Jacobian matrix

The Jacobian matrix ($\text{J}$) is a matrix for a change of variables that relates the partial derivatives of the old variables with respect to the new variables. We are more specifically concerned with the determinant of such a matrix, which allows us to perform a change of variables for double integrals.

The Jacobian has several broad applications, like small signal modelling and solving multivariable non-linear sets of equations. In many problems, the matrix is square (which means we can compute the inverse matrix). In the general case, it’s an $n\times m$ matrix where $n,m\ge 1$.

The set-up

Say, for a rectangular system, we introduce a set of functions that change variables to a different set of variables like the following:

$$x=f(u,v,w)$$

and so on for $y$, $z$. We don’t necessarily have to have a square matrix. For example, we can have a mapping of 3 variables to 2 variables.