1617-257/TUT-R-7

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On 10/28/16, we discussed the following problem:

Suppose F : \mathbb{R}^2 \to \mathbb{R}^2 is C^1 with F(t^4, e^{t^2}) = (4,5) for all t \in \mathbb{R}. Prove that DF(0,1) is not invertible.


Let \gamma(t) := (t^4, e^{t^2}). At the end of the tutorial, a couple of students pointed out that it's also true that DF(\gamma(t)) is not invertible for any t \in \mathbb{R}. It's easy to check that \gamma'(t) \neq 0 when t \neq 0 and that \gamma'(t) is in the kernel of DF(\gamma(t)).

It takes some more care to prove what the problem is asking because one can't immediately deduce that DF(\gamma(t)) has non-trivial kernel with \gamma'(0) = 0. A key observation here is that since F is C^1, invertibility of DF at \gamma(0) implies invertibility of DF at a neighborhood of \gamma(0).