Unformatted text preview: probably encounter in higher math courses. Tangent Planes of Level Surfaces When a surface is de±ned by an equation in x , y and z , it is being presented as a level surface of a function f ( x,y,z ). Theorem 3.5.3 tells us that in theory, we can express the locus of such an equation near a regular point of f as the graph of a function expressing one of the variables in terms of the other two. ²rom this, we can in principle ±nd the tangent plane to the level surface at this point. However, this can be done directly from the de±ning equation, using the gradient or linearization of f . Suppose P ( x ,y ,z ) is a regular point of f , and suppose −→ p ( t ) is a diFerentiable curve in the level surface L ( f,c ) through P (so c = f ( P )),...
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- Fall '08
- Calculus, Continuous function, Contraction Mapping Theorem, level surface