Engineering Calculus Notes 301

Engineering Calculus Notes 301 - 3.5 SURFACES AND THEIR...

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3.5. SURFACES AND THEIR TANGENT PLANES 289 Note that the statement of the general theorem says when we can solve for z in terms of x and y , but an easy argument (Exercise 13 ) shows that we can replace this with any variable whose partial is nonzero at −→ x = −→ x 0 . Proof sketch: This is a straightforward adaptation of the proof of Theorem 3.4.2 for functions of two variables. Recall that the original proof had two parts. The Frst was to show simply that L ( f,c ) R is the graph of a function on B ε ( −→ x 0 ). The argument for this in the three-variable case is almost verbatim the argument in the original proof: assuming that ∂f ∂z > 0 for all −→ x near −→ x 0 , we see that F is strictly increasing along a short vertical line segment through any point ( x ,y ,z ) near −→ x 0 , I ( x ,y ) = { ( x ,y ,z ) | z δ z z + δ } . In particular, assuming
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