA
b
y
dividing
T
into
four
right
triangles
with
legs
of
length
1
and
1
2
.
Use
the
midp
oint
of
the
hyp
otenuses
as
the
test
p
oints.
299
Section
15.1
Q12
Let
R
b
e
the
rectangle
R
=
{
(
x
,
y
)
:
−
2
≤
x
≤
2
,
−
1
≤
y
≤
1
}
.
Let
S
b
e
the
solid
region
ab
ove
R
and
b
elo
w
the
graph
z
=
x
2
y
+
xy
2
.
W
rite
a
function
A
(
x
)
which
gives
the
area
of
the
cross
section
of
S
p
erp
endicula
r
to
the
x
-axis
at
each
value
of
x
.
300
Section
15.2
Double
Integrals
over
General
Regions
Goals:
1
Set
up
double
integrals
over
regions
that
a
re
not
rectangles.
2
Evaluate
integrals
where
the
b
ounds
contain
va
riables.
3
Decide
when
to
mak
e
R
dy
the
outer
integral,
and
compute
the
change
of
b
ounds.
Example
15.2.1
Integrating
Over
a
P
olygon
Let
D
be
the
triangle
with
vertices
(0
,
0),
(4
,
0)
and
(4
,
2).
Calculate
Z
Z
D
4
xy
dA
302
Example
15.2.1
Integrating
Over
a
P
olygon
Let
D
be
the
triangle
with
vertices
(0
,
0),
(4
,
0)
and
(4
,
2).
Calculate
Z
Z
D
4
xy
dA
Click to Load Applet
302
Example
15.2.1
Integrating
Over
a
Polygon
Main
Idea
T
o
find
the
b
ounds
of
a
double
integral
1
Find
the
x
value
where
the
domain
b
egins
and
ends.
These
numb
ers
a
re
the
b
ounds
of
the
outer
integral.
2
Find
the
functions
(of
the
fo
rm
y
=
g
(
x
))
which
define
the
top
and
b
ottom
of
the
domain.
These
functions
are
the
bounds
of
the
inner
integral.
303
Question
15.2.2
What
Are
the
Integral
La
ws
fo
r
Double
Integrals?
Some
single
va
riable
integral
la
ws
apply
to
double
integrals
as
w
ell
(p
rovided
the
integrals
exist).
1
The
sum
rule:
Z
Z
D
f
(
x
,
y
)
+
g
(
x
,
y
)
dA
=
Z
Z
D
f
(
x
,
y
)
dA
+
Z
Z
D
g
(
x
,
y
)
dA
2
The
constant
multiple
rule:
Z
Z
D
cf
(
x
,
y
)
dA
=
c
Z
Z
D
f
(
x
,
y
)
dA
3
If
D
is
the
union
of
tw
o
non-overlapping
sub
domains
D
1
and
D
2
then
Z
Z
D
f
(
x
,
y
)
dA
=
Z
Z
D
1
f
(
x
,
y
)
dA
+
Z
Z
D
2
f
(
x
,
y
)
dA
304
Example
15.2.3
A
Region
Without
a
(Single)
Bottom
Curve
Let
D
b
e
the
region
b
ounded
b
y
y
=
√
x
,
y
=
0
and
y
=
x
−
6.
Calculate
Z
Z
D
(
x
+
y
)
dA
.
305
Example
15.2.3
A
Region
Without
a
(Single)
Bottom
Curve
Let
D
b
e
the
region
b
ounded
b
y
y
=
√
x
,
y
=
0
and
y
=
x
−
6.
Calculate
Z
Z
D
(
x
+
y
)
dA
.
Click to Load Applet
305
Example
15.2.4
Using
Anti-Symmetry
Let
D
be
the
region
x
2
+
y
2
≤
9.
Evaluate
Z
Z
D
3
√
x
p
y
+
3
dA
.
306
Example
15.2.4
Using
Anti-Symmetry
Let
D
be
the
region
x
2
+
y
2
≤
9.
Evaluate
Z
Z
D
3
√
x
p
y
+
3
dA
.
Click to Load Applet
306
Example
15.2.4
Using
Anti-Symmetry
Main
Idea
W
e
can
a
rgue
that
an
integral
Z
Z
D
f
(
x
,
y
)
dA
is
equal
to
zero
when
1
D
is
symmetric
about
some
line
L
.
If
we
folded
it
over
L
,
one
side
of
D
w
ould
lie
exactly
on
the
other
side.
2
f
is
antisymmetric
ab
out
L
.
Fo
r
each
p
oint
(
x
,
y
)
in
D
the
image
of
(
x
,
y
)
across
L
,
denoted
r
L
(
x
,
y
)
has
the
p
rop
erty:
f
(
r
L
(
x
,
y
))
=
−
f
(
x
,
y
)
.
307
Example
15.2.5
Using
Order
to
Manipulate
the
Integrand
Let
D
be
the
triangle
with
vertices
(0
,
0),
(0
,
2)
and
(1
,
2).
Calculate
Z
Z
D
e
(
y
2
)
dA
.
308
Example
15.2.5
Using
Order
to
Manipulate
the
Integrand
Main
Idea
If
w
e
don’t
kno
w
the
anti-derivative
of
an
integrand
with
resp
ect
to
one
va
riable,
try
switching
the
o
rder
of
integration.
Rememb
er
to
change
the
b
ounds
to
o.
309
Application
15.2.6
Area
of
a
Domain
Theo
rem
The
a
rea
of
a
region
D
can
be
calculated:
Z
Z
D
1
dA
.
310
Application
15.2.6
Area
of
a
Domain
Click to Load Applet
Figure:
A
solid
of
height
1
over
a
domain
D
311
Section
15.4
Applications
of
Double
Integrals
Goals:
1
Integrate
a
p
robabilit
y
distribution
to
calculate
a
p
robabilit
y
.
Application
15.4.1
Using
Integrals
to
Compute
Probabilities
Most
p
robabilities
that
p
eople
think
ab
out
a
re
discreet.
A
flipp
ed
coin
has
a
1
2
chance
to
b
e
heads,
1
2
to
b
e
tails.
A
random
M&M
has
a
1
6
chance
to
b
e
red,
1
6
o
range,
1
6
y
ello
w,
1
6
green,
1
6
blue
and
1
6
b
ro
wn.
313
Application
15.4.1
Using
Integrals
to
Compute
Probabilities
On
the
other
hand,
a
p
erson’s
chance
of
b
eing
exactly
68
inches
tall
is
zero.
Even
p
eople
who
sa
y
they
a
re
5
′
8
′′
a
re
slightly
mo
re
o
r
slightly
less.
Instead
w
e
can
ask
what
y
our
chance
is
of
b
eing
b
et
w
een
68
and
69
inches
tall.
Definition
A
function
f
is
a
p
rob
abilit
y
density
function
for
an
event,
if
the
chance
of
an
outcome
b
et
w
een
a
and
b
is
R
b
a
f
(
x
)
dx
.
Click to Load Applet
314
Application
15.4.1
Using
Integrals
to
Compute
Probabilities
Definition
A
function
f
is
a
joint
p
robability
density
function
for
a
pair
random
events
if
the
chance
that
the
outcome
(
x
,
y
)
lies
in
D
is
Z
Z
D
f
(
x
,
y
)
dA
.
315
Application
15.4.1
Using
Integrals
to
Compute
Probabilities
Exercise
Da
rmok
and
Jalad
each
travel
to
the
island
of
T
anagra
and
a
rrive
b
et
w
een
no
on
and
4PM.
Let
(
x
,
y
)
rep
resent
their
resp
ective
a
rrival
times
in
hours
after
no
on.
Supp
ose
the
probabilit
y
that
(
x
,
y
)
falls
in
a
certain
domain
D
which
is
a
subset
of
{
(
x
,
y
)
:
0
≤
x
≤
4
,
0
≤
y
≤
4
}
is
R
R
D
x
32
dydx
.
Calculate
the
p
robabilit
y
that:
1
Da
rmok
a
rrives
after
3PM.
2
Jalad
a
rrives
b
efo
re
1PM.
3
They
b
oth
a
rrive
b
efo
re
2PM.
4
Da
rmok
a
rrives
b
efo
re
Jalad.
5
They
a
rrive
within
an
hour
of
each
other
(set
it
up,
don’t
evaluate).
6
What
do
es
the
distribution
sa
y
ab
out
when
Da
rmok
is
lik
ely
to
a
rrive?
What
ab
out
Jalad?
316
Section
15.2
Summa
ry
Questions
Q1
What
a
re
the
steps
fo
r
writing
a
double
integral
over
a
general
region?
Q2
Ho
w
do
y
ou
decide
whether
dx
or
dy
is
the
inner
variable?
Q3
What
is
antisymmetry
,
and
ho
w
can
w
e
use
it
to
evaluate
integrals?
Q4
Ho
w
can
w
e
use
a
double
integral
to
compute
the
a
rea
of
a
region?
317
Section
15.2
Q8
Let
D
be
the
parallelogram
with
vertices
(0
,
1),
(0
,
4),
(5
,
3)
and
(5
,
6).
Let
f
(
x
,
y
)
b
e
a
continuous
function.
a
Set
up
the
b
ounds
of
integration
of
Z
Z
D
f
(
x
,
y
)
dA
.
b
Could
w
e
save
time
b
y
computing
Z
5
0
Z
4
1
f
(
x
,
y
)
dydx
instead?
Explain.
318
Section
15.2
Q18
Consider
the
integral
Z
6
−
6
Z
0
−
√
36
−
y
2
x
2
dxdy
.
Write
this
integral
in
the
o
rder
dydx
.
319
Section
15.2
Q20
Let
g
(
x
,
y
)
=
x
3
e
y
2
.
Argue
that
Z
4
−
4
Z
3
−
3
g
(
x
,
y
)
dydx
=
0.
320
Section
15.2
Q24
Supp
ose
y
ou
a
re
given
that
f
(
x
,
y
)
=
−
f
(
−
y
,
−
x
).
Over
what
domains
D
can
w
e
argue
b
y
symmetry
that
Z
Z
D
f
(
x
,
y
)
dA
=
0?
Draw
an
example
of
one.
321
Section
15.2
Q32
Consider
the
integral
Z
4
−
4
Z
6
0
x
3
√
y
dydx
a
Sho
w
ho
w
to
app
ro
ximate
the
value
of
this
integral,
dividing
the
domain
into
sub-rectangles
of
length
2
units
and
width
3
units
and
using
the
lo
wer
right
corners
as
test
p
oints.
Y
ou
should
evaluate
any
functions
that
app
ea
r
in
y
our
estimate,
but
y
ou
do
not
need
to
simplify
the
a
rithmetic.
b
Explain
in
a
sentence
o
r
t
w
o
ho
w
y
ou
can
determine
the
exact
value
of
this
integral
without
calculating
any
anti-derivatives.
c
Discuss
what
test
p
oint
y
ou
could
have
pick
ed
in
a
,
such
that
y
our
app
ro
ximation
w
ould
have
computed
the
exact
value
of
the
integral.
Note:
There
a
re
several
relevant
observations
to
make
in
response
to
this
question.
322
Section
15.6
T
riple
Integrals
Goals:
1
Set
up
triple
integrals
over
three-dimensional
domains.
2
Evaluate
triple
integrals.
Question
15.6.1
Ho
w
Do
W
e
Integrate
a
Three-V
a
riable
F
unction?
Definition
Given
a
domain
D
in
three
dimension
space,
and
a
function
f
(
x
,
y
,
z
).
W
e
can
sub
divide
D
into
regions
V
i
is
the
volume
of
the
i
th
region.
(
x
∗
i
,
y
∗
i
,
z
∗
i
)
is
a
p
oint
in
the
i
th
region.
V
is
the
diameter
of
the
la
rgest
region.
W
e
define
the
triple
integral
of
f
over
D
to
b
e
the
following
limit
over
all
p
ossible
divisions
of
D
:
Z
Z
Z
D
f
(
x
,
y
,
z
)
dV
=
lim
V
→
0
n
X
i
=1
f
(
x
∗
i
,
y
∗
i
,
z
∗
i
)
V
i
324
Question
15.6.1
How
Do
We
Integrate
a
Three-Variable
F
unction?
F
ubini’s
theo
rem
applies
to
triple
integrals
as
w
ell.
W
e
write
them
as
iterated
integrals.
Theo
rem
Z
Z
Z
D
f
(
x
,
y
,
z
)
dV
=
Z
x
2
x
1
Z
y
2
y
1
Z
z
2
z
1
f
(
x
,
y
,
z
)
dzdydx
where
z
1
and
z
2
a
re
the
b
ounds
of
z
,
which
may
b
e
functions
of
x
and
y
.
y
1
and
y
2
a
re
the
b
ounds
of
y
,
which
ma
y
b
e
functions
of
x
.
x
1
and
x
2
a
re
the
b
ounds
of
x
.
They
are
numbers.
The
va
riables
of
can
also
b
e
reo
rdered,
with
the
b
ounds
defined
analogously
.
325
Example
15.6.2
Integrating
Over
a
Prism
Let
R
=
{
(
x
,
y
,
z
)
:
0
≤
x
≤
4
,
0
≤
y
≤
2
,
0
≤
z
≤
3
}
.
Compute
Z
Z
Z
R
3
zy
+
x
2
dV
.
326
Example
15.6.2
Integrating
Over
a
Prism
Let
R
=
{
(
x
,
y
,
z
)
:
0
≤
x
≤
4
,
0
≤
y
≤
2
,
0
≤
z
≤
3
}
.
Compute
Z
Z
Z
R
3
zy
+
x
2
dV
.
Click to Load Applet
Figure:
A
Rectangular
Prism
326
Question
15.6.3
Ho
w
Do
W
e
Interp
ret
T
riple
Integrals
Geometrically?
Z
3
0
f
(
x
,
y
,
z
)
dz
computes
the
area
under
the
graph
w
=
f
(
x
,
y
,
z
)
over
each
vertical
segment
of
the
fo
rm
(
x
,
y
)
=
(
x
0
,
y
0
)
in
the
domain.
It
is
a
function
of
x
and
y
.
Click to Load Applet
Figure:
Z
3
0
f
(
x
,
y
,
z
)
dz
,
rep
resented
as
an
area
in
a
zw
-plane
327
Question
15.6.3
How
Do
We
Interpret
T
riple
Integrals
Geometrically?
Z
2
0
Z
3
0
f
(
x
,
y
,
z
)
dzdy
computes
the
volume
under
the
graph
w
=
f
(
x
,
y
,
z
)
over
each
x
=
x
0
cross-section
of
the
domain.
It
is
a
function
of
x
.
Click to Load Applet
Figure:
Z
2
0
Z
3
0
f
(
x
,
y
,
z
)
dzdy
,
represented
as
a
volume
in
yzw
-space
328
Application
15.6.4
T
riple
Integrals
in
Math
and
Science
1
Integrating
a
function
ρ
(
x
,
y
,
z
),
which
gives
the
densit
y
of
an
object
at
each
p
oint,
gives
the
total
mass
of
the
object.
2
Integrating
x
ρ
(
x
,
y
,
z
),
y
ρ
(
x
,
y
,
z
)
and
z
ρ
(
x
,
y
,
z
)
gives
the
center
of
mass
of
the
object.
3
Integrating
a
three-dimensional
p
robabilit
y
distribution
over
a
region
gives
the
p
robabilit
y
that
the
triple
(
X
,
Y
,
Z
)
lies
in
that
region.
4
Integrating
1
dV
over
a
region
gives
the
volume
of
that
region.
329
Application
15.6.4
T
riple
Integrals
in
Math
and
Science
Densit
y
lets
us
visualize
a
triple
integral
without
referring
to
a
fourth
(geometric)
dimension.
Z
3
0
f
(
x
,
y
,
z
)
dz
computes
the
densit
y
of
the
vertical
segments
at
each
(
x
,
y
).
Z
2
0
Z
3
0
f
(
x
,
y
,
z
)
dzdy
computes
the
densit
y
of
the
rectangle
at
each
x
.
Click to Load Applet
Z
4
0
Z
2
0
Z
3
0
f
(
x
,
y
,
z
)
dzdydx
computes
the
total
mass
of
the
p
rism.
