MIT18_024S11_Pset7 - PSET 7 - DUE MARCH 31 1. 9.8:7 (6...

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Unformatted text preview: PSET 7 - DUE MARCH 31 1. 9.8:7 (6 points) Hint: It might help to define a scalar field F (x, y, z) = f (u(x, y, z), v(x, y, z)) where u, v are as needed. 2. Let f : Rm+n Rm be continuously differentiable and let x Rn , y Rm . De note by Df = Df x + Df y the decomposition of the Jacobian such that for h Rn , k Rm , Df (x, y)(h, k) = Df x (x, y)h + Df y (x, y)k. (That is Df x (x, y) : Rn Rm and Df y (x, y) : Rm Rm as mentioned in class.) Suppose for a Rn , b Rm , f (a, b) = 0 and det(Df y (a, b)) = 0. We consider a few steps of the implicit function theorem in this setting. Let F (x, y) : Rm+n Rm+n such that F (x, y) = (x, f (x, y)). Write down the matrix DF as we described it in class. You may write it in block decomposition, but also explain how you produce each block! (2 points) Prove DF is invertible at (a, b). (3 points) Using the inverse function theorem, we know there exists (a, b) V open and F (a, b) W open such that F : V W is invertible with continuously differ entiable inverse G. Let U = {x Rn |(x, 0) W }. Prove that U is open. (3 points) BONUS 1: Prove the existence of a well defined g : U Rm such that f (x, g(x)) = 0 for all x U and show this g is differentiable at a. (6 points) BONUS 2: Prove the formula Dg(a) = -Df y (a, b)-1 Df x (a, b). (3 points) 3. 9.13:17 - part (a) should be a sketch on the (x, y)-plane (8 points) 4. 9.15:8,13 (8 points) 1 MIT OpenCourseWare 18.024 Multivariable Calculus with Theory Spring 2011 For information about citing these materials or our Terms of Use, visit: ...
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This note was uploaded on 01/18/2012 for the course MATH 18.024 taught by Professor Christinebreiner during the Spring '11 term at MIT.

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MIT18_024S11_Pset7 - PSET 7 - DUE MARCH 31 1. 9.8:7 (6...

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