TITLE: How do I take this integral?
QUESTION [0 upvotes]: How do I take the integral of the Jacobian and show this equality?
I'm thinking of doing it component wise, but I feel that I'm overlooking something simpler.

REPLY [1 votes]: Let's say we have $f:\mathbb{R}^n\to\mathbb{R}^n$ and $g:\mathbb{R}\to\mathbb{R}^n$. Then by chain rule we have that 
$$\frac{d}{dt}\Bigr(f(g(t))\Bigr) = J_f(g(t))\cdot g'(t)$$
because the Jacobian is the generalization of $f'$ or the total derivative, in a way that partial derivatives aren't. The crucial difference is that Jacobians will obey the fundamental theorem of calculus under the correct conditions, but arbitrary partials may not.
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