330
Example
15.6.5
Integrating
Over
an
Irregula
r
Region
Let
R
b
e
the
region
above
the
xy
plane,
b
elo
w
the
cylinder
x
2
+
z
2
=
16
and
b
et
w
een
y
=
0
and
y
=
3.
Compute
Z
Z
Z
R
4
yz
dV
.
331
Example
15.6.5
Integrating
Over
an
Irregula
r
Region
Let
R
b
e
the
region
above
the
xy
plane,
b
elo
w
the
cylinder
x
2
+
z
2
=
16
and
b
et
w
een
y
=
0
and
y
=
3.
Compute
Z
Z
Z
R
4
yz
dV
.
Click to Load Applet
Figure:
The
region
b
etw
een
x
2
+
y
2
=
16
and
the
xy
-plane
331
Example
15.6.5
Integrating
Over
an
Irregular
Region
Main
Idea
The
follo
wing
app
roach
will
p
ro
duce
the
b
ounds
of
a
region
with
a
top
surface
and
a
b
ottom
surface.
1
The
z
b
ounds
a
re
given
b
y
the
equations
z
=
f
(
x
,
y
)
and
z
=
g
(
x
,
y
)
of
the
top
and
bottom
surface.
2
The
intersection
of
the
top
and
b
ottom
surface
can
produce
relevant
b
ounds
on
x
and
y
.
We
can
graph
these,
along
with
any
given
b
ounds
involving
x
and
y
.
3
After
dra
wing
the
b
ounded
region
in
the
xy
-plane,
the
x
and
y
b
ounds
a
re
computed
as
fo
r
a
double
integral.
Lik
e
with
double
integrals,
w
e
will
w
ant
to
b
reak
the
region
into
smaller
pieces
in
some
cases.
In
other
cases,
w
e
ma
y
want
to
change
the
o
rder
of
integration.
332
Example
15.6.6
A
Solid
Given
b
y
V
ertices
Supp
ose
w
e
w
ant
to
integrate
over
T
,
the
tetrahedron
(pyramid)
with
vertices
(0
,
0
,
0),
(4
,
0
,
0),
(4
,
2
,
0)
and
(4
,
0
,
2).
How
w
ould
we
set
up
the
b
ounds
of
integration?
333
Example
15.6.6
A
Solid
Given
by
V
ertices
Click to Load Applet
Figure:
z
b
ounds
of
T
Click to Load Applet
Figure:
x
,
y
b
ounds
of
T
334
Example
15.6.7
Changing
the
Order
of
Integration
Supp
ose
D
is
the
bounded
region
enclosed
b
etw
een
the
graph
of
y
=
4
x
2
+
z
2
and
the
plane
y
=
4.
Set
up
the
b
ounds
of
the
integral
Z
Z
Z
D
f
(
x
,
y
,
z
)
dV
.
Figure:
A
region
b
ounded
b
y
a
pa
rab
oloid
and
a
plane
335
Example
15.6.7
Changing
the
Order
of
Integration
Supp
ose
D
is
the
bounded
region
enclosed
b
etw
een
the
graph
of
y
=
4
x
2
+
z
2
and
the
plane
y
=
4.
Set
up
the
b
ounds
of
the
integral
Z
Z
Z
D
f
(
x
,
y
,
z
)
dV
.
Click to Load Applet
Figure:
A
region
b
ounded
b
y
a
pa
rab
oloid
and
a
plane
335
Question
15.6.8
When
Do
es
a
T
riple
Integral
Decomp
ose
as
a
Product?
The
p
ro
duct
theo
rem
from
double
integrals
also
w
o
rks
here:
Theo
rem
If
y
1
,
y
2
,
z
1
and
z
2
a
re
constants,
then
Z
x
2
x
1
Z
y
2
y
1
Z
z
2
z
1
f
(
x
)
g
(
y
)
h
(
z
)
dzdydx
=
Z
x
2
x
1
f
(
x
)
dx
Z
y
2
y
1
g
(
y
)
dy
Z
z
2
z
1
h
(
z
)
dz
336
Question
15.6.8
When
Do
es
a
T
riple
Integral
Decomp
ose
as
a
Product?
Example
Along
with
the
sum
and
constant
multiple
rules
w
e
can
simplify
Z
4
0
Z
2
0
Z
3
0
3
zy
+
x
2
dzdydx
to
obtain
the
follo
wing:
Z
4
0
Z
2
0
Z
3
0
3
zy
dzdydx
+
Z
4
0
Z
2
0
Z
3
0
x
2
dzdydx
=3
Z
4
0
dx
Z
2
0
y
dy
Z
3
0
z
dz
+
Z
4
0
x
2
dx
Z
2
0
dy
Z
3
0
dz
=3
·
4
Z
2
0
y
dy
Z
3
0
z
dz
+
2
·
3
Z
4
0
x
2
dx
337
Section
15.6
Summa
ry
Questions
Q1
What
do
es
F
ubini’s
theo
rem
sa
y
ab
out
integrals
with
dV
?
Q2
Ho
w
is
densit
y
used
to
understand
triple
integrals.
Why
w
asn’t
it
necessa
ry
o
r
app
rop
riate
fo
r
double
integrals?
Q3
Ho
w
do
you
find
the
b
ounds
of
the
inner
variable
in
a
triple
integral?
Q4
Ho
w
to
y
ou
find
the
b
ounds
of
the
other
t
w
o
va
riables?
338
Section
15.6
Q26
Let
R
b
e
the
region
enclosed
b
y
y
=
√
25
−
x
2
,
z
=
6
−
y
and
z
=
√
y
.
Set
up
the
b
ounds
of
R
R
R
R
g
(
x
,
y
,
z
)
dV
.
339
Section
15.6
Q26
Click to Load Applet
Figure:
The
region
enclosed
b
y
x
2
+
y
2
=
25
,
z
=
6
−
y
,
and
z
=
√
y
340
Section
15.6
Q20
Cheng
is
integrating
over
R
,
the
region
given
by
x
2
+
y
2
+
z
2
≤
25.
He
gives
the
follo
wing
setup.
Is
this
valid?
Z
√
25
−
y
2
−
z
2
−
√
25
−
y
2
−
z
2
Z
√
25
−
x
2
−
z
2
−
√
25
−
x
2
−
z
2
Z
√
25
−
x
2
−
y
2
−
√
25
−
x
2
−
y
2
f
(
x
,
y
,
z
)
dzdydx
341
Section
15.6
Q20
Click to Load Applet
341
Section
15.6
Q22
Let
R
=
{
(
x
,
y
,
z
)
:
z
≤
2
x
−
y
,
z
≥
0
,
y
≥
x
2
}
.
Compute
Z
Z
Z
R
xz
dV
.
342
Section
15.6
Q36
Rewrite
the
integral
Z
2
0
Z
2
2
−
x
Z
4
−
x
2
0
f
(
x
,
y
,
z
)
dzdydx
as
an
integral
with
the
differential
dxdzdy
.
343
Section
15.6
Q36
Click to Load Applet
344
Section
15.6
Q38
Let
S
=
{
(
x
,
y
,
z
)
:
x
2
+
y
2
+
z
2
≤
25
}
.
Explain
why
Z
Z
Z
S
x
3
y
4
cos
π
z
dV
cannot
b
e
decomp
osed
as
a
product.
345
Section
15.9
Change
of
Va
riables
in
Multiple
Integrals
Goals:
1
Calculate
a
Jacobian
2
Convert
a
multiva
riable
integral
from
one
co
o
rdinate
system
to
another.
Question
15.9.1
Ho
w
Do
es
u
-Substitution
W
o
rk?
In
a
u
-substitution,
we
do
not
just
change
the
va
riable
and
the
b
ounds.
W
e
also
need
to
account
fo
r
width
changing
from
∆
x
to
∆
u
.
Click to Load Applet
Figure:
Z
π
/
2
0
2
sin(2
x
)
dx
=
Z
π
0
2
sin(
u
)
du
.
Instead
w
e
need
to
divide
to
account
fo
r
the
extra
width
of
each
rectangle.
347
Question
15.9.1
How
Does
u
-Substitution
Work?
W
e
cannot
alw
a
ys
just
divide
b
y
a
constant.
The
ratio
of
widths
ma
y
b
e
different
at
different
values
of
x
and
u
.
Click to Load Applet
Figure:
Z
√
π
0
2
sin(
x
2
)
dx
=
Z
π
0
2
sin(
u
)
du
.
The
w
a
y
w
e
solve
this
is
with
a
differential.
W
e
write
du
=
du
dx
dx
.
348
Question
15.9.1
How
Does
u
-Substitution
Work?
The
rule
du
=
du
dx
dx
can
actually
b
e
used
to
our
advantage.
Example
If
u
=
x
2
,
then
du
=
2
x
dx
.
We
can
use
this
to
p
erform
a
substitution
lik
e:
Z
√
π
0
2
x
sin(
x
2
)
dx
=
Z
π
0
sin(
u
)
du
Note
w
e
have
to
change
three
things
1
The
integrand:
sin(
x
2
)
→
sin(
u
)
2
The
b
ounds:
0
→
0
and
√
π
→
π
3
The
differential:
2
x
dx
→
du
Rema
rk
In
single-va
riable
calculus
w
e
use
substitution
to
handle
integrals
with
difficult
integrands.
In
multiva
riable
calculus,
w
e
will
mostly
use
substitution
to
handle
integrals
with
difficult
domains.
349
Question
15.9.2
Ho
w
Do
es
a
Tw
o-Va
riable
Substitution
Wo
rk?
T
o
p
erfo
rm
a
t
w
o-va
riable
substitution,
it
is
helpful
to
have
the
follo
wing
fo
rmula
fo
r
computing
a
rea.
F
o
rmula
Given
vecto
rs
a
=
⟨
a
x
,
a
y
⟩
and
b
=
⟨
b
x
,
b
y
⟩
,
the
pa
rallelogram
fo
rmed
b
y
a
and
b
has
Area
=
det
a
x
a
y
b
x
b
y
=
|
a
x
b
y
−
b
x
a
y
|
350
Question
15.9.2
How
Does
a
Two-V
ariable
Substitution
Wo
rk?
Main
Idea
In
the
case
of
a
linea
r
substitution
lik
e
r
(
u
,
v
)
=
⟨
a
x
u
+
b
x
v
,
a
y
u
+
b
y
v
⟩
a
unit
uv
square
maps
to
a
pa
rallelogram
with
sides
⟨
a
x
,
a
y
⟩
and
⟨
b
x
,
b
y
⟩
in
the
xy
-plane.
We
can
fix
the
a
rea
distortion
p
ro
duced
by
substituting
x
and
y
fo
r
u
and
v
by
setting
dxdy
=
|
a
x
b
y
−
b
x
a
y
|
dudv
.
Question
Ho
w
can
w
e
handle
non-linea
r
substitutions?
351
Example
15.9.3
P
arabolic
Co
o
rdinates
Our
main
source
of
substitutions
a
re
alternative
co
o
rdinate
systems
fo
r
the
plane.
Here
is
some
pa
rab
olic
graph
pap
er.
Each
p
oint
has
co
o
rdinates
(
σ,
τ
).
The
gold
curves
are
σ
=
0
,
1
,
2
,
3
,
.
.
.
.
The
blue
curves
a
re
τ
=
0
,
1
,
2
,
3
,
.
.
.
.
352
Example
15.9.3
Pa
rabolic
Co
ordinates
F
o
rmula
F
o
r
a
given
p
oint
(
σ,
τ
),
w
e
can
calculate
the
corresponding
(
x
,
y
)
co
o
rdinates:
x
=
σ
τ
y
=
1
2
(
τ
2
−
σ
2
)
W
e
can
exp
ress
this
as
a
function
r
(
σ
,
τ
)
=
σ
τ
,
1
2
(
τ
2
−
σ
2
)
.
353
Example
15.9.3
Pa
rabolic
Co
ordinates
Supp
ose
w
e
w
ant
to
integrate
the
function
f
(
x
,
y
)
=
x
2
over
the
domain
b
elo
w
left.
It’s
easier
to
describ
e
this
domain
in
(
σ,
τ
)
co
ordinates.
1
The
b
ounds
of
integration
a
re
2
≤
σ
≤
5,
and
3
≤
τ
≤
5.
2
W
e
can
substitute
the
integrand:
x
2
=
σ
2
τ
2
.
3
But
Z
5
2
Z
5
3
σ
2
τ
2
d
τ
d
σ
computes
the
volume
over
the
rectangle
(b
elo
w
right),
not
over
our
domain,
which
p
rovides
a
la
rger
base.
354
Example
15.9.3
Pa
rabolic
Co
ordinates
When
w
e
tak
e
a
d
σ
by
d
τ
rectangle
in
a
Cartesian
co
ordinate
system,
ho
w
much
bigger
do
es
it
get
when
w
e
map
it
into
the
pa
rab
olic
co
o
rdinate
system?
It
is
too
difficult
to
compute
it
precisely
.
Instead,
we
can
app
ro
ximate
the
effect
of
d
σ
and
d
τ
b
y
differentials.
∂
r
∂
σ
=
∂
x
∂
σ
,
∂
y
∂
σ
∂
r
∂
τ
=
∂
x
∂
τ
,
∂
y
∂
τ
Click to Load Applet
355
Example
15.9.3
Pa
rabolic
Co
ordinates
W
e
no
w
have
the
final
ingredient
to
rewrite
R
R
D
x
2
dA
where
D
is
the
domain
b
elo
w.
356
Question
15.9.4
Ho
w
Do
es
a
Two-V
a
riable
Substitution
Wo
rk
Generally?
Definition
Given
a
t
w
o-va
riable
vecto
r
function
r
(
u
,
v
)
=
⟨
x
(
u
,
v
)
,
y
(
u
,
v
)
⟩
,
J
=
∂
x
∂
u
∂
y
∂
u
∂
x
∂
v
∂
y
∂
v
is
called
the
Jacobian
matrix
.
The
Jacobian
is
the
absolute
value
of
the
determinant
and
is
denoted:
∂
(
x
,
y
)
∂
(
u
,
v
)
=
|
det
J
|
=
∂
x
∂
u
∂
y
∂
v
−
∂
y
∂
u
∂
x
∂
v
W
e
will
define
the
Jacobian
simila
rly
fo
r
a
three
va
riable
vecto
r
function.
357
Question
15.9.4
How
Does
a
Two-V
ariable
Substitution
Wo
rk
Generally?
Main
Idea
When
p
erfo
rming
tw
o-variable
integration,
if
we
want
to
substitute
u
and
v
for
x
and
y
,
then
our
differential
is
replaced
as
follo
ws:
dxdy
=
∂
(
x
,
y
)
∂
(
u
,
v
)
dudv
=
∂
x
∂
u
∂
y
∂
v
−
∂
y
∂
u
∂
x
∂
v
dudv
Rema
rk
P
a
rab
olic
co
o
rdinates
a
re
not
very
useful.
The
kinds
of
regions
they
describ
e
nicely
almost
never
app
ea
r
naturally
.
They
are
a
used
here
only
to
demonstrate
the
theo
ry
of
multiva
riable
substitution.
358
Section
15.3
Double
Integrals
in
P
olar
Co
o
rdinates
Goals:
1
Convert
integrals
from
Ca
rtesian
to
p
ola
r
co
o
rdinates.
2
Evaluate
integrals
in
p
ola
r
co
o
rdinates.
Question
15.3.1
What
Are
Pola
r
Co
ordinates?
Definition
The
p
ola
r
coordinates
of
a
point
are
denoted
(
r
,
θ
)
where
θ
(“theta”)
is
the
direction
to
the
point
from
the
origin
(measured
anticlo
ckwise
from
the
p
ositive
x
axis).
r
is
the
distance
to
the
p
oint
in
that
direction
(negative
r
means
travel
backw
a
rds).
Unlik
e
Ca
rtesian
co
o
rdinates,
a
p
oint
can
b
e
rep
resented
in
several
different
w
a
ys.
(1
,
0)
=
(1
,
2
π
)
=
(1
,
4
π
).
(1
,
0)
=
(
−
1
,
π
)
(0
,
α
)
=
(0
,
β
)
for
all
α,
β
.
360
Question
15.3.1
What
Are
Pola
r
Co
ordinates?
Exercise
Plot
and
lab
el
the
follo
wing
p
oints
and
sets
in
p
ola
r
co
o
rdinates
A
=
(2
,
π
3
)
B
=
(1
.
