The Jacobian is defined as the absolute value of the determinant (which can be viewed as the scaling factor of the transformation described by a matrix) of the matrix of all partial derivatives of the original variables (i.e. \(x\) and \(y\)) with respect to the new variables (say, \(u\) and \(v\)).

The purpose of the Jacobian is to ensure that the “area” under linear transformations is preserved under change of variables.

Recall in single-valued calculus that when trying to integrate a function by substitution, one distorts the area by making a change of variables.For example, suppose we want to solve the following definite integral:

$$ \int_{a}^{b} (2x + 3)^4 dx $$

Letting \( u=2x+3 \) we get that \( \int_{a}^{b} (2x + 3)^4 dx = \int_{2a+3}^{2b+3}u^4 \left(\frac{1}{2} \right)du = \frac{1}{10} \left(2x+3\right)^5|_{a}^{b} \)

In the example above, when changing the variable of integration from \(x\) to \(u\), the area under this graph will get distorted. In order to preserve the area under this change of variable., we use the Jacobian, \(|J| = |\frac{dx}{du}| = \frac{1}{2}\), to scale the function such that the area is preserved under the change of variable from \(x\) to \(u\). Hence the \(\frac{1}{2}\) in the definite integral above with respect to \(u\). In general, $$\int_{}^{} f\left(x\right)dx = \int_{}^{} f\left[g^{-1}\left(u\right)\right]|J|du$$

where \(u = g\left(x\right)\) and \(g\) is a monotone function.

The Jacobian can be extended to n-dimensions to ensure that the n-dimensional volume defined by some function is preserved under a change of variables. So for example suppose you want to solve the following integral:

$$ \int_{}^{} \int_{}^{} f\left(x,y\right) dx dy$$

You may wish to do a change of variable due to the integration being difficult under a particular co-ordinate system, but simpler in another. Thus suppose that we wish to convert from the \(x\) and \(y\) variables to \(u\) and \(v\) defined by the continuous differentiable functions: \( u = g(x,y) \) and \( v = h(x,y) \). Further suppose that these mappings are injective such that we can define inverses \( x = \phi(u,v) \) and \( y = \psi(u,v) \)

The Jacobian is:

i.e. The absolute values of the determinant of the partial derivatives.