FUNCTIONS OF SEVERAL VARIABLES
The temperature at a point on the surface of the Earth at any given time depends on the longitude and latitude of the point. We can think of as being a function of the two variables and , or as a function of the pair . We indicate this functional dependence by writing .
The volume of a circular cylinder depends on its radius and its height . In fact, we know that . We say that is a function of and , and we write
.
DEFINITION A function of two variables is a rule that assigns to each ordered pair of real numbers in a set a unique real number denoted by
. The set is the domain of and its range is the set of values that
takes on, that is, .
We often write to make explicit the value taken on by at the general point . The variables and are independent variables and is the dependent variable. [Compare this with the notation for functions of a single variable.]
A function of two variables is just a function whose domain is a subset of and whose range is a subset of . One way of visualizing such a function is by means of an arrow diagram (see Figure 1), where the domain is represented as a subset of the
-plane.
If a function is given by a formula and no domain is specified, then the domain of is understood to be the set of all pairs for which the given expression is a well-defined real number.
x, y
f
f
FIGURE 1
y
x
0 0 z
D
f f (a, b)
f (x, y) (x, y)
(a, b)
xy
D
⺢ y y f x z ⺢2
x, y z f x, yx f
f x, y
x, y D f fD
fx, y f x, y D
Vr, h r2h
h r V
Vr2h
h r
V
T f x, y x, y
y x
T y
x T
11.1
PARTIAL DERIVATIVES
So far we have dealt with the calculus of functions of a single variable. But, in the real world, physi- cal quantities often depend on two or more variables, so in this chapter we turn our attention to functions of several variables and extend the basic ideas of differential calculus to such functions.
11
591
EXAMPLE 1 Find the domains of the following functions and evaluate .
(a) (b)
SOLUTION
(a)
The expression for makes sense if the denominator is not 0 and the quantity under the square root sign is nonnegative. So the domain of is
The inequality , or , describes the points that lie on or above the line , while means that the points on the line must be excluded from the domain. (See Figure 2.)
(b)
Since is defined only when , that is, , the domain of is . This is the set of points to the left of the parabola .
(See Figure 3.) ■
EXAMPLE 2 Find the domain and range of . SOLUTION The domain of is
which is the disk with center and radius 3 (see Figure 4). The range of is
Since is a positive square root, . Also
So the range is
■
GRAPHS
Another way of visualizing the behavior of a function of two variables is to consider its graph.
DEFINITION If is a function of two variables with domain D, then the graph of is the set of all points in such that and is in D.
x, y
z f x, y
⺢3
x, y, z
f f
z
0 z 3 0, 3s9 x2 y2 3
? 9 x2 y2 9
z 0 z
z z s9 x2 y2,x, y Dt
0, 0
D x, y
9 x2 y2 0 x, y x2 y2 9t
tx, y s9 x2 y2
x y2
D x, y
x y2 x y fy2 x 0 2
lny2 x
f3, 2 3 ln22 3 3 ln 1 0
x 1 x 1
yx x 1 y 1 0 y x 1
D x, y
x y 1 0, x 1 f ff3, 2 s3 2 1
3 1 s6
2
fx, y x lny2 x
fx, y sx y 1 x 1
f3, 2
FIGURE 2
œ„„„„„„„
x-1 x+y+1 Domain of f(x, y)=
x 0
y
_1 _1
x=1 x+y+1=0
FIGURE 3
Domain of f(x, y)=x ln(¥-x) x 0
y
x=¥
≈+¥=9
3 _3
FIGURE 4
Domain of g(x, y)=œ„„„„„„„„„9-≈-¥
x y
Just as the graph of a function of one variable is a curve with equation so the graph of a function of two variables is a surface with equation . We can visualize the graph of as lying directly above or below its domain in the -plane (see Figure 5).
EXAMPLE 3 Sketch the graph of the function .
