Jacobian

$${\mathbf{J}}_{\mathbf{f}}=\left[\begin{array}{ccc}\frac{\partial \mathbf{f}}{\partial x_1}& \cdots & \frac{\partial \mathbf{f}}{\partial x_n}\end{array}\right]=\left[\begin{array}{ccc}\frac{\partial f_1}{\partial x_1}& \cdots & \frac{\partial f_1}{\partial x_n}\\ ⋮& \ddots & ⋮\\ \frac{\partial f_m}{\partial x_1}& \cdots & \frac{\partial f_m}{\partial x_n}\end{array}\right]$$

Wow! That looks exactly like the Total Derivative definition… Yeah I know.

The total derivative $Df$ and the Jacobian matrix $J_f$ are the same object, just view from slightly different perspectives. The total derivative $Df(x)$ is the linear map that best approximates $f$ near $x$:

$$f(x+h)\approx f(x)+Df(x)h.$$

When you represent that Linear Map as a matrix (in standard coordinates), its matrix representation is exactly the Jacobian:

$$Df(x)=J_f(x).$$

Jacobian Determinant

$$\det {\mathbf{J}}_{\mathbf{f}}=\left|\begin{array}{ccc}\frac{\partial f_1}{\partial x_1}& \cdots & \frac{\partial f_1}{\partial x_n}\\ ⋮& \ddots & ⋮\\ \frac{\partial f_m}{\partial x_1}& \cdots & \frac{\partial f_m}{\partial x_n}\end{array}\right|$$

To accommodate for the change of coordinates the magnitude of the Jacobian determinant arises as a multiplicative factor within the integral. Changing coordinates is basically like performing a Transformation on the rectangular/Cartesian space you start with into some other one. A unit of space is scaled by the determinant of the transformation. This is the same principle. e.g. a double integral in polar coordinates. The r is the jacobian in this case.

Double Integral

$$\iint f(x,y)dA={\int }_{\theta =\alpha }^{\theta =\beta }{\int }_{r=h_1(\theta )}^{r=h_2(\theta )}f(r\cos (\theta ),r\sin (\theta ))rdrd\theta$$Link to original