Proof of the naturality of integration

Question

I have a bit of a problem with the following identity:

Suppose that $U, V \subset \mathbb{R}^n$, are two open sets. Let $x^1,...,x^n$ be a system of coordinates of $U$ and $y^1,...,y^n$ one on $V$. Let $f:V\rightarrow U$ such that $x^i=f^i(y^1,...,y^n)$. Than: $$df^1\wedge ... \wedge df^n=\det(\frac{\partial f^i}{\partial y^j})dy^1\wedge...\wedge dy^n.$$

Is this a legal way to prove it?

$df^1\wedge ... \wedge df^n(v_1,...,v_n)=\det(f^i(v_j))=\det(\frac{\partial y^k}{\partial y^k}f^i(v_j))=\det(\frac{\partial f^i}{\partial y^k}y^k(v_j))$ which leads to the result by applying once again the determinant formula.

I have the impression the last equality is wrong or needs to be justified more in details...

Thanks a lot for your precious help and have a nice day!

Answer 1

I suggest you try to write out $df^{i}$ in concrete terms, then you will see where the equality comes from.

You should read this article carefully.