I am learning more about the change of variables formula and was confused about the role of the determinant of the jacobian. I came across this post, in which I found this particularly interesting:
"The higher dimensional version of the rectangle and interval is the $n$ dimensional rectangle defined by $[a_1,b_1] \ times … \times [a_n,b_n]$ which has hypervolume $\prod_{i=1}^{n} (b_i-a_i)$ If we consider an invertible transformation $T: [a_1,b_1] \times … \times [a_n,b_n] \rightarrow \mathbb{R}^n,$ by which $T(x)=Ax$ and $A$ is some $n \times n$ matrix that has nonzero determinant, we have the general formula
$$\text{Vol}(T([a_1,b_1] \times … \times [a_n,b_n])) = |\det(A)| \text{Vol}([a_1,b_1] \times … \times [a_n,b_n])."$$
My question is if we can derive a similar formula for the determinant of the jacobian to establish a similar rule for general $C^1$ maps
Best Answer
Assuming $A$ is connected, yes, since the function is $C^1$, the determinant of the Jacobian is continuous, and this follows from the change of variables theorem and the mean value theorem for (multiple) integrals. If $A$ has two (or more) connected pieces, as usual it may well fail.