5
,
3
π
)
C
=
(
−
3
,
−
π
4
)
R
=
{
(
r
,
θ
)
:
0
≤
r
≤
2
}
S
=
{
(
r
,
θ
)
:
π
6
≤
θ
≤
π
4
,
r
≥
1
}
361
Question
15.3.1
What
Are
Pola
r
Co
ordinates?
Ca
rtesian
to
P
ola
r
p
(
r
,
θ
)
=
r
cos(
θ
)
i
+
r
sin(
θ
)
j
x
=
r
cos
θ
y
=
r
sin
θ
Notice:
x
2
+
y
2
=
r
2
r
=
p
x
2
+
y
2
θ
=
(
tan
−
1
y
x
x
>
0
tan
−
1
y
x
+
π
x
<
0
A
full
circle
is
0
≤
θ
≤
2
π
.
362
Question
15.3.2
What
Is
the
Jacobian
of
Pola
r
Co
ordinates?
Calculate
the
Jacobian
∂
(
x
,
y
)
∂
(
r
,
θ
)
such
that
dxdy
=
∂
(
x
,
y
)
∂
(
r
,
θ
)
drd
θ
.
363
Question
15.3.2
What
Is
the
Jacobian
of
Polar
Coordinates?
Main
Idea
The
Jacobian
of
p
ola
r
co
o
rdinates
is
r
.
Thus
dydx
=
rdrd
θ
364
Example
15.3.3
Integrating
Over
a
Disc
Let
D
be
the
disk:
x
2
+
y
2
≤
9.
Calculate
Z
Z
D
p
x
2
+
y
2
dA
.
365
Example
15.3.3
Integrating
Over
a
Disc
Let
D
be
the
disk:
x
2
+
y
2
≤
9.
Calculate
Z
Z
D
p
x
2
+
y
2
dA
.
Click to Load Applet
365
Example
15.3.4
Integrating
Over
a
W
edge
Let
D
=
{
(
x
,
y
)
:
x
≥
0
,
x
≤
y
,
x
2
+
y
2
≤
2
}
.
Sketch
D
and
calculate
Z
Z
D
x
2
dA
.
366
Example
15.3.4
../im
gicons/teacher.pdf
Integrating
Over
a
Wedge
T
rig
F
o
rmulas
Higher
p
o
w
ers
of
sine
and
cosine
a
rise
naturally
in
p
ola
r
integrals.
Y
ou’ll
b
e
resp
onsible
fo
r
applying
the
follo
wing
fo
rmulas.
F
o
rmulas
sin
2
θ
=
1
2
−
cos(2
θ
)
2
cos
2
θ
=
1
2
+
cos(2
θ
)
2
sin
3
θ
=
sin
θ
−
cos
2
θ
sin
θ
cos
3
θ
=
cos
θ
−
sin
2
θ
cos
θ
367
Example
15.3.4
Integrating
Over
a
Wedge
Exercise
F
o
r
each
of
the
integrals
b
elo
w,
sk
etch
the
domain
of
integration
then
convert
to
p
ola
r.
Y
ou
need
not
evaluate.
1
Z
Z
D
2
x
−
3
y
2
dydx
where
D
=
{
(
x
,
y
)
:
x
2
+
y
2
≤
16
,
−
y
≤
x
≤
y
}
2
Z
Z
D
x
2
ydydx
where
D
=
{
(
x
,
y
)
:
4
≤
x
2
+
y
2
≤
9
,
y
≤
0
}
3
Z
3
−
3
Z
√
9
−
y
2
0
x
2
+
y
2
dxdy
Which
of
y
our
integrals
can
b
e
solved
using
the
p
ro
duct
fo
rmula?
368
Example
15.3.5
A
Circle
Through
the
Origin
Let
D
be
the
domain
(
x
−
1)
2
+
y
2
≤
1.
Evaluate
Z
Z
D
x
2
+
y
2
dA
.
Click to Load Applet
369
Example
15.3.6
P
olar
Co
o
rdinates
in
T
riple
Integrals
Set
up
the
integral
fo
r
f
(
x
,
y
,
z
)
over
the
region
R
enclosed
b
etw
een
the
graphs
z
=
x
2
+
y
2
and
z
=
p
6
−
x
2
−
y
2
.
370
Example
15.3.6
P
olar
Co
o
rdinates
in
T
riple
Integrals
Set
up
the
integral
fo
r
f
(
x
,
y
,
z
)
over
the
region
R
enclosed
b
etw
een
the
graphs
z
=
x
2
+
y
2
and
z
=
p
6
−
x
2
−
y
2
.
Click to Load Applet
370
Example
15.3.6
Pola
r
Coordinates
in
T
riple
Integrals
Main
Idea
When
setting
up
a
triple
integral,
sometimes
the
domain
of
the
outer
t
w
o
va
riables
(usually
x
and
y
)
is
mo
re
conveniently
written
in
p
ola
r
co
o
rdinates.
Rema
rk
The
co
o
rdinate
system
(
r
,
θ
,
z
)
is
called
the
cylindrical
co
o
rdinate
system.
371
Section
15.8
T
riple
Integrals
in
Spherical
Co
o
rdinates
Goals:
1
W
rite
integrals
in
spherical
co
o
rdinates
Question
15.8.1
What
Are
Spherical
Co
o
rdinates?
Spherical
co
o
rdinates
a
re
a
three
dimensional
co
o
rdinate
system.
Here
ρ
(“rho”)
is
the
(three
dimensional)
distance
from
the
o
rigin.
ϕ
(“phi”)
is
the
angle
the
segment
from
the
o
rigin
mak
es
with
the
p
ositive
z
axis.
θ
is
the
angle
that
the
p
rojection
to
the
xy
-plane
mak
es
with
the
p
ositive
x
-axis.
Click to Load Applet
373
Question
15.8.1
What
Are
Spherical
Coordinates?
The
follo
wing
fo
rmulas
follo
w
from
trigonometry
.
Ca
rtesian
to
Spherical
x
=
ρ
cos
θ
sin
ϕ
y
=
ρ
sin
θ
sin
ϕ
z
=
ρ
cos
ϕ
Notice:
x
2
+
y
2
+
z
2
=
ρ
2
A
full
sphere
is
0
≤
θ
≤
2
π
0
≤
ϕ
≤
π
Click to Load Applet
374
Question
15.8.1
What
Are
Spherical
Co
ordinates?
Exercise
Describ
e
(o
r
dra
w?)
the
follo
wing
regions
in
spherical
co
ordinates.
1
R
=
{
(
ρ,
θ
,
ϕ
)
:
ϕ
=
π
2
}
2
R
=
{
(
ρ,
θ
,
ϕ
)
:
ρ
≤
5
}
3
R
=
{
(
ρ,
θ
,
ϕ
)
:
0
≤
θ
≤
π
4
}
4
R
=
{
(
ρ,
θ
,
ϕ
)
:
ϕ
≥
2
π
3
}
375
Question
15.8.1
What
Are
Spherical
Co
ordinates?
Theo
rem
The
Jacobian
fo
r
spherical
co
o
rdinates
is
ρ
2
sin
ϕ.
376
Example
15.8.2
The
V
olume
of
a
Sphere
Calculate
the
volume
of
a
sphere
of
radius
R
.
377
Example
15.8.3
Converting
to
Spherical
Co
o
rdinates
Convert
the
follo
wing
triple
integral
to
spherical
co
o
rdinates:
Z
3
0
Z
0
−
√
9
−
x
2
Z
√
9
−
x
2
−
y
2
0
yz
2
dzdydx
378
Example
15.8.3
Converting
to
Spherical
Co
o
rdinates
Convert
the
follo
wing
triple
integral
to
spherical
co
o
rdinates:
Z
3
0
Z
0
−
√
9
−
x
2
Z
√
9
−
x
2
−
y
2
0
yz
2
dzdydx
Click to Load Applet
378
Question
15.8.4
When
Do
We
Use
Spherical
Co
o
rdinates?
Spherical
co
o
rdinates
a
re
only
w
o
rth
using
if
the
domain
is
reasonably
w
ell
b
ehaved.
1
In
many
cases,
all
the
b
ounds
of
integration
a
re
constants.
2
The
b
ounds
of
ρ
involve
the
exp
ression
x
2
+
y
2
+
z
2
.
3
The
b
ounds
of
θ
a
re
given
by
inequalities
containing
only
x
and
y
.
Dra
w
these
in
the
plane.
4
The
b
ounds
of
ϕ
a
re
given
b
y
inequalities
concerning
z
.
5
In
some
mo
re
advanced
applications,
the
ρ
b
ounds
ma
y
b
e
a
function
of
ϕ
o
r
θ
,
meaning
ρ
should
b
e
the
inner
va
riable.
379
Question
15.8.4
When
Do
We
Use
Spherical
Co
ordinates?
Exercise
Set
up
the
integrals
of
g
(
x
,
y
,
z
)
over
the
following
regions
using
spherical
co
o
rdinates.
1
The
intersection
of
x
2
+
y
2
+
z
2
≤
4
and
z
≤
0.
2
The
intersection
of
the
sphere
x
2
+
y
2
+
z
2
≤
1
and
the
half-spaces
x
≥
0
and
y
≤
x
.
3
The
intersection
of
the
cone
z
≥
p
x
2
+
y
2
and
the
sphere
x
2
+
y
2
+
z
2
≤
9.
380
Section
16.1
Line
Integrals
Goals:
1
Compute
line
integrals
of
multi
va
riable
functions.
2
Compute
line
integrals
of
vecto
r
functions.
3
Interp
ret
the
physical
implications
of
a
line
integral.
Question
16.1.1
What
Is
a
Line
Integral?
W
e
have
integrated
a
function
over
The
real
numb
er
line
R
b
a
f
(
x
)
dx
The
plane
R
R
D
f
(
x
,
y
)
dA
Three
space
R
R
R
R
f
(
x
,
y
,
z
)
dV
382
Question
16.1.1
What
Is
a
Line
Integral?
Given
a
curve
C
in
the
domain
of
a
multivariable
function
f
,
the
integral
Z
C
f
ds
computes
the
(signed)
a
rea
under
the
graph
of
f
and
over
the
curve
C
.
Click to Load Applet
Figure:
A
tw
o-variable
function
f
(
x
,
y
)
over
a
plane
curve
r
(
t
)
383
Question
16.1.1
What
Is
a
Line
Integral?
W
e
can
use
rectangles
to
app
ro
ximate
this
a
rea.
Click to Load Applet
384
Question
16.1.1
What
Is
a
Line
Integral?
Area
≈
X
f
(
x
∗
i
,
y
∗
i
)
p
∆
x
2
+
∆
y
2
=
X
f
(
x
(
t
∗
i
)
,
y
(
t
∗
i
))
s
∆
x
∆
t
2
+
∆
y
∆
t
2
∆
t
∆
t
→
0
−
−
−
−
→
Z
f
(
x
(
t
)
,
y
(
t
))
s
dx
dt
2
+
dy
dt
2
dt
Alternately:
Z
f
(
r
(
t
))
|
r
′
(
t
)
|
dt
385
Question
16.1.1
What
Is
a
Line
Integral?
W
e
can
also
integrate
with
resp
ect
to
change
in
just
x
or
just
y
.
X
f
(
x
∗
i
,
y
∗
i
)∆
x
=
X
f
(
x
(
t
∗
i
)
,
y
(
t
∗
i
))
∆
x
∆
t
∆
t
∆
t
→
0
−
−
−
−
→
Z
f
(
x
(
t
)
,
y
(
t
))
x
′
(
t
)
dt
Figure:
The
projection
of
the
area
under
f
(
r
(
t
))
into
the
xz
-plane
386
Question
16.1.1
What
Is
a
Line
Integral?
W
e
defined
R
C
fds
as
an
a
rea.
It
can
also
b
e
useful
for
integrating
any
function
that
is
a
rate
with
resp
ect
to
distance:
Example
Over
va
ried
terrain,
if
p
(
x
,
y
)
gives
the
p
rice
p
er
mile
to
build
railroad
tracks
at
p
oint
(
x
,
y
),
then
R
C
p
(
x
,
y
)
ds
gives
the
total
cost
to
construct
a
railroad
follo
wing
C
.
Example
Over
va
ried
terrain,
if
f
(
x
,
y
)
gives
the
fuel
consumption
p
er
mile
traveled
at
the
p
oint
(
x
,
y
),
then
R
C
f
(
x
,
y
)
ds
gives
the
total
fuel
consumption
to
travel
along
C
.
387
Example
16.1.2
A
Line
Integral
Let
C
b
e
the
line
segment
from
(0
,
0)
to
(3
,
4).
Let
f
(
x
,
y
)
=
x
2
+
cos(
π
y
).
Compute
the
line
integral
Z
C
f
(
x
,
y
)
ds
.
388
Example
16.1.2
A
Line
Integral
Let
C
b
e
the
line
segment
from
(0
,
0)
to
(3
,
4).
Let
f
(
x
,
y
)
=
x
2
+
cos(
π
y
).
Compute
the
line
integral
Z
C
f
(
x
,
y
)
ds
.
Click to Load Applet
388
Example
16.1.2
A
Line
Integral
Main
Idea
Y
ou’ll
need
to
kno
w
the
follo
wing
pa
rametrizations
from
Chapter
13
A
line
segment
from
A
to
B
A
circle
of
radius
a
The
graph
of
an
explicit
function
y
=
f
(
x
)
389
Example
16.1.2
A
Line
Integral
Exercise
Consider
t
w
o
curves
defined
b
y
vecto
r
functions:
C
1
:
r
1
(
t
)
=
⟨
5
cos(
t
)
,
5
sin(
t
)
⟩
0
≤
t
≤
2
π
C
2
:
r
2
(
t
)
=
⟨
5
cos(2
π
t
)
,
5
sin(2
π
t
)
⟩
0
≤
t
≤
1
a
Ho
w
a
re
these
curves
related
to
each
other?
What
shap
es
do
they
mak
e?
b
Find
a
pa
rtner.
Each
of
y
ou
should
set
up
one
of
the
follo
wing
line
integrals.
Z
C
1
x
4
−
y
2
ds
Z
C
2
x
4
−
y
2
ds
c
Ho
w
a
re
y
our
line
integrals
related
to
each
other?
Is
there
a
rule
of
calculus
that
seems
to
b
e
applied
here?
390
Application
16.1.3
Arc
Length
What
do
es
integrating
R
C
1
ds
compute?
Click to Load Applet
F
o
rmula
Z
C
1
ds
=
a
rc
length
×
height
=
a
rc
length
391
Application
16.1.3
Arc
Length
What
do
es
integrating
R
C
1
ds
compute?
Click to Load Applet
F
o
rmula
Z
C
1
ds
=
a
rc
length
×
height
=
arc
length
391
Application
16.1.3
Arc
Length
Calculate
the
a
rc
length
of
r
(
t
)
=
(
t
2
−
t
)
i
+
2
3
(2
t
)
3
/
2
j
on
the
interval
0
≤
t
≤
4.
392
Application
16.1.3
Arc
Length
Calculate
the
a
rc
length
of
r
(
t
)
=
(
t
2
−
t
)
i
+
2
3
(2
t
)
3
/
2
j
on
the
interval
0
≤
t
≤
4.
392
Application
16.1.3
../im
gicons/rocket.pdf
Arc
Length
Summa
ry
Questions
Why
do
w
e
convert
to
a
different
differential
when
setting
up
a
line
integral?
What
do
es
ds
mean?
What
is
its
differential
in
terms
of
dt
?
Ho
w
do
w
e
compute
a
rc
length?
393
Section
16.1
V
ector
Fields
Goals:
1
Recognize
real
w
o
rld
phenomena
that
a
re
mo
deled
b
y
vecto
r
fields.
2
Determine
the
geometric
b
ehavior
of
a
vecto
r
field
from
its
equation.
3
Compute
line
integrals
of
a
vecto
r
field
over
a
curve.
Question
16.1.1
What
Is
a
V
ector
Field?
Definition
A
vecto
r
field
in
R
2
is
a
function
that
assigns
a
t
w
o-dimensional
vecto
r
F
(
x
,
y
)
to
each
p
oint
in
R
2
.
