What is the area of the domain corresponding to small change in the function?

Question

Assume we have a function that maps $\vec{x}$ from $\mathbb{R}^n$ to $\vec{c}$ from $\mathbb{R}^m$ where $n$ and $m$ are arbitrary positive integers (we can have all combinations where $n>m$, $n<m$ and $n=m$). Now let's take a vector $\vec{c}$ and a volume element around it $d^m\vec{c}$. Is there a general formula for finding the volume element $d^n\vec{x}(\vec{c})$ of the domain of the function corresponding to all $\vec{x}$ that get mapped into said $d^m\vec{c}$ from the codomain. In the case where $n=m$ and the function is invertible, the answer is easily obtained by just using the Jacobian: $$d^n\vec{x} = \left|\frac{\partial \vec{x}}{\partial \vec{c}}\right|d^n\vec{c},$$ but I am interested is there a generalization for the case of $n\neq m$.

Answer 1

For linear transformations, the generalization of the determinant in this setting is the product of the singular values (thus including any $0$'s, which will always be there when $n<m$). This gives the conversion from the $n$-volume of a ball in the domain to the $m$-volume of a ball in the codomain.

Try to see for yourself whether the product of the singular values of the Jacobian is the right generalization to the setting of differentiable functions, and if so how that generalization works. One place to start is to note that even for linear transformations, this does not play nice with regions in the domain with zero volume when $n>m$.