Engineering Calculus Notes 304

Engineering Calculus Notes 304 - probably encounter in...

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292 CHAPTER 3. REAL-VALUED FUNCTIONS: DIFFERENTIATION Of course, in this case, we can verify the conclusion by solving explicitly: φ ( x,y ) = r 4 x 2 + y 2 1; you should check that the properties of this function are as advertised. However, at (0 , 1 , 0), the situation is diFerent: since ∂f ∂x (0 , 1 , 0) = 0 ∂f ∂y (0 , 1 , 0) = 2 ∂f ∂z (0 , 1 , 0) = 0 we can only hope to solve for y in terms of x and z ; the theorem tells us that in this case ∂y ∂x (0 , 0) = 0 ∂y ∂z (0 , 0) = 0 . We note in passing that Theorem 3.5.3 can be formulated for a function of any number of variables, and the passage from three variables to more is very much like the passage from two to three. However, some of the geometric setup to make this rigorous would take us too far a±eld. There is also a very slick proof of the most general version of this theorem based on the “contraction mapping theorem”; this is the version that you will
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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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