A
vecto
r
field
in
R
3
is
a
function
that
assigns
a
three-dimensional
vecto
r
F
(
x
,
y
,
z
)
to
each
point
in
R
3
.
395
Question
16.1.1
What
Is
a
Vecto
r
Field?
W
e
dra
w
a
vecto
r
field
b
y
attaching
the
vecto
rs
F
(
x
,
y
)
to
the
p
oints
(
x
,
y
)
by
the
tail.
Fo
r
obvious
reasons,
w
e
only
draw
these
vecto
rs
from
a
finite
set
of
p
oints.
396
Question
16.1.1
What
Is
a
Vecto
r
Field?
A
vecto
r
field
is
defined
b
y
comp
onent
functions
P
and
Q
(and
R
in
three
dimensions)
F
(
x
,
y
)
=
P
(
x
,
y
)
i
+
Q
(
x
,
y
)
j
o
r
F
(
x
,
y
)
=
⟨
P
(
x
,
y
)
,
Q
(
x
,
y
)
⟩
.
Click to Load Applet
F
(
x
,
y
,
z
)
=
(0
.
2
x
+
0
.
04
y
)
i
+
(0
.
03
z
−
0
.
1)
j
+
0
.
2
sin(
xz
)
k
397
Question
16.1.1
What
Is
a
Vecto
r
Field?
The
follo
wing
a
re
examples
of
vecto
r
fields:
Wind
sp
eed
at
each
p
oint
on
the
ground.
The
fo
rce
exerted
b
y
gravit
y
(o
r
magnetism
o
r
cha
rge)
at
each
p
oint
in
space.
The
gradient
of
a
differentiable
function.
398
Example
16.1.2
Sk
etching
a
V
ecto
r
Field
Sk
etch
the
t
w
o
dimensional
vecto
r
field
F
(
x
,
y
)
=
y
2
i
−
1
2
j
.
399
Example
16.1.2
Sketching
a
Vecto
r
Field
Main
Idea
W
e
can
alw
a
ys
sk
etch
an
unfamilia
r
vecto
r
field
p
oint
b
y
p
oint
to
get
a
visualization
of
it.
400
Example
16.1.3
Sk
etching
a
V
ecto
r
Field
Ho
w
can
w
e
visualize
the
vecto
r
field
F
(
x
,
y
)
=
x
i
+
y
j
p
x
2
+
y
2
?
401
Example
16.1.3
Sketching
a
Vecto
r
Field
Main
Idea
W
e
can
often
visualize
a
vecto
r
field
b
y
compa
ring
it
to
the
p
osition
vecto
r
at
each
p
oint.
402
Example
16.1.4
Sk
etching
a
V
ecto
r
Field
Ho
w
can
w
e
visualize
the
vecto
r
field
F
(
x
,
y
)
=
−
y
i
+
x
j
?
403
Example
16.1.4
Sketching
a
Vecto
r
Field
404
Question
16.2.1
Ho
w
Do
We
Measure
the
Wo
rk
Done
b
y
a
Vecto
r
Field?
The
dot
p
ro
duct
measures
the
angle
of
t
w
o
vecto
rs,
as
w
ell
as
their
magnitude.
F
·
s
=
|
F
||
s
|
cos
θ
This
mo
dels
the
w
o
rk
done
b
y
a
fo
rce
F
on
a
displacement
s
,
since
only
F
proj
contributes
to
w
o
rk.
W
=
F
proj
·
s
=
F
·
s
Click to Load Applet
405
Question
16.2.1
How
Do
We
Measure
the
Work
Done
by
a
Vecto
r
Field?
The
fo
rmula
W
=
F
·
s
assumes
that
F
is
constant,
and
the
displacement
s
is
along
a
straight
line.
A
vector
field
introduces
the
p
ossibilit
y
that
F
is
different
at
different
p
oints.
T
o
compute
the
wo
rk
done
b
y
a
vecto
r
field,
w
e
use
an
integral.
Definition
The
line
integral
of
the
vecto
r
field
F
(
x
,
y
)
over
the
vector
function
r
(
t
)
is
defined:
Z
C
F
·
d
r
=
Z
F
(
r
(
t
))
·
r
′
(
t
)
dt
This
is
sometimes
called
a
w
o
rk
integral
.
W
e’ll
see
later
that
line
integral
of
a
vecto
r
field
has
applications
b
ey
ond
physics.
406
Question
16.2.1
How
Do
We
Measure
the
Work
Done
by
a
Vecto
r
Field?
If
F
(
x
,
y
)
=
P
(
x
,
y
)
i
+
Q
(
x
,
y
)
j
w
e
can
rewrite
this
integral
without
a
vecto
r
op
eration:
Z
C
F
(
x
,
y
)
·
d
r
=
Z
C
⟨
P
,
Q
⟩
·
x
′
(
t
)
,
y
′
(
t
)
dt
=
Z
C
Px
′
(
t
)
dt
+
Qy
′
(
t
)
dt
=
Z
C
Pdx
+
Qdy
Alternate
Notation
The
line
integral
of
a
vecto
r
field
can
also
b
e
written:
Z
C
P
(
x
,
y
)
dx
+
Z
C
Q
(
x
,
y
)
dy
.
407
Question
16.2.1
How
Do
We
Measure
the
Work
Done
by
a
Vecto
r
Field?
Supp
ose
an
object
travels
once
anticlo
ckwise
a
round
the
unit
circle
and
is
acted
on
b
y
a
fo
rce
field
F
(
x
,
y
)
=
y
2
i
−
1
2
j
.
Do
es
F
do
p
ositive
or
negative
w
o
rk
on
the
object?
Calculate
the
total
w
o
rk
done.
Click to Load Applet
408
Question
16.2.2
Do
es
Choice
of
P
a
rameterization
Matter?
No.
Only
the
path
tak
en
matters.
Theo
rem
If
C
1
and
C
2
a
re
t
w
o
pa
rameterizations
of
the
same
curve
then
Z
C
1
F
(
x
,
y
)
d
r
=
Z
C
2
F
(
x
,
y
)
d
r
T
o
p
rove
this,
let
C
1
b
e
given
b
y
r
1
(
t
).
Then
there
is
some
function
u
(
t
)
such
that
C
2
is
given
b
y
r
2
(
t
)
=
r
1
(
u
(
t
)).
The
proof
is
a
u
-substitution.
This
mak
es
physical
sense,
b
ecause
w
o
rk
do
es
not
ca
re
ab
out
sp
eed,
only
displacement.
Thus
traveling
mo
re
quickly
o
r
slo
wly
along
a
curve
will
not
change
the
total
w
o
rk
done.
409
Question
16.2.2
Does
Choice
of
Parameterization
Matter?
A
Summa
ry
of
Line
Integrals
Notation
Geometric
Interp
retation
Calculation
Z
C
f
(
x
,
y
)
ds
Area
b
elow
the
graph
Z
b
a
f
(
r
(
t
))
|
r
′
(
t
)
|
dt
or
Z
b
a
f
(
x
(
t
)
,
y
(
t
))
q
(
x
′
(
t
))
2
+
(
y
′
(
t
))
2
dt
Z
C
f
(
x
,
y
)
dx
Area
when
projected
onto
xz
plane
Z
b
a
f
(
r
(
t
))
x
′
(
t
)
dt
Z
C
F
(
x
,
y
)
·
d
r
W
ork
done
by
F
Z
b
a
F
(
r
(
t
))
·
r
′
(
t
)
dt
or
Z
b
a
P
(
x
(
t
)
,
y
(
t
))
x
′
(
t
)
+
Q
(
x
(
t
)
,
y
(
t
))
y
′
(
t
)
dt
410
Question
16.2.2
../im
gicons/qm.pdf
Does
Choice
of
Parameterization
Matter?
Summa
ry
Questions
Ho
w
do
w
e
rep
resent
a
vecto
r
field,
graphically?
What
do
es
a
line
integral
of
a
vecto
r
field
measure?
Ho
w
can
w
e
see
whether
a
vecto
r
field
is
doing
p
ositive
o
r
negative
w
o
rk
on
a
path?
What
do
es
d
r
mean?
What
is
its
differential
in
terms
of
dt
?
411
Section
16.3
The
Fundamental
Theo
rem
fo
r
Line
Integrals
Goals:
1
Use
the
fundamental
theo
rem
to
evaluate
line
integrals
of
conservative
vecto
r
fields.
2
Determine
when
a
vecto
r
field
is
conservative.
Question
16.3.1
Do
es
the
Path
of
a
Curve
Matter?
W
e
asserted
p
reviously
that
t
w
o
pa
rameterizations
of
the
same
curve
o
r
vecto
r
function
yield
equal
line
integrals.
Ho
w
ever,
changing
the
course
of
the
curve
will
usually
change
the
value
of
the
integral,
even
if
the
sta
rting
and
ending
p
oints
a
re
left
the
same.
Click to Load Applet
413
Question
16.3.1
Do
es
the
Path
of
a
Curve
Matter?
Exercise
Consider
the
vecto
r
field
F
(
x
,
y
)
=
∇
f
(
x
,
y
),
where
f
(
x
,
y
)
=
x
2
+
y
2
4
.
If
A
=
(
−
4
,
0)
and
B
=
(4
,
0),
what
is
the
w
o
rk
done
by
F
traveling
from
A
to
B
along:
1
A
line
segment?
2
A
semicircle
of
radius
4?
414
Question
16.3.1
Do
es
the
Path
of
a
Curve
Matter?
Exercise
Consider
the
vecto
r
field
F
(
x
,
y
)
=
∇
f
(
x
,
y
),
where
f
(
x
,
y
)
=
x
2
+
y
2
4
.
If
A
=
(
−
4
,
0)
and
B
=
(4
,
0),
what
is
the
w
o
rk
done
by
F
traveling
from
A
to
B
along:
1
A
line
segment?
2
A
semicircle
of
radius
4?
Click to Load Applet
414
Question
16.3.2
What
Is
the
F
undamental
Theo
rem
of
Line
Integrals
Gradient
fields
have
the
follo
wing
property
.
Theo
rem
(The
F
undamental
Theo
rem
of
Line
Integrals)
If
F
=
∇
f
fo
r
some
function
f
,
and
C
travels
from
p
oint
A
to
p
oint
B
,
then
Z
C
F
·
d
r
=
f
(
B
)
−
f
(
A
)
415
Question
16.3.2
What
Is
the
Fundamental
Theo
rem
of
Line
Integrals
The
p
ro
of
follo
ws
from
the
multiva
riable
chain
rule.
Here
is
the
version
fo
r
a
t
w
o-va
riable
function.
F
=
∇
f
=
∂
f
∂
x
i
+
∂
f
∂
y
j
d
r
=
r
′
(
t
)
dt
=
dx
dt
i
+
dy
dt
j
dt
Z
C
F
·
d
r
=
Z
b
a
∂
f
∂
x
dx
dt
+
∂
f
∂
y
dy
dt
dt
=
Z
b
a
d
dt
f
(
r
(
t
))
dt
=
f
(
r
(
b
))
−
f
(
r
(
a
))
=
f
(
B
)
−
f
(
A
)
416
Question
16.3.2
What
Is
the
Fundamental
Theo
rem
of
Line
Integrals
The
p
ro
of
follo
ws
from
the
multiva
riable
chain
rule.
Here
is
the
version
fo
r
a
t
w
o-va
riable
function.
F
=
∇
f
=
∂
f
∂
x
i
+
∂
f
∂
y
j
d
r
=
r
′
(
t
)
dt
=
dx
dt
i
+
dy
dt
j
dt
Z
C
F
·
d
r
=
Z
b
a
∂
f
∂
x
dx
dt
+
∂
f
∂
y
dy
dt
dt
=
Z
b
a
d
dt
f
(
r
(
t
))
dt
=
f
(
r
(
b
))
−
f
(
r
(
a
))
=
f
(
B
)
−
f
(
A
)
416
Question
16.3.2
What
Is
the
Fundamental
Theo
rem
of
Line
Integrals
The
p
ro
of
follo
ws
from
the
multiva
riable
chain
rule.
Here
is
the
version
fo
r
a
t
w
o-va
riable
function.
F
=
∇
f
=
∂
f
∂
x
i
+
∂
f
∂
y
j
d
r
=
r
′
(
t
)
dt
=
dx
dt
i
+
dy
dt
j
dt
Z
C
F
·
d
r
=
Z
b
a
∂
f
∂
x
dx
dt
+
∂
f
∂
y
dy
dt
dt
=
Z
b
a
d
dt
f
(
r
(
t
))
dt
=
f
(
r
(
b
))
−
f
(
r
(
a
))
=
f
(
B
)
−
f
(
A
)
416
Question
16.3.2
What
Is
the
Fundamental
Theo
rem
of
Line
Integrals
The
p
ro
of
follo
ws
from
the
multiva
riable
chain
rule.
Here
is
the
version
fo
r
a
t
w
o-va
riable
function.
F
=
∇
f
=
∂
f
∂
x
i
+
∂
f
∂
y
j
d
r
=
r
′
(
t
)
dt
=
dx
dt
i
+
dy
dt
j
dt
Z
C
F
·
d
r
=
Z
b
a
∂
f
∂
x
dx
dt
+
∂
f
∂
y
dy
dt
dt
=
Z
b
a
d
dt
f
(
r
(
t
))
dt
=
f
(
r
(
b
))
−
f
(
r
(
a
))
=
f
(
B
)
−
f
(
A
)
416
Question
16.3.2
What
Is
the
Fundamental
Theo
rem
of
Line
Integrals
By
our
p
revious
calculation,
F
·
r
′
(
t
)
computes
the
rate
of
change
of
f
(
r
(
t
))
with
respect
to
t
.
This
can
b
e
realized
as
the
change
in
height
of
the
graph
z
=
f
(
x
,
y
).
Click to Load Applet
417
Question
16.3.3
What
is
a
Conservative
V
ecto
r
Field?
Definition
A
vecto
r
field
F
is
conservative
if
line
integrals
dep
end
only
the
endp
oints
of
the
curve.
In
other
w
ords,
whenever
C
1
and
C
2
have
the
same
sta
rting
and
ending
p
oints,
Z
C
1
F
·
d
r
=
Z
C
2
F
·
d
r
.
Example
By
the
F
undamental
Theo
rem
fo
r
Line
Integrals,
every
gradient
field
is
conservative.
But
a
re
any
other
vecto
r
fields
conservative?
418
Question
16.3.4
Ho
w
Do
We
Detect
Whether
a
Vecto
r
Field
is
Conservative?
Theo
rem
F
o
r
a
vecto
r
field
F
=
P
i
+
Q
j
on
a
simply
connected
(no
holes)
domain,
the
follo
wing
a
re
equivalent
(if
one
is
true,
the
others
a
re
true).
1
F
is
conservative.
2
F
=
∇
f
fo
r
some
function
f
.
3
R
C
F
·
d
r
=
0
fo
r
all
closed
curves
(start
and
end
at
same
pt).
4
P
y
=
Q
x
The
function
f
is
sometimes
called
a
p
otential
function
fo
r
F
.
419
Question
16.3.4
How
Do
We
Detect
Whether
a
Vecto
r
Field
is
Conservative?
Here’s
an
outline
of
ho
w
w
e’d
p
rove
the
theo
rem.
F
conservative
F
=
∇
f
Closed
curves
integrate
to
0
P
y
=
Q
x
define
f
(
x
,
y
)
=
Z
(
x
,
y
)
(0
,
0)
F
·
d
r
FTLI
compa
re
to
constant
curve
combine
C
1
and
a
backwa
rd
C
2
f
xy
=
f
yx
ha
rd
420
Question
16.3.4
How
Do
We
Detect
Whether
a
Vecto
r
Field
is
Conservative?
Example
There
a
re
a
couple
cases
of
conservative
fields
that
a
re
easy
to
recognize:
A
constant
field
F
(
x
,
y
)
=
a
i
+
b
j
.
A
sum
of
the
fo
rm
F
(
x
,
y
)
=
P
(
x
)
i
+
Q
(
y
)
j
.
What
is
the
p
otential
function
fo
r
each
of
these?
421
Question
16.3.4
How
Do
W
e
Detect
Whether
a
Vector
Field
is
Conservative?
