12.6 - Section 13.6 Equations of Cylinders and Quadric...

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Section 13.6 Equations of Cylinders and Quadric Surfaces “Other Graphs in 3-space” In this section we shall consider other types of surfaces. In order to sketch such curves, we consider cross-sections with planes (also called traces ). 1. Cylinders Though we have previously considered cylinders as a surface shaped like a pipe, in multivariable calculus, there is a more general deFnition of a cylinder, and there are many other graphs which are considered “cylinders” according to the formula deFnition which do not look like pipes. The formal deFnition of a cylindrical surface is the following: DeFnition 1.1. A cylinder is a surface that consists of all lines (called rulings) that are parallel to a given line and pass through a given curve in some plane. To avoid confusion, we shall usually refer to the cylinders which are shaped like a pipe as pipe cylinders , and general cylinders satisfying this deFnition a cylindrical surface (or cylinder for short). Of course, a pipe cylinder is a cylindrical surface i.e. it consists of parallel lines, all of which all pass through a circle (see illustration below). -2 -1 0 1 2 -2 -1 0 1 2 -1.0 -0.5 0.0 0.5 1.0 There are many other examples of cylindrical surfaces. Example 1.2.
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12.6 - Section 13.6 Equations of Cylinders and Quadric...

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