Multivariate Calculus - Bruce E. Shapiro

Lecture 6
Surfaces in 3D
The text for section 14.6 is no more than a catalog of formulas for different shapes
in 3D. You should be able to recognize these shapes from their equations but you
will not be expected to sketch them during an exam.
Definition 6.1 Let C be any curve that lines in a single plane R, and let L be
any line that intersects C but does not line in R. Then the set of all points on
lines parallel to L that intersect C is called a cylinder. The curve C is called the
generating curve of the cylinder.
Figure 6.1: Cylinders. Left: a right circular cylinder generated by a circle and a
line perpendicular to the cylinder. Center: a circular cylinder generated by a circle
and a line that is not perpendicular to the circle. Both circular cylinders extend to
infinity on the top and bottom of the figure. Right: a parabolic cylinder, generated
by a parabola and a line that is not in the plane of the parabola. The sheets of the
parabola extend to infinity to the top, bottom, and right of the figure.
Definition 6.2 A simpler definition of a cylinder (actually, this is a right cylinder
oriented parallel to one of the coordinate axes) is to consider any curve
in a plane, such as the xy plane. This is an equation in two-variables, x and y.
Then consider the same formula as describing a surface in 3D. This is the cylinder
generated by the curve.
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