Exercise
Supp
ose
w
e
have
a
function
f
(
x
,
y
)
such
that
f
x
(
x
,
y
)
=
3
x
2
−
2
xy
.
a
Find
three
different
p
ossible
exp
ressions
fo
r
f
.
b
Compa
re
y
our
exp
ressions
with
someone
nea
r
y
ou.
Can
y
ou
p
ro
duce
an
entire
family
of
p
ossible
f
s?
c
Do
es
one
memb
er
of
y
ou
family
have
the
pa
rtial
derivative
f
y
=
cos
y
−
x
2
?
If
not,
should
y
ou
expand
y
our
family?
Ho
w?
422
Example
16.3.5
Appliying
the
Fundamental
Theo
rem
for
Line
Integrals
a
Is
F
=
(3
x
2
−
2
xy
)
i
+
(cos
y
−
x
2
)
j
conservative?
b
What
is
its
p
otential
function?
c
If
C
is
a
path
from
(1
,
0)
to
(3
,
0),
what
is
R
C
F
·
d
r
?
423
Example
16.3.5
Appliying
the
Fundamental
Theorem
for
Line
Integrals
Exercise
F
o
r
each
vecto
r
field,
determine
whether
it
is
conservative.
If
it
is,
find
a
p
otential
function.
a
F
1
=
(
√
xy
−
y
)
i
+
(
√
xy
−
x
)
j
(fo
r
x
,
y
≥
0)
b
F
2
=
(
e
y
+
2
x
)
i
+
(
xe
y
−
4
y
3
)
j
424
Example
16.3.5
Appliying
the
Fundamental
Theorem
for
Line
Integrals
Summa
ry
Questions
What
do
es
is
mean
fo
r
a
vecto
r
field
to
b
e
conservative?
What
is
the
relationship
b
et
w
een
the
gradient
and
a
conservative
vecto
r
field?
Ho
w
do
w
e
test
that
a
vecto
r
field
is
conservative?
What
do
es
the
F
undamental
Theo
rem
of
Line
Integrals
sa
y?
425
Section
16.4
Green’s
Theo
rem
ggb/greensappro
x
Goals:
1
Use
Green’s
Theo
rem
to
replace
a
line
integral
with
a
double
integral
o
r
vice
versa.
Question
16.4.1
What
is
Green’s
Theo
rem?
The
fundamental
theo
rem
of
calculus
sa
ys
that
integrating
(adding
up
in
small
pieces)
a
rate
of
change
on
the
interval
[
a
,
b
]
gives
the
total
change
b
et
w
een
the
b
ounda
ry
p
oints
a
and
b
.
Z
b
a
f
′
(
x
)
dx
=
f
(
b
)
−
f
(
a
)
W
e
will
attempt
to
find
a
simila
r
co
rresp
ondence
fo
r
t
w
o-dimensional
domains.
427
Question
16.4.1
What
is
Green’s
Theorem?
Theo
rem
(Green’s
Theo
rem)
Supp
ose
D
is
a
simply
connected
(no
holes),
b
ounded
region
and
r
(
t
)
defines
C
,
a
piecewise
smo
oth
curve
that
traces
the
b
ounda
ry
of
D
counterclo
ckwise.
If
F
=
P
i
+
Q
j
is
a
vecto
r
field,
then
Z
C
F
·
d
r
=
Z
Z
D
∂
Q
∂
x
−
∂
P
∂
y
dA
428
Question
16.4.1
What
is
Green’s
Theorem?
T
o
avoid
mentioning
vecto
rs,
y
our
textb
o
ok
uses
the
notation
Z
C
Pdx
+
Qdy
Thus
w
e
can
also
write
Green’s
Theo
rem
Theo
rem
(Green’s
Theo
rem,
Alternate
Version)
Z
C
Pdx
+
Qdy
=
Z
Z
D
∂
Q
∂
x
−
∂
P
∂
y
dA
429
Question
16.4.1
What
is
Green’s
Theorem?
What
is
the
significance
of
∂
Q
∂
x
−
∂
P
∂
y
?
If
∂
Q
∂
x
>
0,
then
the
upw
ard
w
ork
on
the
right
side
of
C
outw
eighs
the
up
w
a
rd
w
o
rk
on
the
left
side
of
C
.
If
∂
P
∂
y
<
0,
then
the
rightw
ard
w
ork
on
the
b
ottom
of
C
outw
eighs
the
right
w
a
rd
w
o
rk
on
the
top
of
C
.
Click to Load Applet
430
Question
16.4.1
What
is
Green’s
Theorem?
T
o
Prove
Green’s
Theo
rem
w
e
app
ro
ximate
the
line
integral
ab
out
C
by
summing
line
integrals
a
round
∆
x
by
∆
y
rectangles.
Notice
that
the
interio
r
edges
cancel
each
other
out.
The
remaining
outer
edges
app
ro
ximate
C
.
Z
C
F
·
d
r
=
lim
∆
x
,
∆
y
→
0
n
X
i
=1
Z
C
i
F
·
d
r
i
Click to Load Applet
431
Question
16.4.1
What
is
Green’s
Theorem?
In
o
rder
to
app
ro
ximate
the
line
integral
a
round
a
∆
x
by
∆
y
rectangle,
w
e
linea
rize
F
.
We’ll
use
differential
notation.
Unless
noted
otherwise,
all
functions
evaluated
at
(
x
,
y
).
L
(
x
+
dx
,
y
+
dy
)
=
F
+
d
F
=
F
+
F
x
dx
+
F
y
dy
=
P
+
P
x
dx
+
P
y
dy
i
+
Q
+
Q
x
dx
+
Q
y
dy
j
432
Question
16.4.1
What
is
Green’s
Theorem?
Let’s
pa
rameterize
the
edges
of
a
∆
x
by
∆
y
rectangle.
Fo
r
each
segment,
0
≤
t
≤
1.
(
x
+
t
∆
x
)
i
+
y
j
(
x
+
t
∆
x
)
i
+
(
y
+
∆
y
)
j
x
i
+
(
y
+
t
∆
y
)
j
(
x
+
∆
x
)
i
+
(
y
+
t
∆
y
)
j
(
x
,
y
)
(
x
+
∆
x
,
y
)
(
x
+
∆
x
,
y
+
∆
y
)
(
x
,
y
+
∆
y
)
433
Question
16.4.1
What
is
Green’s
Theorem?
Let’s
pa
rameterize
the
edges
of
a
∆
x
by
∆
y
rectangle.
Fo
r
each
segment,
0
≤
t
≤
1.
(
x
+
t
∆
x
)
i
+
y
j
(
x
+
t
∆
x
)
i
+
(
y
+
∆
y
)
j
x
i
+
(
y
+
t
∆
y
)
j
(
x
+
∆
x
)
i
+
(
y
+
t
∆
y
)
j
(
x
,
y
)
(
x
+
∆
x
,
y
)
(
x
+
∆
x
,
y
+
∆
y
)
(
x
,
y
+
∆
y
)
433
Question
16.4.1
What
is
Green’s
Theorem?
Let’s
pa
rameterize
the
edges
of
a
∆
x
by
∆
y
rectangle.
Fo
r
each
segment,
0
≤
t
≤
1.
(
x
+
t
∆
x
)
i
+
y
j
(
x
+
t
∆
x
)
i
+
(
y
+
∆
y
)
j
x
i
+
(
y
+
t
∆
y
)
j
(
x
+
∆
x
)
i
+
(
y
+
t
∆
y
)
j
(
x
,
y
)
(
x
+
∆
x
,
y
)
(
x
+
∆
x
,
y
+
∆
y
)
(
x
,
y
+
∆
y
)
433
Question
16.4.1
What
is
Green’s
Theorem?
When
w
e
replace
F
with
L
and
compute
the
line
integral,
w
e
get
convenient
cancellation.
Here
a
re
the
top
and
b
ottom
segments.
Z
1
0
L
(
x
+
t
∆
x
,
y
)
·
(∆
x
i
)
dt
−
Z
1
0
L
(
x
+
t
∆
x
,
y
+
∆
y
)
·
(∆
x
i
)
dt
=
∆
x
Z
1
0
P
+
P
x
(
t
∆
x
)
dt
−
Z
1
0
P
+
P
x
(
t
∆
x
)
+
P
y
(∆
y
)
dt
=
∆
x
Z
1
0
−
P
y
∆
ydt
=
−
P
y
∆
y
∆
x
Simila
rly
the
left
and
right
segments
sum
to
Q
x
∆
x
∆
y
.
434
Question
16.4.1
What
is
Green’s
Theorem?
Finally
w
e
return
to
our
o
riginal
limit
app
ro
ximation.
Note
∆
y
∆
x
is
the
a
rea
of
a
∆
x
by
∆
y
rectangle,
so
our
expression
conforms
to
the
limit
definition
of
a
double
integral.
Z
C
F
·
d
r
=
lim
∆
x
,
∆
y
→
0
n
X
i
=1
Z
C
i
F
·
d
r
i
=
lim
∆
x
,
∆
y
→
0
n
X
i
=1
(
Q
x
−
P
y
)∆
y
∆
x
=
lim
∆
x
,
∆
y
→
0
n
X
i
=1
(
Q
x
−
P
y
)∆
A
=
Z
Z
D
(
Q
x
−
P
y
)
dA
435
Example
16.4.2
Applying
Green’s
Theorem
Let
F
(
x
,
y
)
=
(
y
2
−
3
x
)
i
+
xy
j
.
Let
C
b
e
the
path
that
travels
along
the
line
segment
from
(2
,
4)
to
(
−
1
,
1)
and
then
back
to
(2
,
4)
along
the
pa
rab
ola
y
=
x
2
.
Compute
R
C
F
·
d
r
.
436
Example
16.4.2
Applying
Green’s
Theorem
Main
Idea
Dra
w
y
our
closed
curves
to
see
what
region
they
b
ound.
Don’t
fo
rget
to
check
whether
they
travel
counterclo
ckwise.
437
Example
16.4.2
Applying
Green’s
Theorem
Exercise
Let
F
(
x
,
y
)
=
(3
y
−
e
x
)
i
+
(2
x
−
sin
y
)
j
.
Let
C
b
e
a
circle
of
radius
3
traveling
counterclo
ckwise
once
a
round
the
o
rigin.
a
Set
up
the
line
integral
R
C
F
·
d
r
as
a
single-variable
integral
of
t
.
b
If
F
w
ere
conservative
what
w
ould
the
value
of
this
integral
b
e?
Is
F
conservative?
c
Ho
w
w
ould
y
ou
apply
Green’s
theo
rem
to
the
integral?
What
is
its
value?
d
What
w
ould
R
C
F
·
d
r
b
e
if
C
traveled
clo
ckwise
instead?
438
Example
16.4.3
Example
2
Let
C
b
e
a
semicircle
from
(2
,
0)
to
(
−
2
,
0)
ab
ove
the
x
-axis.
Compute
Z
C
(
x
2
−
y
3
)
dx
+
(
x
3
+
e
y
2
)
dy
439
Example
16.4.3
Using
Green’s
Theorem
on
a
Non-Closed
Curve
Main
Idea
Green’s
Theo
rem
can
b
e
used
to
replace
a
w
o
rk
integral
over
a
complex
path
with
a
w
o
rk
integral
over
a
simpler
path.
440
Example
16.4.3
Using
Green’s
Theorem
on
a
Non-Closed
Curve
Summa
ry
of
Line
Integral
Metho
ds
First
decide
whether
y
ou’re
taking
the
line
integral
of
a
function:
f
(
x
,
y
)
o
r
a
vecto
r
field:
F
(
x
,
y
).
1
F
unction
f
a
Pa
rameterize
C
and
set
up
the
integral,
replacing
dx
,
dy
,
ds
with
the
app
rop
riate
differential.
b
Evaluate
the
integral.
2
V
ecto
r
Field
F
a
Is
F
conservative?
Use
FTLI.
b
Can
y
ou
dra
w
the
curve
C
?
Is
it
closed?
Use
Green’s.
c
If
neither
w
orks,
parameterize
C
and
set
up
the
integral,
replacing
d
r
with
r
′
(
t
)
dt
,
and
evaluate.
441
Example
16.4.3
../im
gicons/teacher.pdf
Using
Green’s
Theorem
on
a
Non-Closed
Curve
Summa
ry
What
kind
of
integrals
can
w
e
evaluate
with
Green’s
theo
rem?
Why
w
ould
w
e
ever
w
ant
to
replace
a
single
integral
with
a
double
integral?
442
Section
12.4
The
Cross
Pro
duct
Goals:
1
Calculate
the
determinant
of
a
3
×
3
matrix.
2
Calculate
the
cross
p
ro
duct
of
t
w
o
vecto
rs.
3
Understand
the
geometric
relationship
b
et
w
een
tw
o
vectors
and
their
cross
p
ro
duct.
Question
12.4.1
Ho
w
Do
W
e
Compute
a
Determinant?
Definition
A
matrix
is
a
rectangula
r
a
rra
y
of
values
(usually
numb
ers).
An
m
×
n
matrix
has
m
rows
and
n
columns.
If
a
matrix
has
the
same
numb
er
of
ro
ws
and
columns,
it
is
sqaure
.
Examples
a
2
×
4
matrix
3
0
4
−
2
4
2
0
1
a
3
×
1
matrix
2
0
5
a
squa
re
3
×
3
matrix
1
3
0
0
2
2
3
1
1
444
Question
12.4.1
How
Do
We
Compute
a
Determinant?
A
determinant
is
a
numb
er
that
w
e
can
compute
and
asso
ciate
to
a
squa
re
matrix.
If
the
matrix
has
a
name
(lik
e
M
),
we
use
the
notation
det
M
or
|
M
|
.
We
can
also
write
det
1
3
0
0
2
2
3
1
1
o
r
1
3
0
0
2
2
3
1
1
445
Question
12.4.1
How
Do
We
Compute
a
Determinant?
The
determinant
of
a
2
×
2
matrix
is
calculated
by
the
fo
rmula
a
b
c
d
=
ad
−
b
c
The
fo
rmulas
fo
r
la
rger
matrices
a
re
derived
from
those
of
smaller
mino
r
matrices.
a
b
c
d
e
f
g
h
i
=
a
e
f
h
i
−
b
d
f
g
i
+
c
d
e
g
h
446
Question
12.4.1
How
Do
We
Compute
a
Determinant?
The
determinant
of
a
2
×
2
matrix
is
calculated
by
the
fo
rmula
a
b
c
d
=
ad
−
b
c
The
fo
rmulas
fo
r
la
rger
matrices
a
re
derived
from
those
of
smaller
mino
r
matrices.
a
b
c
d
e
f
g
h
i
=
a
e
f
h
i
−
b
d
f
g
i
+
c
d
e
g
h
446
Question
12.4.1
How
Do
We
Compute
a
Determinant?
The
determinant
of
a
2
×
2
matrix
is
calculated
by
the
fo
rmula
a
b
c
d
=
ad
−
b
c
The
fo
rmulas
fo
r
la
rger
matrices
a
re
derived
from
those
of
smaller
mino
r
matrices.
a
b
c
d
e
f
g
h
i
=
a
e
f
h
i
−
b
d
f
g
i
+
c
d
e
g
h
446
Question
12.4.1
How
Do
We
Compute
a
Determinant?
The
determinant
of
a
2
×
2
matrix
is
calculated
by
the
fo
rmula
a
b
c
d
=
ad
−
b
c
The
fo
rmulas
fo
r
la
rger
matrices
a
re
derived
from
those
of
smaller
mino
r
matrices.
a
b
c
d
e
f
g
h
i
=
a
e
f
h
i
−
b
d
f
g
i
+
c
d
e
g
h
446
Example
12.4.2
A
3
by
3
Determinant
Calculate
1
3
0
0
2
2
3
1
1
447
Question
12.4.3
What
Geometric
Measurement
Do
es
the
Determinant
Compute?
The
absolute
value
of
the
determinant
of
a
matrix
is
the
volume
of
the
pa
rallelepip
ed
constructed
from
the
ro
w
(o
r
column)
vecto
rs.
1
3
0
0
2
2
3
1
1
=
18
Click to Load Applet
448
Question
12.4.4
What
Is
the
Cross
Pro
duct?
Definition
The
cross
p
ro
duct
is
a
product
of
three-dimensional
vectors
u
and
v
,
whose
output
is
also
a
three
dimensional
vecto
r
denoted
u
×
v
.
