Question

Implicit Function Theorem in Two Variables: Let g: R2 → R be a smooth function. Set {(z, y) E R2 | g(z, y) = 0} S Suppose g(a, b)-0 so that (a, b) E S and dg(a, b)メO. Then there exists an open neighborhood of (a, b) say V such that SnV is the image of a smooth parameterized curve. (1) Verify the implicit function theorem using the two examples above. 2) Since dg(a,b) 0, argue that it suffices to assume (a, b)メ0. (3) Prove the implicit function theorem by following these steps: (a) Define G: R2R2 by G(x,y) = (z, g(z,y)) Compute dG (b) Use the inverse function theorem to find a local inverse, i.e. there exists open set U, V so that has inverse H:U-V (c) Since G fixes the first variable, show that the inverse H also fixes the first variable. That is, H is of the form H(z, y) = (z, φ(x, y)) for some function φ(z,y). (d) Define f(x) = y(x,0) for x sufficiently close to a (that is we want (z,0) E U. Show that this is the correct f by analyzing the inverse function relations H(G(x,g) (r, y) and G(H(x, y) (r, y). Think about why the implicit function theorem allows us to prove that level sets are smooth curves (provided the conditions of the theorem). The general case of > 2 variables is the regular value theorem.

Answer 1

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