What goes wrong in this derivation of polar coordinates?

Question

Given a vector in the standard cartesian basis \begin{equation} v = \left( \begin{aligned} x\\ y \end{aligned}\right) \end{equation}

We may write its components in terms of the polar coordinates as follows: $$ v=\left( \begin{aligned} r \cos\theta\\ r \sin\theta \end{aligned}\right) $$

To put this in terms of the spherical basis, we can apply the jacobian transform: $$ \frac{\partial (x,y)}{\partial (r,\theta)} = \left(\begin{array}[cc] \cos\cos\theta & \sin\theta \\ -r \sin\theta & r \cos\theta \end{array}\right) $$

Which when applied to the vector $v$ gives

$$ \left( \begin{aligned} r \\ 0 \end{aligned}\right) $$

Clearly $\theta$ is not always $0$, what on earth am I doing wrong?

Answer 1

We have $v = (r\cos \theta, r\sin \theta) = f(r,\theta)$. To recover $(r,\theta)$, we simply apply the inverse map $f^{-1}(f(r,\theta))=(r,\theta)$. The inverse map is given by $f^{-1}(x,y) = (\sqrt{x^2+y^2},\tan^{-1}(y/x))$.

Answer 2

Your jacobian transform converts one set of Cartesian coordinates to another. Specifically, it converts coordinates of a tangent space at a particular point aligned with your original Cartesian basis vectors $\hat x, \hat y$ to coordinates of the same tangent space based on the polar basis vectors at that point, $\hat r, \hat \theta.$

The polar basis vector $\hat r$ of the tangent space at any given point is a unit vector in the same direction as that point's position vector. The position vector is therefore equal to $r \hat r$ where $r$ is the distance from the origin. That is what your calculations have confirmed.

Specifically, the $\hat \theta$ component of the position vector at any point is zero according to the way the polar basis vectors are defined at that point.

If you repeat this procedure at another point in a different direction from the origin, you will again find that its position vector is $r \hat r,$ which may seem paradoxical until you realize that you are working in a different tangent space there and the $\hat r$ of that tangent space is not the same as the $\hat r$ of the tangent space at the first point.

To put it another way, although you may be able to use a single Cartesian basis in exactly the same way at every point in the plane, there is no single "polar basis" that you can use in exactly the same way at every point in the plane. The polar basis at Cartesian coordinates $(1,0)$ is different from the polar basis at Cartesian coordinates $(0,1).$

There is no linear transformation between the Cartesian coordinates of a point and the polar coordinates of a point.