SOLUTION The graph of has the equation , or , which represents a plane. To graph the plane we first find the intercepts. Putting
in the equation, we get as the -intercept. Similarly, the -intercept is 3 and the -intercept is 6. This helps us sketch the portion of the graph that lies in
the first octant (Figure 6). ■
The function in Example 3 is a special case of the function
which is called a linear function. The graph of such a function has the equation , or , so it is a plane. In much the same way that linear functions of one variable are important in single-variable calculus, we will see that linear functions of two variables play a central role in multivariable calculus.
EXAMPLE 4 Sketch the graph of .
SOLUTION The graph has equation . We square both sides of this
equation to obtain , or , which we recognize as
an equation of the sphere with center the origin and radius 3. But, since , the graph of is just the top half of this sphere (see Figure 7).
■ EXAMPLE 5 Find the domain and range and sketch the graph of
.
SOLUTION Notice that is defined for all possible ordered pairs of real num- bers , so the domain is , the entire xy-plane. The range of h is the set of all nonnegative real numbers. [Notice that and , so for all x and y.]
The graph of h has the equation , which is the elliptic paraboloid that we sketched in Example 4 in Section 10.6. Horizontal traces are ellipses and
vertical traces are parabolas (see Figure 8). ■
Computer programs are readily available for graphing functions of two variables.
In most such programs, traces in the vertical planes and are drawn for equally spaced values of and parts of the graph are eliminated using hidden line removal.
k
y k x k
z 4x2 y2
hx, y 0 y2 0
x2 0 0,
⺢2
x, y hx, y
hx, y 4x2 y2
V FIGURE 7 Graph of g(x, y)= 9-≈-¥œ„„„„„„„„„
0 (0, 3, 0) (0, 0, 3)
(3, 0, 0) y
z
x
t z2 9 x2 y2z s9 xx2 y22 z y22 9 z 0 tx, y s9 x2 y2
V
ax by z c 0 z ax by c
fx, y ax by c
z x 2 x y
y z 0
3x 2y z 6 z 6 3x 2y
f
fx, y 6 3x 2y xy
D
f
z f x, yy f x, ff S CS
FIGURE 5
f(x, y) 0
z
y x
D S
{x, y, f(x, y)}
(x, y, 0)
FIGURE 8
Graph of h(x, y)=4≈+¥
z
x y FIGURE 6
(2, 0, 0)
(0, 3, 0) z
y x
(0, 0, 6)
Figure 9 shows computer-generated graphs of several functions. Notice that we get an especially good picture of a function when rotation is used to give views from dif- ferent vantage points. In parts (a) and (b) the graph of is very flat and close to the -plane except near the origin; this is because is very small when or is large.
LEVEL CURVES
So far we have two methods for visualizing functions: arrow diagrams and graphs. A third method, borrowed from mapmakers, is a contour map on which points of con- stant elevation are joined to form contour curves, or level curves.
DEFINITION The level curves of a function of two variables are the curves with equations , where is a constant (in the range of ).
A level curve is the set of all points in the domain of at which takes on a given value . In other words, it shows where the graph of has height .
You can see from Figure 10 the relation between level curves and horizontal traces.
The level curves are just the traces of the graph of in the horizontal plane projected down to the xy-plane. So if you draw the level curves of a function
z k fx, y k f
k f
k
f f
fx, y k
f k
fx, y k f
FIGURE 9
(c) f(x, y)=sin x+sin y z
x y
x
z
y
(d) f(x, y)=sin x sin y xy (a) f(x, y)=(≈+3¥)e_≈_¥
z
x y
(b) f(x, y)=(≈+3¥)e_≈_¥
x z
y x ex2 y2
xy
f
and visualize them being lifted up to the surface at the indicated height, then you can mentally piece together a picture of the graph. The surface is steep where the level curves are close together. It is somewhat flatter where they are farther apart.
One common example of level curves occurs in topographic maps of mountainous regions, such as the map in Figure 11. The level curves are curves of constant eleva- tion above sea level. If you walk along one of these contour lines you neither ascend nor descend. Another common example is the temperature function introduced in the opening paragraph of this section. Here the level curves are called isothermals and join locations with the same temperature. Figure 12 shows a weather map of the world indicating the average January temperatures. The isothermals are the curves that sep- arate the shaded bands.
