jacobian_dets - MAS 4105Nov. 28, 1994 Jacobian Determinants...

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MAS 4105—Nov. 28, 1994 Jacobian Determinants and Multiple Integrals. The terminology in this handout is the same as in the earlier “Jacobians, Directional Derivatives, and the Chain Rule” handout. In the current handout, we consider differen- tiable functions f : R n R n (i.e. functions for which the dimensions of the domain and target spaces are equal.) Thus, for any a R n , the Jacobian matrix J f ( a ) is a square matrix, and we can consider its determinant and (potentially) its inverse. Exercise. 1. Suppose that f : R n R n is differentiable and has a differentiable inverse function f - 1 ; i.e. both f f - 1 and f - 1 f are the identity map of R n . (Note: do not assume that f is linear!) Show that for all a R n , J f - 1 ( f ( a )) = ( J f ( a )) - 1 . (Hint: Chain Rule.) The discussion I give below does not prove anything. It is merely meant as motivation for why a certain theorem says a certain thing. In order to keep the main ideas from being lost in details that are better left to Advanced Calculus, I’ll frequently make assumptions that I won’t attempt to justify here. I’ll also use imprecise language from time to time. Consider a “box” (parallelepiped) P in R n spanned by linearly independent vectors v 1 ,..., v n . Let P a be the translate of this box by the vector a : P a = { a + t 1 v 1 + ... + t n v n | 0 t i 1 , i = 1 , 2 ,...,n } . (You will probably want to have a pencil and paper by your side as you read this, to draw diagrams of all the objects involved.) We want to compare the volume of the image f ( P a ) to the volume of P a . We saw in class that, for linear maps L , L ( P ) is also a box, and that vol n ( L ( P )) = | det
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jacobian_dets - MAS 4105Nov. 28, 1994 Jacobian Determinants...

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