Say I have the following transformation
$$u = xy, v = \frac{y}{x}$$
for my double integral:
$$\int_{a}^{b} \int_{c}^{d} f(x,y)dxdy$$
To successfully transform the coordinates I need to know the Jacobian of the transformation. However, it is not linear and thus impossible to express by a matrix. How can I find the Jacobian then?
The Jacobian is
$$ \frac{\partial (u,v)}{\partial (x,y)} = \left|\begin{matrix} \displaystyle\frac{\partial u}{\partial x} & \displaystyle\frac{\partial u}{\partial y} \\ \displaystyle\frac{\partial v}{\partial x} & \displaystyle\frac{\partial v}{\partial y} \end{matrix}\right| = \left|\begin{matrix} y & x \\ \displaystyle -\frac{y}{x^2} & \displaystyle\frac{1}{x} \end{matrix}\right| = \frac{2y}{x} = 2v $$
The Jacobian is just a matrix of partial derivatives. It has nothing to do with whether the transformation itself is linear.