The
cross
p
ro
duct
is
defined
as
follo
ws
on
the
standa
rd
basis
vecto
rs:
i
×
j
=
k
j
×
k
=
i
k
×
i
=
j
j
×
i
=
−
k
k
×
j
=
−
i
i
×
k
=
−
j
i
×
i
=
j
×
j
=
k
×
k
=
0
Notice
that
the
cross
p
ro
duct
of
t
w
o
vecto
rs
is
a
vecto
r,
whereas
the
dot
p
ro
duct
is
a
numb
er.
449
Question
12.4.4
What
Is
the
Cross
Pro
duct?
In
o
rder
to
finish
defining
the
cross
p
ro
duct,
w
e
need
the
follo
wing
algeb
raic
p
rop
erties:
1
The
cross
p
ro
duct
is
asso
ciative
with
scala
r
multiplication:
(
a
u
)
×
v
=
u
×
(
a
v
)
=
a
(
u
×
v
)
2
The
cross
p
ro
duct
distributes
across
vecto
r
sums:
(
u
1
+
u
2
)
×
v
=
u
1
×
v
+
u
2
×
v
u
×
(
v
1
+
v
2
)
=
u
×
v
1
+
u
×
v
2
450
Example
12.4.5
A
Cross
Pro
duct
from
Standard
Basis
V
ectors
If
u
=
2
i
+
3
j
+
4
k
and
v
=
i
+
2
j
−
3
k
,
compute
u
×
v
.
451
Synthesis
12.4.6
The
Cross
Pro
duct
as
a
Determinant
F
o
rmula
If
u
=
⟨
u
1
,
u
2
,
u
3
⟩
and
v
=
⟨
v
1
,
v
2
,
v
3
⟩
then
u
×
v
=
u
2
u
3
v
2
v
3
i
−
u
1
u
3
v
1
v
3
j
+
u
1
u
2
v
1
v
2
k
If
w
e’re
a
bit
slopp
y
and
allo
w
our
matrix
to
have
vecto
rs
as
entries,
we
can
write
mo
re
compactly:
u
×
v
=
i
j
k
u
1
u
2
u
3
v
1
v
2
v
3
452
Example
12.4.7
The
Cross
Pro
duct
b
y
Determinant
Calculate
⟨
2
,
0
,
3
⟩
×
⟨
3
,
1
,
1
⟩
.
453
Question
12.4.8
What
Is
the
Geometric
Significance
of
the
Cross
Product?
The
direction
of
u
×
v
is
given
by
the
follo
wing
facts:
u
×
v
is
orthogonal
to
both
u
and
v
.
If
y
our
right
hand
traces
a
circle
from
u
through
v
,
then
your
thumb
p
oints
in
the
direction
of
u
×
v
.
Click to Load Applet
454
Question
12.4.8
What
Is
the
Geometric
Significance
of
the
Cross
Pro
duct?
If
θ
is
the
angle
b
etw
een
u
and
v
,
the
length
satisfies
the
fo
rmula
|
u
×
v
|
=
|
u
||
v
|
sin
θ.
|
u
×
v
|
is
also
the
a
rea
of
the
parallelogram
defined
b
y
u
and
v
.
Click to Load Applet
455
Example
12.4.9
Using
Geometry
to
Describ
e
a
Cross
Pro
duct
If
u
=
4
k
and
v
is
in
the
xy
-plane,
then
what
can
we
sa
y
ab
out
u
×
v
?
456
Application
12.4.10
T
o
rque
In
physics,
to
rque
measures
the
tendency
of
a
rigid
b
o
dy
to
rotate
a
round
a
fixed
o
rigin.
If
w
e
apply
the
fo
rce
F
at
the
p
osition
r
from
the
o
rigin,
the
to
rque
is
τ
=
r
×
F
.
Viewing
to
rque
as
a
vecto
r
is
very
useful.
F
o
r
example,
if
mo
re
than
one
fo
rce
is
applied,
the
to
rques
can
b
e
added
to
compute
a
total
to
rque
on
the
object.
Click to Load Applet
457
Application
12.4.11
The
No
rmal
Equation
of
a
Plane
Find
an
equation
of
the
plane
that
contains
the
p
oints
(2
,
1
,
1),
(3
,
4
,
−
1)
and
(0
,
5
,
2).
458
Application
12.4.11
../im
gicons/rocket.pdf
The
Normal
Equation
of
a
Plane
Summa
ry
Questions
What
do
the
cross
p
ro
duct
and
dot
p
ro
duct
have
in
common?
Ho
w
a
re
they
different?
W
ould
y
ou
rather
use
the
mino
r
matrices
o
r
the
distributive
metho
d
to
compute
a
cross
p
ro
duct?
Why?
Can
a
cross
p
ro
duct
b
e
used
to
compute
the
angle
b
et
w
een
t
w
o
vecto
rs?
W
ould
y
ou
p
refer
to
use
the
dot
p
ro
duct?
Why?
459
Section
16.5
Curl
and
Divergence
Goals:
1
Compute
the
curl
and
divergence
of
a
vecto
r
field.
2
Interp
ret
curl
and
divergence
geometrically
.
Question
16.6.1
What
Is
the
Derivative
of
a
V
ector
Field?
If
w
e
compa
re
the
fundamental
theo
rem
of
calculus
to
the
fundamental
theo
rem
of
line
integrals,
f
(
b
)
−
f
(
a
)
=
Z
b
a
f
′
(
x
)
dx
f
(
B
)
−
f
(
A
)
=
Z
C
∇
f
·
d
r
w
e
see
that
∇
f
fills
the
role
of
a
derivative
of
the
multiva
riable
function
f
in
this
context.
In
this
section
we
try
to
define
some
derivative-lik
e
op
erations
of
vecto
r
fields.
461
Question
16.6.1
What
Is
the
Derivative
of
a
Vecto
r
Field?
Notation
W
e
define
the
gradient
operator
∇
(“del”),
which
behaves
in
some
w
a
ys
lik
e
a
vecto
r.
Dep
ending
on
our
choice
of
dimension
w
e
can
have
∇
=
∂
∂
x
,
∂
∂
y
∇
=
∂
∂
x
,
∂
∂
y
,
∂
∂
z
462
Question
16.6.1
What
Is
the
Derivative
of
a
Vecto
r
Field?
Given
a
function
f
(
x
,
y
),
w
e
can
reexamine
∇
f
in
terms
of
the
gradient
op
erato
r:
∇
f
=
∂
∂
x
,
∂
∂
y
f
=
∂
∂
x
f
,
∂
∂
y
f
=
∂
f
∂
x
,
∂
f
∂
y
463
Question
16.6.2
What
Is
the
Divergence
of
a
V
ector
Field?
The
∇
op
erato
r
is
mo
re
exciting
when
w
e
apply
it
to
vecto
r
fields.
Definition
The
divergence
of
a
vecto
r
field
F
=
P
i
+
Q
j
at
a
p
oint
(
x
0
,
y
0
)
measures
the
extent
to
which
F
p
oints
aw
ay
from
(
x
0
,
y
0
).
The
divergence
function
is
div
F
=
∇
·
F
=
∂
∂
x
P
+
∂
∂
y
Q
Divergence
is
defined
analogously
fo
r
3-dimensional
vecto
r
fields.
Notice
that
∇
·
F
(
x
0
,
y
0
)
is
a
numb
er,
not
a
vecto
r.
464
Question
16.6.2
What
Is
the
Divergence
of
a
Vecto
r
Field?
The
Geometric
Significance
of
Divergence
∇
·
F
(
x
0
,
y
0
)
is
p
ositive
when
F
is
mostly
pointing
aw
ay
from
(
x
0
,
y
0
)
(a
source).
It
is
negative
when
the
F
is
mostly
pointing
tow
ards
from
(
x
0
,
y
0
)
(a
sink).
It
is
easiest
to
recognize
divergence
when
F
(
x
0
,
y
0
)
is
the
zero
vecto
r.
∇
·
F
>
0
∇
·
F
<
0
ggb/divergencezero.
png
∇
·
F
≈
0
465
Question
16.6.2
What
Is
the
Divergence
of
a
Vecto
r
Field?
When
F
(
x
0
,
y
0
)
is
not
zero,
it
can
b
e
ha
rder
to
tell
whether
F
is
expanding
a
w
a
y
from
(
x
0
,
y
0
)
o
r
accelerating
to
w
a
rd
it.
If
F
(
x
0
,
y
0
)
=
v
w
e
can
subtract
the
constant
vecto
r
v
from
F
.
Click to Load Applet
∇
·
F
=
∇
·
(
F
−
v
)
466
Question
16.6.2
What
Is
the
Divergence
of
a
Vector
Field?
Exercise
If
F
=
xz
i
+
xyz
j
−
y
2
k
,
compute
∇
·
F
at
(
−
2
,
2
,
1).
What
do
es
it
mean?
467
Question
16.6.3
What
Is
the
Curl
of
a
V
ector
Field?
Recall
that
∂
Q
∂
x
−
∂
P
∂
y
measured
the
amount
that
a
vecto
r
field
curled
a
round
a
p
oint.
Green’s
theorem
related
this
to
the
line
integral
of
a
curve
a
round
that
p
oint.
This
is
a
sp
ecial
case
of
our
other
“derivative,”
the
curl
of
F
.
468
Question
16.6.3
What
Is
the
Curl
of
a
Vecto
r
Field?
Definition
The
curl
is
defined
fo
r
a
three-dimensional
vecto
r
field
F
=
⟨
P
,
Q
,
R
⟩
.
It
is
denoted
∇
×
F
and
computed
as
follows:
curl
F
=
∇
×
F
=
i
j
k
∂
∂
x
∂
∂
y
∂
∂
z
P
Q
R
=
∂
R
∂
y
−
∂
Q
∂
z
i
−
∂
R
∂
x
−
∂
P
∂
z
j
+
∂
Q
∂
x
−
∂
P
∂
y
k
469
Question
16.6.3
What
Is
the
Curl
of
a
Vecto
r
Field?
Notice
∇
×
F
(
x
0
,
y
0
,
z
0
)
is
a
vecto
r.
∇
×
F
(
x
0
,
y
0
,
z
0
)
is
related
to
the
line
integrals
of
F
around
the
p
oint
(
x
0
,
y
0
,
z
0
).
The
length
indicates
ho
w
la
rge
such
integrals
can
b
e,
as
a
multiple
of
the
a
rea
they
enclose.
The
direction
is
p
erp
endicula
r
to
the
plane
in
which
these
line
integrals
a
re
maximized,
o
r
a
round
which
F
curls
most
strongly
.
Physically
,
if
y
ou
attached
a
freely
rotating
imp
eller
to
(
x
0
,
y
0
,
z
0
)
in
the
fo
rce
field
F
,
∇
×
F
(
x
0
,
y
0
,
z
0
)
w
ould
b
e
the
axis
a
round
which
the
imp
eller
w
ould
spin
the
fastest
(direction
determined
b
y
right-hand-rule).
470
Question
16.6.3
What
Is
the
Curl
of
a
Vector
Field?
Exercise
If
F
=
xz
i
+
xyz
j
−
y
2
k
,
compute
∇
×
F
at
(
−
2
,
2
,
1).
What
do
es
it
mean?
471
Synthesis
16.6.4
V
ector
V
ersion
of
Green’s
Theo
rem
Green’s
theo
rem
is
t
w
o
dimensional,
so
w
e
assume
F
(
x
,
y
,
z
)
=
P
(
x
,
y
)
i
+
Q
(
x
,
y
)
j
+
0
k
.
Most
of
the
terms
in
the
curl
are
zero.
Sp
ecifically
,
∇
×
F
=
∂
Q
∂
x
−
∂
P
∂
y
k
.
Theo
rem
(Green’s
Theo
rem)
Supp
ose
D
is
a
simply
connected,
b
ounded
region
in
the
plane
z
=
0
and
r
(
t
)
defines
C
,
a
piecewise
smo
oth
curve
that
traces
the
b
ounda
ry
of
D
counterclo
ckwise.
If
F
=
⟨
P
,
Q
,
0
⟩
is
a
vector
field,
then
Z
C
F
·
d
r
=
Z
Z
D
(
∇
×
F
)
·
k
dA
472
Synthesis
16.6.5
Comp
osing
∇
Op
erato
rs
W
e
can
also
comp
ose
∇
op
erato
rs
together.
Here
are
some
examples
Example
∇
2
f
=
∇
·
(
∇
f
)
takes
the
divergence
of
the
gradient
vector
field.
Example
∇
·
(
∇
×
F
)
computes
the
divergence
of
the
curl
of
F
.
Theo
rem
A
vecto
r
field
G
on
a
simply
connected
3-dimensional
domain
is
equal
to
∇
×
F
for
some
F
,
if
and
only
if
∇
·
G
(
x
,
y
,
z
)
=
0
for
all
(
x
,
y
,
z
).
473
Synthesis
16.6.5
../im
gicons/molecule.pdf
Composing
∇
Operators
Summa
ry
Questions
Ho
w
do
y
ou
compute
divergence
and
curl?
Ho
w
do
y
ou
interp
ret
divergence
geometrically?
On
what
vecto
r
fields
can
y
ou
compute
curl?
Divergence?
If
someone
handed
y
ou
t
w
o
functions
and
tells
y
ou
one
is
the
curl
of
a
vecto
r
field
and
the
other
is
the
divergence,
ho
w
could
y
ou
tell
which
is
which?
474
Section
16.7
P
arametric
Surfaces
and
Their
Areas
Goals:
1
P
a
rameterize
a
surface.
2
Compute
tangent
vecto
rs
to
a
pa
rametric
surface.
Question
16.7.1
Ho
w
Do
W
e
P
a
rametrize
a
Surface?
Definition
A
pa
rametric
surface
is
the
set
of
points
defined
by
t
wo-va
riable
pa
rametric
equations,
usually
in
three-space.
r
(
u
,
v
)
=
x
(
u
,
v
)
i
+
y
(
u
,
v
)
j
+
z
(
u
,
v
)
k
Where
u
and
v
are
both
parameters.
Lik
e
with
curves,
w
e
write
this
as
a
vecto
r
function
so
w
e
can
do
calculus,
but
w
e
visualize
it
as
a
set
of
p
oints.
Click to Load Applet
476
Question
16.7.1
How
Do
We
Pa
rametrize
a
Surface?
F
o
rmula
The
graph
of
an
explicit
function
z
=
f
(
x
,
y
)
can
be
pa
rameterized
by
substituting
x
=
u
y
=
v
z
=
f
(
u
,
v
)
r
(
u
,
v
)
=
u
i
+
v
j
+
f
(
u
,
v
)
k
Click to Load Applet
477
Question
16.7.1
How
Do
We
Pa
rametrize
a
Surface?
F
o
rmula
If
a
plane
p
contains
the
p
oint
(
x
0
,
y
0
,
z
0
)
and
the
vecto
rs
a
and
b
,
then
a
pa
rametrization
of
p
is
r
(
u
,
v
)
=
⟨
x
0
,
y
0
,
z
0
⟩
+
u
a
+
v
b
Click to Load Applet
478
Synthesis
16.7.2
P
arametrizations
from
Other
Co
ordinate
Systems
Another
source
of
pa
rametrizations
comes
from
co
ordinate
systems
we’ve
lea
rned.
Constant
multiples
and
constant
terms
stretch
and
shift
the
surface.
Click to Load Applet
r
(
u
,
v
)
=
(cos
u
sin
v
)
i
+
(sin
u
sin
v
)
j
+
(cos
v
)
k
0
≤
u
≤
2
π
0
≤
v
≤
π
479
Synthesis
16.7.2
P
arametrizations
from
Other
Co
ordinate
Systems
Another
source
of
pa
rametrizations
comes
from
co
ordinate
systems
we’ve
lea
rned.
Constant
multiples
and
constant
terms
stretch
and
shift
the
surface.
