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TangPlaneGrad

TangPlaneGrad - Department of Mathematics University of...

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Department of Mathematics University of Southern California February 25, 2011 Tangent Plane/Gradient Notes For a function F = F ( x, y, z ), when it is held constant, say, F ( x, y, z )= k, it gives a relationship between x , y ,and z . This may be viewed as z ( x, y ), or x ( y,z ), or y ( z,x ), and forms a surface. Such a family of surfaces for various values of k is called level surfaces. Now consider an arbitrary curve con f ned to a particular level surface. Such a curve may be described parametrically as r ( t )= <x ( t ) ,y ( t ) ,z ( t ) > Based on our earlier work on three-dimensional curves, we know that r I ( t )= <x I ( t ) ,y I ( t ) ,z I ( t ) > is a tangent to the curve. Now, we are considering the surface F ( x, y, z )= k on which r ( t ) lies. If we remain con f ned to the curve (which is con f ned to the level surface), we may express the curve as F ( x ( t ) ,y ( t ) ,z ( t )) = k. Now, di
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