FUNCTIONS OF SEVERAL VARIABLES
367
Describe the level curves (contour map) of f(x, y) = x
2 + y
i .
41.32
41.33
41.34
41.35
41.36
41.37
In general, the level curves of a function f(x, y) are the family of curves f(x, y) = k, z = 0. Here, the
level curves are the circles x
2 + y
2 = r
2
(write k = r
2 > 0) of radius r with center at the origin (Fig. 41-16).
There is one through every point of the jcy-plane except the origin (unless we consider the origin as the level curve
Jt
2 + y
2 =0).
Describe the level curves of f(x, y) — y — x.
The level curves form the family of parallel lines y - x = k with slope 1 (Fig. 41-17). There is one level
curve through every point.
Fig. 41-17
Fig. 41-18
Describe the level curves of f(x, v) = y — x
3 .
The level curves form the family of cubic curves y = x
3 + k (Fig. 41-18). There is one level curve through
every point.
Describe the level curves of f(x, y) = y/x
2 .
See Fig. 41-19. The level curves form the family of curves y = fcr
2
. When k > 0, we get a parabola that
opens upward; when k<0, the parabola opens downward. When k = 0, we get the x-axis. In each
case, the level curve is "punctured" at the origin, since y/x
2 is undefined when x — 0. There is a level curve
through every point not on the y-axis.
Fig. 41-19
Fig. 41-20
Describe the level curves oi f(x, y) = y
2 - x
2 .
See Fig. 41-20. The level curves, y
2 — x
2 = k are two sets of rectangular hyperbolas (corresponding to
k > 0 and k < 0), plus two straight lines (k - 0) to which all the hyperbolas are asymptotic. There is a
single level curve through every point (A:, y) except (0,0), which lies on two.
Describe the level curves of f(x, y) = 4*
2 + 9y
2
.
See Fig. 41-21. The level curves form a family of ellipses 4o:
2 + 9y
2 = k
2 , or
There is a level curve through every point (considering the origin to be a level curve consisting of a single point).