836 � CHAPTER 14 FUNCTIONS OF SEVERAL VARIABLES

Example 10 For the function g(x, y, z) = √x2 + y2 + z2, the level surfaces are

concentric spheres

x2 + y2 + z2 = c2. �

Example 11 As our final example we take the function

f (x, y, z) = |z|x2 + y2

.

We extend this function to the origin by defining it to be zero there. At other points ofthe z-axis we leave f undefined.

In the first place note that f takes on only nonnegative values. Since f is zero onlywhen z = 0, the 0-level surface is the xy-plane. To find the other level surfaces, we takec > 0 and set f (x, y, z) = c. This gives

|z|x2 + y2

= c and thus |z| = c(x2 + y2)

(Figure 14.3.11). Each of these surfaces is a double-paraboloid of revolution.† �

y

x

z

level surface: |z| = c(x2 + y2), (c > 0)

Figure 14.3.11 † It is surface 5 of the last section together with its mirror image below the xy-plane.

EXERCISES 14.3

Identify the level curves f (x, y) = c and sketch the curvescorresponding to the indicated values of c.

1. f (x, y) = x − y; c = −2, 0, 2.

2. f (x, y) = 2x − y; c = −2, 0, 2.

3. f (x, y) = x2 − y; c = −1, 0, 1, 2.

4. f (x, y) = 1

x − y2; c = −2,−1, 1, 2.

5. f (x, y) = x

x + y; c = −1, 0, 1, 2.

6. f (x, y) = y

x2; c = −1, 0, 1, 2.

7. f (x, y) = x3 − y; c = −1, 0, 1, 2.

8. f (x, y) = exy; c = 12 , 1, 2, 3.

9. f (x, y) = x2 − y2; c = −2,−1, 0, 1, 2.

10. f (x, y) = x2; c = 0, 1, 4, 9.

11. f (x, y) = y2; c = 0, 1, 4, 9.

12. f (x, y) = x(y − 1); c = −2,−1, 0, 1, 2.

13. f (x, y) = ln (x2 + y2); c = −1, 0, 1.

14. f (x, y) = ln( y

x2

); c = −2,−1, 0, 1, 2.

15. f (x, y) = ln y

x2; c = −2,−1, 0, 1, 2.

16. f (x, y) = x2y2; c = −4,−1, 0, 1, 4.

17. f (x, y) = x2

x2 + y2, c = 0, 1

4 , 12 .

18. f (x, y) = ln y

x; c = −2,−1, 0, 1, 2.

Identify the c-level surface and sketch it.

19. f (x, y, z) = x + 2y + 3z, c = 0.

20. f (x, y, z) = x2 + y2, c = 4.

21. f (x, y, z) = z(x2 + y2)−1/2, c = 1.

22. f (x, y, z) = x2/4+ y2/6+ z2/9, c = 1.

23. f (x, y, z) = 4x2 + 9y2 − 72z, c = 0.

24. f (x, y, z) = z2 − 36x2 − 9y2, c = 1.

25. Identify the c-level surfaces of

f (x, y, z) = x2 + y2 − z2

taking (i) c < 0, (ii) c = 0, (iii) c > 0.

26. Identify the c-level surfaces of

f (x, y, z) = 9x2 − 4y2 + 36z2

taking (i) c < 0, (ii) c = 0, (iii) c > 0.

Find an equation for the the level curve of f that contains thepoint P.

27. f (x, y) = 1− 4x2 − y2; P(0, 1).

28. f (x, y) = (x2 + y2) exy; P(1, 0).

29. f (x, y) = y2 tan−1 x; P(1, 2).

30. f (x, y) = (x2 + y) ln [2− x + ey]; P(2, 1).

14.3 GRAPHS; LEVEL CURVES AND LEVEL SURFACES � 837

Find an equation for the level surface of f that contains thepoint P.

31. f (x, y, z) = x2 + 2y2 − 2xyz; P(− 1, 2, 1).

32. f (x, y, z) = √x2 + y2 − ln z; P(3, 4, e).

�c 33. Use a graphing utility to draw (a) the surfaces and (b) thedefault level curves.

(a) f (x, y) = 3x + y3. (b) f (x, y) = x2 + 1

y2 + 4.

�c 34. Use a graphing utility to draw the level surfaces correspond-ing to the values of c.

(a) f (x, y, z) = x + 2y + 4z; c = 0, 4, 8.

(b) f (x, y, z) = x + y

1+ z2; c = −2, 0, 2.

�c 35. Use a CAS to find the level curve/surface at the point P.

(a) f (x, y) = 3x + 2y + 1

4x2 + 9; P(2, 4).

(b) f (x, y, z) = x2 + 2y2 − z2; P(2,−3, 1).

