I have an expression for a magnetic field that I would like to express in (2D) polar coordinates as $\mathbf{B} = B_\theta(r)\mathbf{\hat{\theta}}$, where $B_\theta(r)$ is some known function of $r$. However, I would like to express this vector in terms of its Cartesian components $B_x$ and $B_y$.
I know how to express ($x$, $y$) in terms of ($r$, $\theta$). However, I do not know how to convert the vector components. I read that for an arbitrary vector, $$ \begin{pmatrix} F_x \\ F_y \end{pmatrix} = \begin{pmatrix} \cos\theta & -r\sin\theta \\ \sin\theta & r\cos\theta \end{pmatrix} \begin{pmatrix} F_r \\ F_\theta \end{pmatrix} $$ However, it then seems that $F_x^2 + F_y^2 = F_r^2 + r^2 F_\theta^2 \neq F_r^2 + F_\theta^2$, so it seems like the transformation is changing the magnitude of the vector, which seems wrong.
What am I getting wrong about this problem? How do I express $B_\theta$ in terms of $B_x$ and $B_y$?
I think that you're confusing polar coordinate / cartesian coordinates change, vs the change of coordinates in a rotated basis. In a rotated basis of angle $\theta$, the change of vector coordinates is just
$$ \begin{pmatrix} F_x \\ F_y \end{pmatrix} = \begin{pmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{pmatrix} \begin{pmatrix} F_r \\ F_\theta \end{pmatrix} $$ and magnitudes are the same.