In this Quora Answer, a very intuitive explanation for the Jacobian is provided, there is however a step I don't understand:
He takes this square:
And via a polar coordinates transformation transforms it into this parallelogram:
I get the idea, what I don't get however is how the new coordinates are calculated. He says
This square transforms into a parallelogram with the coordinate change. The point $(x,y)$ transforms to the point $(r,\theta)$. The point $(x + \mathrm{d}x,y)$ becomes $( r(x + \mathrm{d}x,y) , \theta(x + \mathrm{d}x,y))$. Because it's a small square, we can find these coordinates using calculus. The new point is $\left(r + \frac{\partial r}{\partial x}\mathrm{d}x, \theta + \frac{\partial \theta}{\partial x}\mathrm{d}x\right)$
But if the polar transformation is $ x = r\cdot cos(\theta) , y = r\cdot sin(\theta) $, shouldn't $(x,y) \rightarrow ( r\cdot cos(\theta), r\cdot sin(\theta)) $ ? Why is it $(x,y) \rightarrow (r,\theta) $ instead? And what about the partial derivatives?
The formula you gave below describes the relationship between Cartesian coordinate (x,y) and polar coordinate $(r, \theta)$. They are not supposed to be written in this way, i.e. in bracket.
$$(x,y) \rightarrow ( r\cdot cos(\theta), r\cdot sin(\theta))$$
The correct way to write this is,
$$x=r\cdot cos(\theta)$$ $$y=r\cdot sin(\theta)$$
Reversely, you can write the relationship between polar coordinate $(r,\theta)$ and Cartesian coordinate (x,y) as below. note: care should be taken for calculating $\theta$.
$$r=\sqrt {x^2+y^2}$$ $$\theta = \tan^{-1}{\frac yx}$$
As you can see polar coordinate r and $\theta$ are functions of x and y. Theoretically, you can do partial derivative and then calculate Jacobin matrix.
You may ask yourself again if you got the idea of transforming a square to a parallelogram. A better way to comprehend this is to assume coordinates of the square's vertices can transform them to polar coordinate system and draw it in $(r, \theta)$ plane.