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I've been working through differential geometry problem sets I found online (associated with MATH 481 at UIUC) and am struggling to show that T(MxN) is diffeomorphic to TM x TN.

My intuition is that I could set up the product of projection maps from MxN to M and MxN to N to form the diffeomorphism. However, I seem to be stuck on the implementation (which probably indicates gaps in my understanding of previous material).

Thanks for your assistance.