Vectors transforming under change of coordinates

Question

I was watching a lecture on tensors and the professor said that a defining feature of a vector $v$ is that it transforms under a coordinate transformation $x^{\mu} \rightarrow x^{\mu'}$ as

$$v^{\mu'}(x^{\mu'}) \equiv \frac{\partial x^{\mu'} }{\partial x^\mu} v^{\mu}(x^{\mu}(x^{\mu'})) $$

Basically the last last term on the right hand side is the $x^{\mu}$ coordinates expressed in terms of $x^{\mu'}$ coordinates.

I am trying to understand this in $\mathbb R^2$ for the Cartesian and Polar coordinates. If the $x^{\mu'} \equiv (x,y)$ and $x^{\mu} \equiv (r,\theta)$, then I get

$$\bigg(\begin{matrix}x\\y \end{matrix}\bigg) \equiv \bigg[ \begin{matrix} \cos\theta&-r\sin\theta\\ \sin\theta&r\cos\theta \end{matrix}\bigg] \bigg(\begin{matrix}r\\\theta \end{matrix}\bigg)$$

which doesn't work out to be true. Can you please tell me what I'm missing here?

Answer 1

This transformation equation applies to a vector $v$ that "lives at" at the point $x$ (i.e., is in the tangent space at $x$). The point $x$ is described by the coordinates $(x, y)$ or $(r, \theta)$, but these coordinate tuples are not vectors at $x$. Examples of vectors at $x$ are $(dx, dy)$ and $(dr, d\theta)$, and your matrix equation applies to them:

$$\bigg(\begin{matrix}dx\\dy \end{matrix}\bigg) = \bigg[ \begin{matrix} \cos\theta&-r\sin\theta\\ \sin\theta&r\cos\theta \end{matrix}\bigg] \bigg(\begin{matrix}dr\\d\theta \end{matrix}\bigg)$$