FUNCTIONS OF SEVERAL VARIABLES
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Fig. 41-26
Describe the level curves of f(x, y) =
See Fig. 41-27. The level curves are
= k, or y = kx + (2k + 1). This is a family of punctured
straight lines. All these lines pass through the point (-2,1), which is excluded from them. There is a level
curve through every point not on the vertical line x = -2.
Fig. 41-27
41.44
41.45
41.46
41.47
41.43
Describe the level surfaces of f(x, y, z) = 3x - 2y + z.
The level surfaces are the planes 3x - 2y + z = k. Since they all have the vector (3, -2,1) as normal
vector, they form the family of parallel planes perpendicular to that vector. There is a level surface through
every point.
Describe the level surfaces of f(x, y, z) =
The level surfaces form a family of ellipsoids
or
There is a level surface through
in which the axes along the x-axis, v-axis, and z-axis are in the proportion 5:3:1.
every point (except the origin if one does not count a point as a level surface).
Describe the level surfaces of f(x, y, z) = 3x
2 + 5y
2 - z
2 .
The level surfaces 3x
2 + 5y
2 - z
2 = k > 0 are hyperboloids of one sheet around the z-axis. The level
surfaces 3x
2 + 5y
2 - z
2 = k s 0 are hyperboloids of two sheets around the z-axis. There is a level surface
through every point.
Describe the level surfaces of f(x, y,z)The level surfaces are x
2 + y
2 + z
2 = k
2 >0, concentric spheres with center at the origin. Every point
except the origin lies on a level surface.