Chapter Eleven
Partial Derivatives
Section 11.1
Functions of Several Variables

Goals

Study functions of two or more variables from
four points of view

Discuss visual representations

Describe functions of three or more variables
Four Points of View

We can study functions of two or more
variables from four viewpoints:

Verbally

Numerically

Algebraically

Visually
Example

The wind-chill index is often used to describe the
apparent severity of the cold.

The index W depends on the actual temperature
T and the wind speed v.

A table of values of W(T, v).

f(–5, 50) = –15.
Domain and Range

Recall that for a function f(x, y) given by an
algebraic formula, the domain consists of all
pairs (x, y) for which the expression for f(x, y) is
a well-defined real number.

As an example we find the domain and range of
g x , y  

9x y :
2
2
Solution The domain of g is
D  x , y |9  x  y  0  x , y | x  y  9
2

2
2
2
This is the disk with center (0, 0) and radius 3.

The range of g is
z| z 

9  x  y , x , y   D
2
Since z is a positive square root, z ≥ 0. Also
9x y 9 
2

2
2
9x y 3
2
2
So the range is z |0  z  3  0 ,3 .
Visual Representations

One way to visualize a function of two variables is
through its graph.

For example, we sketch the graph of
g x , y  
9x y :
2
2

Solution Squaring both sides of this equation gives x2 +
y2 + z2 = 9, which is an equation of the sphere with center
the origin and radius 3.

Since z ≥ 0, the graph of g is just
the top half of this sphere.
Level Curves

Another way to see functions is a contour map on which
points of constant elevation are joined to form level
curves:

The level curves f(x, y) = k are traces of the graph of f in
the plane z = k projected down to the xy-plane:

A common example of level curves occurs in
topographical maps of mountainous regions, as
shown on the next slide.

The level curves are curves of constant elevation
above sea level.

If we walk along one of these contour lines we
neither ascend nor descend.
Example

The figure shows a contour map for a function.

Use it to estimate f(1, 3) and f(4, 5).

The point (1, 3) lies part way between the level curves
with z-values 70 and 80.

So we estimate that f (1, 3) ≈ 73.

Similarly, we estimate that f (4, 5) ≈ 56.
Example

Sketch the level curves of the function
g x , y  

9  x  y for k  0 ,1,2 ,3
2
2
Solution The level curves are
9  x  y  k or x  y  9  k
2
2
2
2
2

This is a family of concentric circles with center (0, 0) and
2
radius
9k .

The cases k = 0, 1, 2, 3 are shown:
Example

Sketch some level curves of the function h(x, y) = 4x2 + y2.

Solution The level curves are
4 x  y  k or
2
2
x
2
k/4

y
2
k
1
which, for k > 0, describes a family of ellipses with
semiaxes
k / 2 and k .

Shown below are these level curves lifted up to the
graph of h:
Three or More Variables

A function of three variables, f, is a rule that
assigns to each ordered triple (x, y, z) in a
domain D in space a unique real number
denoted by f(x, y, z).

For instance, the temperature T at a point
on the surface of the Earth depends on the
longitude x and latitude y of the point and
on the time t, so T = f(x, y, t).
Example

Find the domain of f if
f(x, y, z) = ln (z – y) + xy sin z.

Solution The expression for f(x, y, z) is
defined as long as z – y > 0, so the domain
of f is the half-space
Level Surfaces

To gain insight into a function f of three
variables we can examine its level surfaces,
which are the surfaces with equations
f(x, y, z) = k, where k is a constant.

If the point (x, y, z) moves along a level
surface, the value of f(x, y, z) remains
fixed.
Example

Find the level surfaces of the function
f(x, y, z) = x2 + y2 + z2

Solution The level surfaces are
x2 + y2 + z2 = k, where k ≥ 0.

These form a family of concentric spheres with
radius k , as the next slide shows :
More Variables

Functions of any number of variables can be considered.

A function of n variables is a rule that assigns a number
z = f(x1, x2,…, xn) to an n-tuple (x1, x2,…, xn) of real
numbers.

Sometimes we use vector notation in order to write
functions more compactly:

If x  x , x , , x , we often write
f(x) in place of f(x1, x2,…, xn).

So there are three ways of viewing a function of n
variables: As a function of…
1
2
n

n real variables x1, x2,…, xn ;

a single point variable (x1, x2,…, xn);

a single vector variable x  x1 , x2 , , xn
Review

Four ways of viewing functions of two
variables

Visual representations


Graphs

Level curves
Functions of three or more variables

Level surfaces