Engineering Calculus Notes 299

Engineering Calculus Notes 299 - 2.1 This is a reasonable...

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3.5. SURFACES AND THEIR TANGENT PLANES 287 vanishes. Similarly, the function f ( x,y,z ) = x 2 + y 2 z 2 can be seen, following the analysis in § 3.4 , to have as its level sets L ( f,c ) a family of hyperboloids 9 —of one sheet for c > 0 and two sheets for c < 0. (See Figure 3.14 .) Figure 3.14: Level Sets of f ( x,y,z ) = x 2 + y 2 z 2 For c = 0, the level set is given by the equation x 2 + y 2 = z 2 which can be rewritten in polar coordinates r 2 = z 2 ; we recognize this as the conical surface we used to study the conics in
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Unformatted text preview: § 2.1 . This is a reasonable surface, except at the origin, which again is the only place where the gradient grad f vanishes. This might lead us to expect an analogue of Theorem 3.4.2 for functions of three variables. Before stating it, we introduce a useful bit of notation. By 9 Our analysis in § 3.4 clearly carries through if 1 is replaced by any positive number | c |...
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