I'm still learning calculus and Jacobian, and I am confused because I don't intuitively understand what Jacobian actually does and how to interpret its result. For example, I consider the Jacobian for convertion from polar to Cartesian coordinate system.
The conversion from polar to Cartesian coordinates is done like this: $$ x(r,\theta)=r\cos(\theta) \\ y(r,\theta)=r\sin(\theta) $$
The Jacobian of the transformation is defined like this: $$ J = \begin{bmatrix} \frac{\partial x}{\partial r} & \frac{\partial x}{\partial \theta} \\ \frac{\partial y}{\partial r} & \frac{\partial y}{\partial \theta} \\ \end{bmatrix} = \begin{bmatrix} \cos(\theta) & -r\sin(\theta) \\ \sin(\theta) & r\cos(\theta) \\ \end{bmatrix} $$ Each row of which is transposed gradient of functions $x$ and $y$.
From linear algebra course I learned that a column in a matrix represents new coordinates of i-th basis vector. But from the Jacobian above I see that at least the first column does not represent the new coordinates of the vector $\hat{i}$ in Cartesian coordinate system after applying the transformation as I expected because it does not depend on coordinate $r$ in polar coordinate system as it should (I think).
Question 1: If the columns do not represent the change in basis vectors in Cartesian coordinate system, then what they do represent?
Also I struggle to interpret the resulting matrix of linear transformation after substituting the coordinates from polar coordinate system.
For example $(r, \theta)=(2,\pi)$. The coordinates in Cartesian coordinate system should be equal to $(x,y)=(2\cos(\pi),2\sin(\pi))=(-2, 0)$.
$$ J = \begin{bmatrix} \cos(\pi) & -2\sin(\pi) \\ \sin(\pi) & 2\cos(\pi) \\ \end{bmatrix} = \begin{bmatrix} -1 & 0 \\ 0 & -2 \\ \end{bmatrix} $$
Question 2: How to interpret the Jacobian above? What transformation the matrix does?
Well, as I expected, the vectors are rotated by 180 degrees (since $\theta=\pi$), but the $\hat{j}$ vector is stretched by 2 instead of vector $\hat{i}$ as I expected.
Question 3: How we defined the order of variables $r$ and $\theta$? Why $r$ corresponds to $x$ and not to $y$? What will happen if I swap the variables $r$ and $\theta$ (meaning swapping the order of them in the functions $x$ and $y$, also changing the order of colunms in the Jacobian)? If I swap the variables $r$ and $\theta$ then the columns of the Jacobian matrix will also swap, swapping the $\hat{i}$ and $\hat{j}$ vectors, which changes the result.
Sorry for such a newbie questions. Hope for your help!