FIGURE 12 World mean sea-level temperatures in January in degrees Celsius
Tarbuck, Atmosphere: Introduction to Meteorology, 4th Edition,
© 1989. Reprinted by permission of Pearson Education, Inc., Upper Saddle River, NJ.
FIGURE 10 0
k=35 k=40
k=20 k=25 k=30 k=45
LONESOME MTN.
5000 4500
4500 4000
5000
5500
LonesomeCreek
A
B
FIGURE 11 x y
z 45
f(x, y)=20
Visual 11.1A animates Figure 10 by showing level curves being lifted up to graphs of functions.
EXAMPLE 6 A contour map for a function is shown in Figure 13. Use it to esti- mate the values of and .
SOLUTION The point (1, 3) lies partway between the level curves with -values 70 and 80. We estimate that
Similarly, we estimate that
■ EXAMPLE 7 Sketch the level curves of the function for the values , , , .
SOLUTION The level curves are
This is a family of lines with slope . The four particular level curves with
, , , and are , , , and
. They are sketched in Figure 14. The level curves are equally spaced parallel lines because the graph of is a plane (see Figure 6).
■ EXAMPLE 8 Sketch the level curves of the function
SOLUTION The level curves are
This is a family of concentric circles with center and radius . The cases , , , are shown in Figure 15. Try to visualize these level curves lifted up to form a surface and compare with the graph of (a hemisphere) in Figure 7.
(See TEC Visual 11.1A.)
■
EXAMPLE 9 Sketch some level curves of the function . SOLUTION The level curves are
which, for , describes a family of ellipses with semiaxes and . Fig- ure 16(a) shows a contour map of h drawn by a computer with level curves corre-
sk sk 2 k 0
x2 k4 y2
k 1 or
4x2 y2 k
hx, y 4x2 y2
0 k=3
k=2 k=1
k=0
3 y
x
FIGURE 15 Contour map of g(x, y)=œ„„„„„„„„„9-≈-¥
t 3
2 1
k 0 0, 0 s9 k2
x2 y2 9 k2 s9 x2 y2 k or
k 0, 1, 2, 3 tx, y s9 x2 y2 for
V
f
3x 2y 6 00 6 12 3x 2y 12 0 3x 2y 6 0 3x 2y 0
k 6 32
3x 2y k 6 0 or
6 3x 2y k 12
6 0
k 6 fx, y 6 3x 2y
f4, 5 56 f1, 3 73
z f4, 5
f1, 3 f
FIGURE 13
FIGURE 14 Contour map of f(x, y)=6-3x-2y
y
x
0 1
1 2 3 4 5
2 3 4 5
50
50 60
70 80 60
70 80
x y
k=120 k=6 k=0 k=_
6
sponding to . Figure 16(b) shows these level curves lifted up to the graph of h (an elliptic paraboloid) where they become horizontal traces.
We see from Figure 16 how the graph of h is put together from the level curves.
■ Figure 17 shows some computer-generated level curves together with the corre- sponding computer-generated graphs. Notice that the level curves in part (c) crowd together near the origin. That corresponds to the fact that the graph in part (d) is very steep near the origin.
FIGURE 17 (c) Level curves of f(x, y)= _3y
≈+¥+1 (d) f(x, y)= _3y
≈+¥+1 (b) Two views of f(x, y)=_xye_≈_¥
(a) Level curves of f(x, y)=_xye_≈_¥
y
x
z
y x
z
x y
z
x y
(a) Contour map x y
y z
x
(b) Horizontal traces are raised level curves The graph of h(x, y)=4≈+¥
is formed by lifting the level curves.
FIGURE 16
k 0.25, 0.5, 0.75, . . . , 4
Visual 11.1B demonstrates the connection between surfaces and their contour maps.
FUNCTIONS OF THREE OR MORE VARIABLES
A function of three variables, , is a rule that assigns to each ordered triple in a domain a unique real number denoted by . For instance, the temperature 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 we could write .
