Jeffim: Created page with "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 ..."

2016-10-29T20:42:12Z

Created page with "On 10/28/16, we discussed the following problem: Suppose <math>F : \mathbb{R}^2 \to \mathbb{R}^2</math> is <math>C^1</math> with <math>F(t^4, e^{t^2}) = (4,5)</math> for all ..."

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

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

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

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

1617-257/TUT-R-7 - Revision history

Jeffim: Created page with "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 ..."

Jeffim: Created page with "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 ..."