Let $J_x$ denote the Jacobian operator of the function $f: \mathbb{R}^n \to \mathbb{R}^n$.
What is the proper statement of the fundamental theorem of calculus for this case?
That is can we write \begin{align} f(x_2)-f(x_1) =\int (J_x f) \cdot dx \end{align}
where $\int$ is some proper integral that integrates over a path from $x_1$ to $x_2$.
A reference would be greatly appreciated.
Technically, the Jacobian is involved in integration only when a change of variables is involved. For example, when changing from Cartesian to polar coordinates, you'd have $x = r \cos \theta$ and $y = r \sin \theta$. Instead of $dy dx$ as your integrating factor, you'd have the product of the Jacobian and $dr d\theta$.