EXAMPLE 10 Find the domain of if . SOLUTION The expression for is defined as long as , so the domain of is
This is a half-space consisting of all points that lie above the plane . ■ It’s very difficult to visualize a function of three variables by its graph, since that would lie in a four-dimensional space. However, we do gain some insight into by examining its level surfaces, which are the surfaces with equations , where is a constant. If the point moves along a level surface, the value of
remains fixed.
EXAMPLE 11 Find the level surfaces of the function . SOLUTION The level surfaces are , where . These form a family of concentric spheres with radius . (See Figure 18.) Thus, as
varies over any sphere with center , the value of remains fixed. ■ Functions of any number of variables can be considered. A function of n variables
is a rule that assigns a number to an -tuple of
real numbers. We denote by the set of all such n-tuples. For example, if a company uses different ingredients in making a food product, is the cost per unit of the ingredient, and units of the ingredient are used, then the total cost of the ingre- dients is a function of the variables :
The function is a real-valued function whose domain is a subset of . Some- times we will use vector notation in order to write such functions more compactly: If , we often write in place of . With this nota- tion we can rewrite the function defined in Equation 1 as
where and denotes the dot product of the vectors c and x in .
In view of the one-to-one correspondence between points in and their position vectors in , we have three ways of looking at a function f defined on a subset of :
1. As a function of real variables 2. As a function of a single point variable 3. As a function of a single vector variable We will see that all three points of view are useful.
x x1, x2, . . . , xn
x1, x2, . . . , xn x1, x2, . . . , xn
n
⺢n Vn
x x1, x2, . . . , xn x1, x2, . . . , xn ⺢n Vn
c x c c1, c2, . . . , cn
fx c x
fx1, x2, . . . , xn fx
x x1, x2, . . . , xn
⺢n f
C f x1, x2, . . . , xn c1x1 c2x2 cnxn
1
x1, x2, . . . , xn
n
C ith
xi
ith ci
n
⺢nz f x1, x2, . . . , xn n x1, x2, . . . , xn fx, y, z
O
x, y, z
sk k 0
x2 y2 z2 k
fx, y, z x2 y2 z2
fx, y, zk x, y, z
fx, y, z kf f
z y D x, y, z ⺢3
z yf
z y 0 fx, y, z
fx, y, z lnz y xy sin z f
T f x, y, t
T
fx, y, z
D ⺢3 f x, y, z
FIGURE 18
≈+¥+z@=9
x
y z
≈+¥+z@=1
≈+¥+z@=4
22. Two contour maps are shown. One is for a function whose graph is a cone. The other is for a function t whose graph is a paraboloid. Which is which, and why?
Locate the points and in the map of Lonesome Moun- tain (Figure 11). How would you describe the terrain near ? Near ?
24. Make a rough sketch of a contour map for the function whose graph is shown.
25–32 ■ Draw a contour map of the function showing several level curves.
25. 26.
27. 28.
30.
31. 32.
■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■
33–34 ■ Sketch both a contour map and a graph of the function and compare them.
33.
34.
■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■
fx, y s36 9x2 4y2 fx, y x2 9y2
fx, y yx2 y2 fx, y sy2 x2
fx, y y sec x fx, y yex
29.
fx, y eyx fx, y y ln x
fx, y x3 y fx, y y 2x2
z
x y B A
B 23. A
I II
x x
y y
1. Let . f
(a) Evaluate . (b) Find the domain of . (c) Find the range of .
2. Let .
(a) Evaluate . (b) Evaluate .
(c) Find and sketch the domain of . (d) Find the range of .
3. Let .
(a) Evaluate . (b) Find the domain of . (c) Find the range of .
4. Let .
(a) Evaluate . (b) Find the domain of . (c) Find the range of .
5–12 ■ Find and sketch the domain of the function.
5. 6.
8.
10.
11.
12.
■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■
13–20 ■ Sketch the graph of the function.
13. 14.
