Jacobians
This page is dedicated to explaining what the Jacobian matrix is, and also how to use it. In general, this topic is useful for Vector Calculus and making some surface integrals easier to do.
Basic Definition
The Jacobian Matrix is just a matrix that takes the partial derivatives of each element of a function (which is in the form of a vector. So, if F is a function that maps from < u , v > to < x , y >, then the Jacobian of this function would looks like this:
$ \begin{bmatrix} \frac{\partial x}{\partial u} & \frac{\partial x}{\partial v} \\ \frac{\partial y}{\partial u} & \frac{\partial y}{\partial v} \end{bmatrix} $
To help illustrate this, let's do an example:
Example #1) Let's take the Transformation: $ T(u,v) = <u*\cos v,r*\sin v>. $ What would be the Jacobian Matrix of this Transformation?
Solution: Note that $ x=u*\cos v \longrightarrow \frac{\partial x}{\partial u}= \cos v and \frac{\partial x}{\partial v} = -u*\sin v y=u*\sin v \longrightarrow \frac{\partial y}{\partial u}= \sin v and \frac{\partial y}{\partial v} = u*\cos v $
Therefore the Jacobian matrix is
$ \begin{bmatrix} \frac{\partial x}{\partial u} & \frac{\partial x}{\partial v} \\ \frac{\partial y}{\partial u} & \frac{\partial y}{\partial v} \end{bmatrix}= \begin{bmatrix} \cos v & -u*\sin v \\ \sin v & u*\cos v \end{bmatrix} $
Now after doing
