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Topic: Name for elementary result in multivariable calculus?
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Paul

Posts: 387
Registered: 7/12/10
Name for elementary result in multivariable calculus?
Posted: May 11, 2013 12:18 PM
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Baby Rudin's theorem 9.32 says:
Suppose m, n, r are non-negative integers, m>=r, n>=r, F is a continuously differentiable mapping of an open set E of R^n into R^m and the derivative of F always has rank r.
Fix a in E, put A = the derivative of F at a, let Y1 = the range of A and let P be a projection in R ^ m whose range is Y1. Let Y2 be the null space of P.
Then there are open sets U and V in R^n with a in U, U contained in E, and there is a 1-1 continuously differentiable mapping H of V onto U (whose inverse is also continuously differentiable) such that F(H(x)) = A(x) + phi(A(x)) (x in V) where phi is a continuously differentiable mapping of the open set A(V) contained in Y1 into Y2.

I'd like to find an alternative account of this result which must be standard, but Rudin doesn't give the name of this theory.

Could anyone give a name of this (or a similar) theorem so that I can research it?

Thanks,

Paul Epstein



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