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

To further my comment, the Jacobian can be thought of intuitively as follows.

For a sufficiently nice function $T:\mathbb{R}^n\to\mathbb{R}^n$, we can locally approximate $T$ around a point $x$ with the linear function $$T(x)+\mathbf{DT}(\Delta x)$$ where $\Delta x$ is the change in the vector $x$ and $\mathbf{DT}$ is the derivative matrix of $T$.

From linear algebra, a linear transformation $L$ with determinant $\det L$ distorts the volumes of objects by a factor of $|\det L|$. Thus, around the point $x$, the function $T$ will distort volumes by a factor of $|\det \mathbf{DT}|$. But this, by definition, is the Jacobian (perhaps without the absolute value, depending on who you ask). This is also why the Jacobian appears in change of variables for multiple integration - it plays the same role that the $x'(u)$ plays in $u$-substitution.

Answer 2

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.