CHAPTER 41
Functions of Several Variables
MULTIVARIATE FUNCTIONS AND THEIR GRAPHS
41.1
Sketch the cylinder
See Fig. 41-1. The surface is generated by taking the ellipse
parallel to the z-axis (z is the missing variable).
in the ry-plane and moving it
Fig. 41-1
Fig. 41-2
Describe and sketch the cylinder z = y .
See Fig. 41-2. Take the parabola in the _yz-plane z = y and move it parallel to the jt-axis (x is the missing
variable).
41.2
41.3
Describe and sketch the graph of 2x + 3>y = 6.
See Fig. 41-3. The graph is a plane, obtained by taking the line 2x + 3y = 6, lying in the *.y-plane, and
moving it parallel to the z-axis.
Fig. 41-3
Fig. 41-4
41.4
Write the equation for the surface obtained by revolving the curve z = y
2
(in the yz-plane) about the z-axis.
When the point (0, y*, z*) on z - y2 is rotated about the z-axis (see Fig. 41-4), consider any resulting
Hence, * +y" = (y*) = z* = z. So, the resultpoint (x, y, z). Clearly, z = z* and
ing points satisfy the equation z = x + y2.
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