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Math Help - Rank of Functions

  1. #1
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    Rank of Functions

    Let f:{M}\rightarrow{N} and g:{N}\rightarrow{K} be two differentiable functions and M, N and K be manifolds. If the rank of g at f(P) is equal to the dimension of N, show that f and g\circ f have the same rank at P.

    Something tells me that I have to use the fundemental equation of dimension, but I can't work it out.
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  2. #2
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    The "rank" of f:A-> B is the dimension of f(A) as a subspace of B. Since the rank of g:B-> C is the same as the dimension of C, g(B) is all of C and, in particular, g(f(A)) has the same dimension as f(A).
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