Click to Load Applet
r
(
u
,
v
)
=
(
a
cos
u
sin
v
+
x
0
)
i
+
(
b
sin
u
sin
v
+
y
0
)
j
+
(
c
cos
v
+
z
0
)
k
0
≤
u
≤
2
π
0
≤
v
≤
π
479
Synthesis
16.7.2
Parametrizations
from
Other
Co
ordinate
Systems
Exercise
Describ
e
the
surfaces
with
the
follo
wing
pa
rametric
equations.
a
r
(
u
,
v
)
=
3
cos
u
i
+
3
sin
u
j
+
v
k
0
≤
u
≤
2
π
,
0
≤
v
≤
5
b
r
(
u
,
v
)
=
(3
−
3
u
−
3
v
)
i
+
(6
u
+
2
v
)
j
+
(2
−
9
v
)
k
c
r
(
u
,
v
)
=
u
cos
π
4
sin
v
i
+
u
sin
π
4
sin
v
j
+
u
cos
v
k
0
≤
u
≤
5
,
0
≤
v
≤
π
480
Question
16.7.3
What
Are
the
T
angent
Vecto
rs
of
a
P
arametric
Surface?
Definition
The
pa
rtial
derivatives
r
u
(
u
0
,
v
0
)
and
r
v
(
u
0
,
v
0
)
a
re
tangent
vecto
rs
to
the
surface
S
.
The
exp
ression
a
r
u
(
u
0
,
v
0
)
+
b
r
v
(
u
0
,
v
0
)
p
ro
duces
a
general
tangent
vecto
r
to
S
at
r
(
u
0
,
v
0
).
W
e
can
use
these
tangent
vecto
rs
to
p
ro
duce
the
follo
wing
linea
rization
of
S
at
r
(
u
0
,
v
0
).
L
(
u
,
v
)
=
r
(
u
0
,
v
0
)
+
r
u
(
u
0
,
v
0
)(
u
−
u
0
)
+
r
v
(
u
0
,
v
0
)(
v
−
v
0
)
Some
algeb
ra
sho
ws
that
this
is
the
equation
of
a
tangent
plane
.
481
Question
16.7.3
../im
gicons/qm.pdf
What
Are
the
T
angent
Vectors
of
a
Parametric
Surface?
Summa
ry
Questions
Ho
w
do
w
e
pa
rameterize
a
plane?
Ho
w
do
w
e
pa
rametrize
the
graph
z
=
f
(
x
,
y
)?
Ho
w
do
w
e
pa
rametrize
a
sphere
o
r
a
cylinder?
What
is
the
relationship
b
et
w
een
the
tangent
vecto
rs
and
the
tangent
plane
of
a
surface?
482
Section
16.8
Surface
Integrals
Goals:
1
Understand
the
geometric
significance
of
the
different
surface
integrals.
2
Set
up
and
evaluate
surface
integrals.
Question
16.8.1
Ho
w
do
W
e
Integrate
on
a
Pa
rametric
Surface?
Just
lik
e
with
line
integrals,
w
e’d
lik
e
to
find
w
a
ys
of
integrating
a
function
f
(
x
,
y
,
z
)
on
a
surface
S
that
do
not
dep
end
on
the
choice
of
pa
rameterization.
Definition
An
integral
dS
is
computed
with
resp
ect
to
the
geometric
area
on
the
surface.
W
e
divide
S
into
regions
and
let
∆
S
i
b
e
the
a
rea
of
the
i
th
region.
(
x
∗
i
,
y
∗
i
,
z
∗
i
)
b
e
a
test
p
oint
in
the
i
th
region.
D
b
e
the
la
rgest
diameter
of
any
of
the
regions
(the
longest
distance
b
et
w
een
t
w
o
p
oints
in
the
region).
W
e
then
define
the
surface
integral
:
Z
Z
S
f
(
x
,
y
,
z
)
dS
=
lim
D
→
0
X
i
f
(
x
∗
i
,
y
∗
i
,
z
∗
i
)∆
S
i
484
Question
16.8.1
How
do
We
Integrate
on
a
Pa
rametric
Surface?
It’s
easier
to
cho
ose
sub
regions
defined
b
y
a
change
in
u
and
a
change
in
v
.
Still,
we
ma
y
not
know
the
a
rea
of
such
a
region,
so
we
use
our
old
trick
and
linea
rize
r
(
u
,
v
)
at
a
p
oint
and
use
the
area
of
the
pa
rallelogram
given
b
y
∆
u
and
∆
v
.
The
area
is
thus
dS
=
|
r
u
×
r
v
|
dudv
Click to Load Applet
485
Example
16.8.2
Surface
Area
Z
Z
S
1
dS
computes
the
area
of
a
surface.
Compute
the
surface
area
of
a
sphere
of
radius
R
.
486
Example
16.8.2
Surface
Area
Exercise
Let
L
b
e
the
surface
r
(
u
,
v
)
=
3
cos
u
i
+
3
sin
u
j
+
v
k
0
≤
u
≤
2
π
,
0
≤
v
≤
5
.
Set
up
(but
do
not
evaluate)
the
surface
integral
Z
Z
L
x
2
zdS
.
487
Question
16.8.3
Ho
w
Do
w
e
Compute
Flow
Through
a
Surface?
Motivational
Example
Supp
ose
w
ater,
traveling
at
3
meters
p
er
second
is
passing
through
a
circula
r
op
ening
of
radius
4m.
1
Ho
w
much
w
ater
flo
ws
through
the
op
ening
p
er
second?
2
What
if
the
w
ater
is
not
flo
wing
p
erp
endicula
r
to
the
circle
?
488
Question
16.8.3
Ho
w
Do
w
e
Compute
Flow
Through
a
Surface?
Motivational
Example
Supp
ose
w
ater,
traveling
at
3
meters
p
er
second
is
passing
through
a
circula
r
op
ening
of
radius
4m.
1
Ho
w
much
w
ater
flo
ws
through
the
op
ening
p
er
second?
2
What
if
the
w
ater
is
not
flo
wing
p
erp
endicula
r
to
the
circle?
488
Question
16.8.3
How
Do
we
Compute
Flow
Through
a
Surface?
If
the
w
ater
is
flo
wing
with
velo
cit
y
v
and
n
is
normal
to
the
opening
S
with
length
equal
to
the
a
rea
of
S
,
then
the
flow
rate
is
v
·
n
.
Click to Load Applet
489
Question
16.8.3
How
Do
we
Compute
Flow
Through
a
Surface?
W
e
can
generalize
this
p
roblem
further:
3
What
if
the
velo
cit
y
of
the
w
ater
is
not
constant,
but
dep
ends
on
the
lo
cation
where
it
is
measured?
4
What
if
the
op
ening
isn’t
a
flat
shap
e,
but
a
surface
in
3
dimensions?
490
Question
16.8.3
How
Do
we
Compute
Flow
Through
a
Surface?
W
e
can
generalize
this
p
roblem
further:
3
What
if
the
velo
cit
y
of
the
w
ater
is
not
constant,
but
dep
ends
on
the
lo
cation
where
it
is
measured?
4
What
if
the
op
ening
isn’t
a
flat
shap
e,
but
a
surface
in
3
dimensions?
490
Question
16.8.3
How
Do
we
Compute
Flow
Through
a
Surface?
Definition
The
flux
integral
of
F
through
S
is
denoted
Z
Z
S
F
·
d
S
.
F
o
r
a
pa
rameterization
r
(
u
,
v
)
we
define
d
S
=
(
r
u
×
r
v
)
dudv
.
F
·
d
S
measures
the
flo
w
of
F
through
the
pa
rallelogram
dS
.
Click to Load Applet
491
Example
16.8.4
A
Flux
Integral
Let
F
(
x
,
y
,
z
)
=
x
i
−
y
j
b
e
a
flo
w
of
w
ater
and
let
N
b
e
a
net
given
b
y
N
=
{
(
x
,
y
,
z
)
:
z
=
x
2
−
y
2
,
x
2
+
y
2
≤
1
}
.
a
Compute
Z
Z
N
F
·
d
S
.
b
What
do
es
the
sign
of
y
our
answ
er
mean?
Click to Load Applet
492
Synthesis
16.8.5
Orientation
Dep
ending
on
our
choice
of
pa
rameterization,
r
u
×
r
v
could
p
oint
in
one
of
t
w
o
no
rmal
directions.
If
a
surface
has
t
w
o
sides,
then
cho
osing
a
no
rmal
direction
defines
an
o
rientation
of
the
surface.
Note
that
in
general
not
all
surfaces
have
t
w
o
sides.
Surfaces
without
t
w
o
sides
a
re
non-o
rientable
.
Click to Load Applet
493
Synthesis
16.8.5
Orientation
W
e
can
connect
surface
integrals
to
flux
integrals
via
the
follo
wing
definition
Definition
Given
a
surface
S
,
the
unit
normal
vector
n
to
S
at
the
point
P
is
the
no
rmal
vecto
r
of
length
1
at
P
whose
direction
agrees
with
the
o
rientation
of
the
surface.
Notice
that
at
the
p
oint
r
(
u
,
v
),
n
=
r
u
×
r
v
|
r
u
×
r
v
|
so
Z
Z
S
F
·
d
S
=
Z
Z
F
(
r
)
·
(
r
u
×
r
v
)
dudv
=
Z
Z
F
(
r
)
·
n
|
r
u
×
r
v
|
dudv
=
Z
Z
S
F
·
ndS
This
gives
us
an
alternative
notation
fo
r
flux
integrals.
494
Synthesis
16.8.5
../im
gicons/molecule.pdf
Orientation
Summa
ry
Questions
What
a
re
the
t
w
o
kinds
of
surface
integrals?
What
do
they
compute?
What
exp
ression
do
w
e
substitute
fo
r
the
differentials
dS
and
d
S
?
Ho
w
many
different
o
rientations
can
a
connected
surface
have?
Do
es
this
change
if
the
surface
consists
of
t
w
o
o
r
mo
re
disconnected
pa
rts?
495
Section
16.9
Stok
es’
Theo
rem
Goals:
1
Use
Stok
es’
Theo
rem
to
evaluate
integrals.
Question
16.9.1
Ho
w
Do
We
Extrap
olate
Green’s
Theorem
to
Surfaces?
Green’s
Theo
rem
related
a
line
integral
a
round
C
to
a
double
integral
of
a
region
b
ounded
b
y
C
.
In
three
dimensions,
a
curve
C
might
b
ound
a
surface
S
.
We
can
attempt
to
apply
the
method
of
Green’s
to
this
situation.
497
Question
16.9.1
How
Do
We
Extrapolate
Green’s
Theorem
to
Surfaces?
Lik
e
in
t
w
o
dimensions,
given
a
sub
division
of
S
into
smaller
pieces
with
p
erimeter
curves
r
i
,
w
e
have
Z
C
F
·
d
r
=
X
i
Z
C
i
F
·
d
r
i
W
e
app
ro
ximate
the
C
i
with
pa
rallelograms
from
the
linea
rization
of
S
.
This
lets
us
write
the
line
integrals
in
terms
of
the
pa
rameterization
of
S
.
Click to Load Applet
498
Question
16.9.1
How
Do
We
Extrapolate
Green’s
Theorem
to
Surfaces?
Let
s
(
u
,
v
)
be
a
parameterization
of
S
.
We
will
parameterize
the
edges
of
pa
rallelogram
that
results
from
a
change
of
∆
u
and
∆
v
in
the
linea
rization
of
s
(
u
,
v
).
The
domains
are
all
0
≤
t
≤
1.
r
′
(
t
)
z
}|
{
s
+
t
s
u
∆
u
s
+
s
v
∆
v
+
t
s
u
∆
u
r
(
t
)
=
s
+
t
s
v
∆
v
s
+
s
u
∆
u
+
t
s
v
∆
v
F
+
F
u
t
∆
u
F
+
F
u
t
∆
u
+
F
v
∆
v
F
+
F
u
∆
u
+
F
v
t
∆
v
F
(
r
(
t
))
=
F
+
F
v
t
∆
v
s
s
+
s
u
∆
u
s
+
s
u
∆
u
+
s
v
∆
v
s
+
s
v
∆
v
W
e
app
ro
ximate
F
(
r
i
(
t
))
b
y
L
(
u
,
v
)
=
F
+
F
u
du
+
F
v
dv
.
499
Question
16.9.1
How
Do
We
Extrapolate
Green’s
Theorem
to
Surfaces?
Let
s
(
u
,
v
)
be
a
parameterization
of
S
.
We
will
parameterize
the
edges
of
pa
rallelogram
that
results
from
a
change
of
∆
u
and
∆
v
in
the
linea
rization
of
s
(
u
,
v
).
The
domains
are
all
0
≤
t
≤
1.
r
′
(
t
)
z
}|
{
s
+
t
s
u
∆
u
s
+
s
v
∆
v
+
t
s
u
∆
u
r
(
t
)
=
s
+
t
s
v
∆
v
s
+
s
u
∆
u
+
t
s
v
∆
v
F
+
F
u
t
∆
u
F
+
F
u
t
∆
u
+
F
v
∆
v
F
+
F
u
∆
u
+
F
v
t
∆
v
F
(
r
(
t
))
=
F
+
F
v
t
∆
v
s
s
+
s
u
∆
u
s
+
s
u
∆
u
+
s
v
∆
v
s
+
s
v
∆
v
W
e
app
ro
ximate
F
(
r
i
(
t
))
b
y
L
(
u
,
v
)
=
F
+
F
u
du
+
F
v
dv
.
499
Question
16.9.1
How
Do
We
Extrapolate
Green’s
Theorem
to
Surfaces?
Let
s
(
u
,
v
)
be
a
parameterization
of
S
.
We
will
parameterize
the
edges
of
pa
rallelogram
that
results
from
a
change
of
∆
u
and
∆
v
in
the
linea
rization
of
s
(
u
,
v
).
The
domains
are
all
0
≤
t
≤
1.
r
′
(
t
)
z
}|
{
s
+
t
s
u
∆
u
s
+
s
v
∆
v
+
t
s
u
∆
u
r
(
t
)
=
s
+
t
s
v
∆
v
s
+
s
u
∆
u
+
t
s
v
∆
v
F
+
F
u
t
∆
u
F
+
F
u
t
∆
u
+
F
v
∆
v
F
+
F
u
∆
u
+
F
v
t
∆
v
F
(
r
(
t
))
=
F
+
F
v
t
∆
v
s
s
+
s
u
∆
u
s
+
s
u
∆
u
+
s
v
∆
v
s
+
s
v
∆
v
W
e
app
ro
ximate
F
(
r
i
(
t
))
b
y
L
(
u
,
v
)
=
F
+
F
u
du
+
F
v
dv
.
499
Question
16.9.1
How
Do
We
Extrapolate
Green’s
Theorem
to
Surfaces?
Let
s
(
u
,
v
)
be
a
parameterization
of
S
.
We
will
parameterize
the
edges
of
pa
rallelogram
that
results
from
a
change
of
∆
u
and
∆
v
in
the
linea
rization
of
s
(
u
,
v
).
The
domains
are
all
0
≤
t
≤
1.
r
′
(
t
)
z
}|
{
s
+
t
s
u
∆
u
s
+
s
v
∆
v
+
t
s
u
∆
u
r
(
t
)
=
s
+
t
s
v
∆
v
s
+
s
u
∆
u
+
t
s
v
∆
v
F
+
F
u
t
∆
u
F
+
F
u
t
∆
u
+
F
v
∆
v
F
+
F
u
∆
u
+
F
v
t
∆
v
F
(
r
(
t
))
=
F
+
F
v
t
∆
v
s
s
+
s
u
∆
u
s
+
s
u
∆
u
+
s
v
∆
v
s
+
s
v
∆
v
W
e
app
ro
ximate
F
(
r
i
(
t
))
b
y
L
(
u
,
v
)
=
F
+
F
u
du
+
F
v
dv
.
499
Question
16.9.1
How
Do
We
Extrapolate
Green’s
Theorem
to
Surfaces?
Let
s
(
u
,
v
)
be
a
parameterization
of
S
.
We
will
parameterize
the
edges
of
pa
rallelogram
that
results
from
a
change
of
∆
u
and
∆
v
in
the
linea
rization
of
s
(
u
,
v
).
