14
Partial Differentiation
In singlevariable calculus we were concerned with functions that map the real numbers
R
to
R
, sometimes called “real functions of one variable”, meaning the “input” is a single real
number and the “output” is likewise a single real number. In the last chapter we considered
functions taking a real number to a vector, which may also be viewed as functions
f
:
R
→
R
3
, that is, for each input value we get a position in space.
Now we turn to functions
of several variables, meaning several input variables, functions
f
:
R
n
→
R
. We will deal
primarily with
n
= 2 and to a lesser extent
n
= 3; in fact many of the techniques we
discuss can be applied to larger values of
n
as well.
A function
f
:
R
2
→
R
maps a pair of values (
x, y
) to a single real number. The three
dimensional coordinate system we have already used is a convenient way to visualize such
functions: above each point (
x, y
) in the
x

y
plane we graph the point (
x, y, z
), where of
course
z
=
f
(
x, y
).
EXAMPLE 14.1
Consider
f
(
x, y
) = 3
x
+ 4
y
−
5. Writing this as
z
= 3
x
+ 4
y
−
5 and
then 3
x
+4
y
−
z
= 5 we recognize the equation of a plane. In the form
f
(
x, y
) = 3
x
+4
y
−
5
the emphasis has shifted: we now think of
x
and
y
as independent variables and
z
as a
variable dependent on them, but the geometry is unchanged.
EXAMPLE 14.2
We have seen that
x
2
+
y
2
+
z
2
= 4 represents a sphere of radius 2.
We cannot write this in the form
f
(
x, y
), since for each
x
and
y
in the disk
x
2
+
y
2
<
4 there
are two corresponding points on the sphere. As with the equation of a circle, we can resolve
323
324
Chapter 14 Partial Differentiation
this equation into two functions,
f
(
x, y
) =
radicalbig
4
−
x
2
−
y
2
and
f
(
x, y
) =
−
radicalbig
4
−
x
2
−
y
2
,
representing the upper and lower hemispheres. Each of these is an example of a function
with a restricted domain: only certain values of
x
and
y
make sense (namely, those for
which
x
2
+
y
2
≤
4) and the graphs of these functions are limited to a small region of the
plane.
EXAMPLE 14.3
Consider
f
=
√
x
+
√
y
. This function is defined only when both
x
and
y
are nonnegative. When
y
= 0 we get
f
(
x, y
) =
√
x
, the familiar square root function
in the
x

z
plane, and when
x
= 0 we get the same curve in the
y

z
plane.
Generally
speaking, we see that starting from
f
(0
,
0) = 0 this function gets larger in every direction
in roughly the same way that the square root function gets larger.
For example, if we
restrict attention to the line
x
=
y
, we get
f
(
x, y
) = 2
√
x
and along the line
y
= 2
x
we
have
f
(
x, y
) =
√
x
+
√
2
x
= (1 +
√
2)
√
x
.
10.0
7.5
5.0
0
0.0
y
2.5
1
2.5
5.0
x
2
7.5
0.0
10.0
3
4
5
6
Figure 14.1
f
(
x, y
) =
√
x
+
√
y
(JA)
A computer program that plots such surfaces can be very useful, as it is often difficult
to get a good idea of what they look like.
Still, it is valuable to be able to visualize
relatively simple surfaces without such aids. As in the previous example, it is often a good
idea to examine the function on restricted subsets of the plane, especially lines. It can also
be useful to identify those points (
x, y
) that share a common
z
value.