�c 36. Use a CAS to draw the surface and the level curves.

(a) f (x, y) = (x2 − y2) e(−x2−y2); −2 ≤ x ≤ 2,− 2 ≤ y ≤ 2.

(b) f (x, y) = xy3 − yx3; −5 ≤ x ≤ 5, −5 ≤ y ≤ 5.

37. The magnitude of the gravitational force exerted by a bodyof mass M situated at the origin on a body of mass m locatedat the point (x, y, z) is given by

F(x, y, z) = GmM

x2 + y2 + z2

where G is the universal gravitational constant. If m and Mare constants, describe the level surfaces of F . What is thephysical significance of these surfaces?

38. The strength E of an electric field at a point (x, y, z) dueto an infinitely long charged wire lying along the y-axis isgiven by

E(x, y, z) = k√x2 + z2

where k is a positive constant. Describe the level surfacesof E.

39. A thin metal plate is situated in the xy-plane. The tempera-ture T (in ◦C) at the point (x, y) is inversely proportional tothe square of its distance from the origin.

(a) Express T as a function of x and y.

(b) Describe the level curves and sketch a representa-tive set. NOTE: The level curves of T are calledisothermals; all points on an isothermal have the sametemperature.

(c) Suppose the temperature at the point (1, 2) is 50◦. Whatis the temperature at the point (4, 3)?

40. The formula

V (x, y) = k√r2 − x2 − y2

,

where k and r are positive constants, gives the electric poten-tial (in volts) at a point (x, y) in the xy-plane. Describe thelevel curves of V and sketch a representative set. NOTE:The level curves of V are called the equipotential curves;all points on an equipotential curve have the same electricpotential.

In Exercises 41–46, a function f , together with a set of levelcurves for f , is given. Figures A–F are the surfaces z = f (x, y)(in some order). Match f and its system of level curves with itsgraph z = f (x, y).

41. f (x, y) = y2 − y3.

y

x

42. f (x, y) = sin x, 0 ≤ x ≤ 2π .

y

x

43. f (x, y) = cos√

x2 + y2,−10 ≤ x ≤ 10,−10 ≤ y ≤ 10.

x

y

838 FUNCTIONS OF SEVERAL VARIABLES

44. f (x, y) = 2x2 + 4y2.

y

x

45. f (x, y) = xye−(x2+y2)/2.

y

x

46. f (x, y) = sin x sin y.

y

x0

Az

yx

B

y

x

zC

y

x

z

D

y

x

zE

z

y

x

Fz

y

x

� CHAPTER 14

14.4 PARTIAL DERIVATIVES � 839

� PROJECT 14.3 Level Curves and Surfaces

Computer systems such as Derive, Maple, and Mathematica areable to map the level curves of a function f = f (x, y). In thisproject you are asked to map the level curves of a given functionover a given rectangle and then you are asked to “visualize” thesurface z = f (x, y). For example, the level curves of

f (x, y) = x2y2 e−(x2+y2)

on the rectangle: −3 ≤ x ≤ 3,−3 ≤ y ≤ 3 are

y

x

This map of level curves suggests that the surface z = f (x, y)has either “peaks” or “pits” symmetrically placed in the fourquadrants. The graph of the surface shown to the right confirmsthis conjecture.

z

x

y

Problem 1. Make a map of the level curves of f (x, y) =1

x2 + y2over the rectangle: −3 ≤ x ≤ 3,−3 ≤ y ≤ 3. Try to

visualize the graph of the surface from your map of the levelcurves. Then graph the surface z = f (x, y) to confirm yourvisualization.

Problem 2. Repeat Problem 1 for the function f (x, y) =2y

x2 + y2 + 1over the rectangle −5 ≤ x ≤ 5,−5 ≤ y ≤ 5.

Problem 3. Repeat Problem 1 for the function f (x, y) =cos x cos y e−(1/4)

√x2+y2

over the rectangle −2π ≤ x ≤ 2π ,−2π ≤ y ≤ 2π .

Problem 4. Repeat Problem 1 for the function f (x, y) = −xy

ex2+y2

over the rectangle: −2 ≤ x ≤ 2,−2 ≤ y ≤ 2.

� 14.4 PARTIAL DERIVATIVES

Functions of Two Variables

Let f be a function of x and y; for example

f (x, y) = 3x2y − 5x cosπy.

The partial derivative of f with respect to x is the function fx obtained by differentiatingf with respect to x, treating y as a constant. In this case

fx(x, y) = 6xy − 5 cosπy.

The partial derivative of f with respect to y is the function fy obtained by differentiatingf with respect to y, treating x as a constant. In this case

fy(x, y) = 3x2 + 5πx sin πy.

These partial derivatives are formally defined as limits: