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

   9 

x

2 

y

2 :  Solution The domain of g is

D

    | 9 

x

2 

y

2  0      |

x

2 

y

2  9   This is the disk with center (0, 0) and radius 3.

 The range of g is 

z

|

z

 9 

x

2 

y

2 , 

D

  Since z is a positive square root, z ≥ 0. Also  9 

x

2 

y

2 So the range is  9  

z

| 0 

z

9 

x

2 

y

2  3     .

 3

Visual Representations

   One way to visualize a function of two variables is through its graph.

For example, we sketch the graph of

g

 9 

x

2 

y

2 : Solution Squaring both sides of this equation gives x 2

y

2 + z 2 + = 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

 9 

x

2 

y

2 for

k

 0 , 1 , 2 , 3  Solution The level curves are   9 

x

2 

y

2 

k

or

x

2 

y

2  9 

k

2 This is a family of concentric circles with center (0, 0) and radius 9 

k

2 .

The cases k = 0, 1, 2, 3 are shown:

Example

   Sketch some level curves of the function h(x, y) = 4x 2 + y 2 .

Solution The level curves are 4

x

2 

y

2 

k

or

k x

/ 2 4 

y

2  1

k

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) = x 2 + y 2 + z 2   Solution The level surfaces are

x

2 + y 2 + z 2 = 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(x 1 , x 2 ,…, x

n

) to an n-tuple (x 1 , x 2 ,…, x

n

) of real numbers.

Sometimes we use vector notation in order to write functions more compactly:   If

x



x

1 ,

x

2 ,  ,

x n

, we often write f(x) in place of f(x 1 , x 2 ,…, x

n

).

So there are three ways of viewing a function of n variables: As a function of…    n real variables x 1 , x 2 ,…, x

n

; a single point variable (x 1 , x 2 ,…, x

n

); a single vector variable

x



x

1 ,

x

2 ,  ,

x n

Review

 Four ways of viewing functions of two variables  Visual representations  Graphs  Level curves  Functions of three or more variables  Level surfaces