[Math] Uses of Jacobian of a map on $\mathbb{R}^n$.

Question

For a map $f:\mathbb{R}^n\to\mathbb{R}^n$, Jacobian matrix of $f$ is defined as $$\begin{bmatrix}
\frac{\partial f_1}{x_1}& \frac{\partial f_1}{\partial x_2}& \ldots \frac{\partial f_1}{\partial x_n}\\
\ldots & \ldots & \ldots \\
\frac{\partial f_n}{x_1}& \frac{\partial f_n}{\partial x_2}& \ldots \frac{\partial f_n}{\partial x_n}
\end{bmatrix}.$$
And determinant of this matrix is called Jacobian of $f$. I know that in case of a continuous function on $\mathbb{R}^n$ (and for certain classes of functions on $\mathbb{C}^n$) also), non vanishing of Jacobian at a point implies local injectivity at that particular point. Beside this, can anyone help me why and how this Jacobian is useful? Any historical reasons, why this was introduced. I searched online about it but could not gather much information. If anyone can give me some information about this.

Answer 1

One immediate application of great consequence that comes to mind is the Jacobian of the endomorphism of the tangent space of an embedded surface in 3-space, known as the shape operator (or the Weingarten map). This Jacobian (i.e., the determinant) happens to be the extremely important invariant called the Gaussian curvature of the surface.

More generally, the Jacobian appears in change of variable formulas for multiple integrals. One can't do multivariate calculus without it.