The
domains
are
all
0
≤
t
≤
1.
r
′
(
t
)
z
}|
{
s
+
t
s
u
∆
u
s
+
s
v
∆
v
+
t
s
u
∆
u
r
(
t
)
=
s
+
t
s
v
∆
v
s
+
s
u
∆
u
+
t
s
v
∆
v
F
+
F
u
t
∆
u
F
+
F
u
t
∆
u
+
F
v
∆
v
F
+
F
u
∆
u
+
F
v
t
∆
v
F
(
r
(
t
))
=
F
+
F
v
t
∆
v
s
s
+
s
u
∆
u
s
+
s
u
∆
u
+
s
v
∆
v
s
+
s
v
∆
v
W
e
app
ro
ximate
F
(
r
i
(
t
))
b
y
L
(
u
,
v
)
=
F
+
F
u
du
+
F
v
dv
.
499
Question
16.9.1
How
Do
We
Extrapolate
Green’s
Theorem
to
Surfaces?
Z
C
i
F
·
d
r
i
≈
Z
1
0
(
F
+
F
u
t
∆
u
)
·
(
s
u
∆
u
)
dt
+
Z
1
0
(
F
+
F
u
∆
u
+
F
v
t
∆
v
)
·
(
s
v
∆
v
)
dt
−
Z
1
0
(
F
+
F
v
∆
v
+
F
u
t
∆
u
)
·
(
s
u
∆
u
)
dt
−
Z
1
0
(
F
+
F
v
t
∆
v
)
·
(
s
v
∆
v
)
dt
=
−
Z
1
0
F
v
∆
v
·
(
s
u
∆
u
)
dt
+
Z
1
0
F
u
∆
u
·
(
s
v
∆
v
)
dt
=
Z
1
0
(
F
u
∆
u
)
·
(
s
v
∆
v
)
−
(
F
v
∆
v
)
·
(
s
u
∆
u
)
dt
=
Z
1
0
(
F
u
·
s
v
−
F
v
·
s
u
)∆
v
∆
udt
=
(
F
u
·
s
v
−
F
v
·
s
u
)∆
v
∆
u
500
Question
16.9.1
How
Do
We
Extrapolate
Green’s
Theorem
to
Surfaces?
Z
C
i
F
·
d
r
i
≈
Z
1
0
(
F
+
F
u
t
∆
u
)
·
(
s
u
∆
u
)
dt
+
Z
1
0
(
F
+
F
u
∆
u
+
F
v
t
∆
v
)
·
(
s
v
∆
v
)
dt
−
Z
1
0
(
F
+
F
v
∆
v
+
F
u
t
∆
u
)
·
(
s
u
∆
u
)
dt
−
Z
1
0
(
F
+
F
v
t
∆
v
)
·
(
s
v
∆
v
)
dt
=
−
Z
1
0
F
v
∆
v
·
(
s
u
∆
u
)
dt
+
Z
1
0
F
u
∆
u
·
(
s
v
∆
v
)
dt
=
Z
1
0
(
F
u
∆
u
)
·
(
s
v
∆
v
)
−
(
F
v
∆
v
)
·
(
s
u
∆
u
)
dt
=
Z
1
0
(
F
u
·
s
v
−
F
v
·
s
u
)∆
v
∆
udt
=
(
F
u
·
s
v
−
F
v
·
s
u
)∆
v
∆
u
500
Question
16.9.1
How
Do
We
Extrapolate
Green’s
Theorem
to
Surfaces?
Z
C
i
F
·
d
r
i
≈
Z
1
0
(
F
+
F
u
t
∆
u
)
·
(
s
u
∆
u
)
dt
+
Z
1
0
(
F
+
F
u
∆
u
+
F
v
t
∆
v
)
·
(
s
v
∆
v
)
dt
−
Z
1
0
(
F
+
F
v
∆
v
+
F
u
t
∆
u
)
·
(
s
u
∆
u
)
dt
−
Z
1
0
(
F
+
F
v
t
∆
v
)
·
(
s
v
∆
v
)
dt
=
−
Z
1
0
F
v
∆
v
·
(
s
u
∆
u
)
dt
+
Z
1
0
F
u
∆
u
·
(
s
v
∆
v
)
dt
=
Z
1
0
(
F
u
∆
u
)
·
(
s
v
∆
v
)
−
(
F
v
∆
v
)
·
(
s
u
∆
u
)
dt
=
Z
1
0
(
F
u
·
s
v
−
F
v
·
s
u
)∆
v
∆
udt
=
(
F
u
·
s
v
−
F
v
·
s
u
)∆
v
∆
u
500
Question
16.9.1
How
Do
We
Extrapolate
Green’s
Theorem
to
Surfaces?
Z
C
i
F
·
d
r
i
≈
Z
1
0
(
F
+
F
u
t
∆
u
)
·
(
s
u
∆
u
)
dt
+
Z
1
0
(
F
+
F
u
∆
u
+
F
v
t
∆
v
)
·
(
s
v
∆
v
)
dt
−
Z
1
0
(
F
+
F
v
∆
v
+
F
u
t
∆
u
)
·
(
s
u
∆
u
)
dt
−
Z
1
0
(
F
+
F
v
t
∆
v
)
·
(
s
v
∆
v
)
dt
=
−
Z
1
0
F
v
∆
v
·
(
s
u
∆
u
)
dt
+
Z
1
0
F
u
∆
u
·
(
s
v
∆
v
)
dt
=
Z
1
0
(
F
u
∆
u
)
·
(
s
v
∆
v
)
−
(
F
v
∆
v
)
·
(
s
u
∆
u
)
dt
=
Z
1
0
(
F
u
·
s
v
−
F
v
·
s
u
)∆
v
∆
udt
=
(
F
u
·
s
v
−
F
v
·
s
u
)∆
v
∆
u
500
Question
16.9.1
How
Do
We
Extrapolate
Green’s
Theorem
to
Surfaces?
Z
C
i
F
·
d
r
i
≈
Z
1
0
(
F
+
F
u
t
∆
u
)
·
(
s
u
∆
u
)
dt
+
Z
1
0
(
F
+
F
u
∆
u
+
F
v
t
∆
v
)
·
(
s
v
∆
v
)
dt
−
Z
1
0
(
F
+
F
v
∆
v
+
F
u
t
∆
u
)
·
(
s
u
∆
u
)
dt
−
Z
1
0
(
F
+
F
v
t
∆
v
)
·
(
s
v
∆
v
)
dt
=
−
Z
1
0
F
v
∆
v
·
(
s
u
∆
u
)
dt
+
Z
1
0
F
u
∆
u
·
(
s
v
∆
v
)
dt
=
Z
1
0
(
F
u
∆
u
)
·
(
s
v
∆
v
)
−
(
F
v
∆
v
)
·
(
s
u
∆
u
)
dt
=
Z
1
0
(
F
u
·
s
v
−
F
v
·
s
u
)∆
v
∆
udt
=(
F
u
·
s
v
−
F
v
·
s
u
)∆
v
∆
u
500
Question
16.9.1
How
Do
We
Extrapolate
Green’s
Theorem
to
Surfaces?
(
F
u
·
s
v
−
F
v
·
s
u
)
=
∂
P
∂
x
∂
x
∂
u
+
∂
P
∂
y
∂
y
∂
u
+
∂
P
∂
z
∂
z
∂
u
∂
x
∂
v
+
∂
Q
∂
x
∂
x
∂
u
+
∂
Q
∂
y
∂
y
∂
u
+
∂
Q
∂
z
∂
z
∂
u
∂
y
∂
v
+
∂
R
∂
x
∂
x
∂
u
+
∂
R
∂
y
∂
y
∂
u
+
∂
R
∂
z
∂
z
∂
u
∂
z
∂
v
−
∂
P
∂
x
∂
x
∂
v
+
∂
P
∂
y
∂
y
∂
v
+
∂
P
∂
z
∂
z
∂
v
∂
x
∂
u
−
∂
Q
∂
x
∂
x
∂
v
+
∂
Q
∂
y
∂
y
∂
v
+
∂
Q
∂
z
∂
z
∂
v
∂
y
∂
u
−
∂
R
∂
x
∂
x
∂
v
+
∂
R
∂
y
∂
y
∂
v
+
∂
R
∂
z
∂
z
∂
v
∂
z
∂
u
501
Question
16.9.1
How
Do
We
Extrapolate
Green’s
Theorem
to
Surfaces?
(
F
u
·
s
v
−
F
v
·
s
u
)
=
∂
R
∂
y
−
∂
Q
∂
z
∂
y
∂
u
∂
z
∂
v
−
∂
z
∂
u
∂
y
∂
v
+
∂
P
∂
z
−
∂
R
∂
x
∂
x
∂
u
∂
z
∂
v
−
∂
z
∂
u
∂
x
∂
v
+
∂
Q
∂
x
−
∂
P
∂
y
∂
x
∂
u
∂
y
∂
v
−
∂
y
∂
u
∂
x
∂
v
=
(
∇
×
F
)
·
(
s
u
×
s
v
)
502
Question
16.9.1
How
Do
We
Extrapolate
Green’s
Theorem
to
Surfaces?
(
F
u
·
s
v
−
F
v
·
s
u
)
=
∂
R
∂
y
−
∂
Q
∂
z
∂
y
∂
u
∂
z
∂
v
−
∂
z
∂
u
∂
y
∂
v
+
∂
P
∂
z
−
∂
R
∂
x
∂
x
∂
u
∂
z
∂
v
−
∂
z
∂
u
∂
x
∂
v
+
∂
Q
∂
x
−
∂
P
∂
y
∂
x
∂
u
∂
y
∂
v
−
∂
y
∂
u
∂
x
∂
v
=
(
∇
×
F
)
·
(
s
u
×
s
v
)
502
Question
16.9.1
How
Do
We
Extrapolate
Green’s
Theorem
to
Surfaces?
Using
this
computation
w
e
can
see
what
happ
ens
as
w
e
let
the
size
of
our
sub
divisions
app
roach
0.
Z
C
F
·
d
r
=
X
i
Z
C
i
F
·
d
r
i
=
lim
∆
u
,
∆
v
→
0
X
i
(
F
u
·
s
v
−
F
v
·
s
u
)∆
v
∆
u
=
lim
∆
u
,
∆
v
→
0
X
i
(
∇
×
F
)
·
(
s
u
×
s
v
)∆
v
∆
u
=
Z
Z
S
(
∇
×
F
)
·
d
S
503
Question
16.9.1
How
Do
We
Extrapolate
Green’s
Theorem
to
Surfaces?
Using
this
computation
w
e
can
see
what
happ
ens
as
w
e
let
the
size
of
our
sub
divisions
app
roach
0.
Z
C
F
·
d
r
=
X
i
Z
C
i
F
·
d
r
i
=
lim
∆
u
,
∆
v
→
0
X
i
(
F
u
·
s
v
−
F
v
·
s
u
)∆
v
∆
u
=
lim
∆
u
,
∆
v
→
0
X
i
(
∇
×
F
)
·
(
s
u
×
s
v
)∆
v
∆
u
=
Z
Z
S
(
∇
×
F
)
·
d
S
503
Question
16.9.1
How
Do
We
Extrapolate
Green’s
Theorem
to
Surfaces?
Using
this
computation
w
e
can
see
what
happ
ens
as
w
e
let
the
size
of
our
sub
divisions
app
roach
0.
Z
C
F
·
d
r
=
X
i
Z
C
i
F
·
d
r
i
=
lim
∆
u
,
∆
v
→
0
X
i
(
F
u
·
s
v
−
F
v
·
s
u
)∆
v
∆
u
=
lim
∆
u
,
∆
v
→
0
X
i
(
∇
×
F
)
·
(
s
u
×
s
v
)∆
v
∆
u
=
Z
Z
S
(
∇
×
F
)
·
d
S
503
Question
16.9.1
How
Do
We
Extrapolate
Green’s
Theorem
to
Surfaces?
Using
this
computation
w
e
can
see
what
happ
ens
as
w
e
let
the
size
of
our
sub
divisions
app
roach
0.
Z
C
F
·
d
r
=
X
i
Z
C
i
F
·
d
r
i
=
lim
∆
u
,
∆
v
→
0
X
i
(
F
u
·
s
v
−
F
v
·
s
u
)∆
v
∆
u
=
lim
∆
u
,
∆
v
→
0
X
i
(
∇
×
F
)
·
(
s
u
×
s
v
)∆
v
∆
u
=
Z
Z
S
(
∇
×
F
)
·
d
S
503
Question
16.9.1
How
Do
We
Extrapolate
Green’s
Theorem
to
Surfaces?
Definition
A
pa
rametrized
surface
S
with
a
b
ounda
ry
curve
C
has
p
ositive
o
rientation
if
the
rotation
of
C
and
the
direction
of
d
S
ob
ey
the
right
hand
rule.
Theo
rem
(Stokes’
Theorem)
If
S
is
a
smo
oth
surface
that
is
b
ounded
b
y
a
simple
closed
b
ounda
ry
curve
C
with
p
ositive
o
rientation
and
F
is
a
vecto
r
field
with
continuous
pa
rtial
derivatives,
then
Z
C
F
·
d
r
=
Z
Z
S
(
∇
×
F
)
·
d
S
If
the
surface
is
negatively
o
riented,
Stok
es’
Theo
rem
can
b
e
salvaged
b
y
intro
ducing
a
minus
sign.
504
Question
16.9.1
How
Do
We
Extrapolate
Green’s
Theorem
to
Surfaces?
Much
lik
e
Green’s
theo
rem,
Stok
es’
theo
rem
is
understo
o
d
as
adding
up
the
extent
to
which
the
vecto
r
field
curls
a
round
each
p
oint
to
get
the
total
w
o
rk
a
round
the
b
ounda
ry
.
The
dot
product
measures
the
extent
to
which
the
overall
curl
of
F
takes
place
in
the
surface.
Click to Load Applet
505
Example
16.9.2
Applying
Stok
es’
Theo
rem
Let
C
b
e
the
curve
given
by
r
(
t
)
=
cos(
t
)
i
+
sin(
t
)
j
+
(cos
2
(
t
)
−
sin
2
(
t
))
k
.
Let
F
(
x
,
y
,
z
)
=
xy
k
.
Ho
w
do
es
Stok
es’
Theo
rem
apply
to
Z
C
F
·
d
r
?
506
Example
16.9.2
Applying
Stok
es’
Theo
rem
Let
C
b
e
the
curve
given
by
r
(
t
)
=
cos(
t
)
i
+
sin(
t
)
j
+
(cos
2
(
t
)
−
sin
2
(
t
))
k
.
Let
F
(
x
,
y
,
z
)
=
xy
k
.
Ho
w
do
es
Stok
es’
Theo
rem
apply
to
Z
C
F
·
d
r
?
Click to Load Applet
506
Example
16.9.2
Applying
Stokes’
Theo
rem
Exercise
Supp
ose
that
F
is
a
conservative
vector
field
on
R
3
and
C
is
a
smo
oth
curve
that
b
ounds
a
p
ositively-o
riented
surface
S
.
a
Can
the
F
undamental
Theo
rem
of
Line
Integrals
compute
R
C
F
·
d
r
?
What
is
the
value?
b
Use
y
our
cha
racterization
of
F
from
a
to
compute
∇
×
F
.
507
Example
16.9.3
Stok
es’s
Theo
rem
on
a
Closed
Surface
Let
F
(
x
,
y
,
z
)
be
a
smo
oth
vector
field
on
R
3
.
Let
S
b
e
a
sphere.
What
is
R
R
S
(
∇
×
F
)
·
d
S
?
508
Example
16.9.3
Stokes’s
Theorem
on
a
Closed
Surface
Exercise
Supp
ose
S
is
part
of
the
paraboloid
z
=
16
−
x
2
−
y
2
ab
ove
the
xy
-plane,
and
C
is
its
b
oundary
.
Is
there
an
easier
w
a
y
to
compute
R
R
S
F
·
d
S
?
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509
Application
16.9.4
F
arada
y’s
La
w
of
Induction
F
a
rada
y’s
la
w
of
induction
sa
ys
that
the
change
in
magnetic
field
through
a
surface
S
induces
an
electromotive
force
through
a
wire
on
its
b
ounda
ry
C
.
Physicists
measure
the
induced
voltage
b
y
integrating
the
change
in
magnetic
field
d
S
.
510
Application
16.9.4
../im
gicons/rocket.pdf
Fa
raday’s
Law
of
Induction
Summa
ry
Questions
What
t
w
o
t
yp
es
of
integrals
do
es
Stok
es’
Theo
rem
equate?
What
do
es
p
ositive
o
rientation
mean?
511
>
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