15. 16.
17. 18.
19.
20.
■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■
21. A contour map for a function is shown. Use it to esti- mate the values of and . What can you say about the shape of the graph?
y
x 0 1
1 70 60 50 40 30 20 10 f3, 2
f3, 3
f fx, y s16 x2 16y2 fx, y sx2 y2
fx, y 3 x2 y2 fx, y 4x2 y2 1
fx, y cos x fx, y y2 1
fx, y y fx, y 6 3x 2y
fx, y, z ln16 4x2 4y2 z2 fx, y, z s1 x2 y2 z2
fx, y sx2 y2 1 ln4 x2 y2 fx, y sy x2
1 x2 9.
fx, y sy x lny x
fx, y ln9 x2 9y2 7.
fx, y sxy fx, y sx y
t t
t2, 2, 4
tx, y, z ln25 x2 y2 z2 f
f f2, 1, 6
fx, y, z eszx2y2 f
f
fe, 1
f1, 1
fx, y lnx y 1
f
f f2, 0
fx, y x2e3 x y
EXERCISES
11.1
z
y x
A B C z
y x
z
x y
Graphs and Contour Maps for Exercises 41–46
z
x y
D z E F
y x
z
y x
I II III
IV V VI
x y
x y
x y
x y
x y
x y
47–50 ■ Describe the level surfaces of the function.
48.
49.
50.
■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■
51–52 ■ Describe how the graph of is obtained from the graph of .
(a) (b)
(c) (d)
52. (a) (b)
(c)
■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■
;53. Use a computer to investigate the family of functions . How does the shape of the graph depend on ?
;54. Graph the functions
and
In general, if t is a function of one variable, how is the graph of
obtained from the graph of t?
fx, y t
(
sx2 y2)
fx, y 1 sx2 y2
fx, y sin
(
sx2 y2)
fx, y lnsx2 y2
fx, y esx2y2 fx, y sx2 y2
c
fx, y ec x2y2
tx, y f x 3, y 4 tx, y f x, y 2
tx, y f x 2, y
tx, y 2 f x, y
tx, y f x, y tx, y 2f x, y
tx, y f x, y 2 51.
f
t fx, y, z x2 y2
fx, y, z x2 y2 z2 fx, y, z x2 3y2 5z2 fx, y, z x 3y 5z 47.
35. A thin metal plate, located in the -plane, has temperature at the point . The level curves of are called isothermals because at all points on an isothermal the tem- perature is the same. Sketch some isothermals if the temper- ature function is given by
36. If is the electric potential at a point in the -plane, then the level curves of are called equipotential curves because at all points on such a curve the electric potential is the same. Sketch some equipotential curves if
, where is a positive constant.
;37– 40 ■ Use a computer to graph the function using various domains and viewpoints. Get a printout of one that, in your opinion, gives a good view. If your software also produces level curves, then plot some contour lines of the same function and compare with the graph.
37.
38.
39. (monkey saddle)
40. (dog saddle)
■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■
41– 46 ■ Match the function (a) with its graph (labeled A – F on page 600) and (b) with its contour map (labeled I–VI). Give reasons for your choices.
42.
43. 44.
45. 46.
■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■
z x y
1 x2 y2 z 1 x21 y2
z sin x sin y z sinx y
z ex cos y z sinxy
41.
fx, y xy3 yx3 fx, y xy2 x3
fx, y 1 3x2 y2e1x2y2 fx, y excos y
Vx, y csr2 x2 y2 c V xy
x, y
Vx, y
Tx, y 1001 x2 2y2
x, y T Tx, y
xy
LIMITS AND CONTINUITY
The limit of a function of two or more variables is similar to the limit of a function of a single variable. We use the notation
to indicate that the values of approach the number L as the point approaches the point along any path that stays within the domain of . In other words, we can make the values of as close to L as we like by taking the point sufficiently close to the point , but not equal to . A more precise def- inition follows.
a, b
a, b
x, y a, b fx, yfx, y f x, y
x, y l a, blim fx, y L
