Math 260, Spring 2012
Jerry L. Kazdan
Class Outline: March 1, 2012
Functions of Several Variables
Reading:
Marsden and Tromba,
Vector Calculus
, Chapter 2, Chapter 3.1–3.4 and
Sections 8.1-8.2 in the notes
http://www.math.upenn.edu/
∼
kazdan/260S12/notes/math21/math21-2012-2up.pdf
Exam. 2
will be on Tuesday, March 13 during class. As usual, it will be closed book,
no calculators or cell phones etc., but you may use one 3
×
5 card with notes on both
sides.
The Exam will cover the mater covered through class on Tuesday, Feb.
28, with
emphasis on the material since Exam. 1.
1.
The Chain Rule
a)
Simplest Case
Say you have a function
f
(
X
) :=
f
(
x,y,z
) which might
givde the temperature at a point
X
= (
x,y,z
) in
R
3
.
If we have a curve
X
(
t
) = (
x
(
t
)
,y
(
t
)
,z
(
t
)), then
h
(
t
) :=
f
(
X
(
t
)) gives the temperature at points
of the curve. We want to compute
dh/dt
. So we need the
chain rule
. For
h
(
t
) =
f
(
x
(
t
)
,y
(
t
)
,z
(
t
)) it states
dh
(
t
)
dt
=
∂f
∂x
dx
dt
+
∂f
∂y
dy
dt
+
∂f
∂z
dz
dt
=
∇
f
(
X
(
t
))
·
X
′
(
t
)
.
Example
f
(
X
) :=
x
2
y
+
e
2
yz
and the curve is
X
(
t
) = (cos
t,
sin
t,
2
−
t
). If
h
(
t
) :=
f
(
X
(
t
)) then
∇
f
(
X
) = (2
xy,x
2
+ 2
ze
2
yz
,
2
ye
2
yz
)
and
X
′
(
t
) = (
−
sin
t,
cos
t,
−
1)
.
Thus, for instance at
t
= 0,
X
(0) = (1
,
0
,
2),
X
′
(0) = (0
,
1
,
−
1) and
∇
f
(
X
(0)) = (0
,
5
,
0) so
h
′
(0) = 5.
Application
Say we have a surface in
R
3
on which
f
(
X
) is constant,
f
(
x,y,z
) =
c
.
This is called a
level surface
of
f
(if
f
(
X
) specifies the
temperature, this surface is often called an
isotherm
). Let
X
(
t
) be a smooth
curve on this surface, so
f
(
X
(
t
)) =
c
. Since
c
is a constant, if we take the
derivative of this we find
Theorem
On any level surface,
∇
f
(
X
)
·
X
′
(
t
) = 0
so the vector
∇
f
(
X
)
is
orthogonal to the surface
.
Proof
Since
X
(
t
) is a curve on the surface, its tangent vector,
X
′
(
t
) is tan-
gent to the surface. But
∇
f
(
X
)
·
X
′
(
t
) = 0 means that
∇
f
(
X
) is orthogonal
to all these tangent vectors and hence to the surface.
Example
Find the tangent plane to the surface
z
=
x
2
+
y
3
at the point
X
0
:= (
−
1
,
1
,
2).