1617-257/TUT-R-7

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.

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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 $\gamma (0)$.