I'm struggling to understand the significance of these matrices based on partial derivatives, when the book I'm reading didn't make sense I found this article: https://en.m.wikiversity.org/wiki/PlanetPhysics/Transformation_Between_Cartesian_Coordinates_and_Polar_Coordinates
Computing the transformations at the end via programming didn't clear things up and when I looked at the equation for theta at the very bottom I found that it is always zero! $-xy/r^2+yx/r^2$
However the book uses the exact same equations... What am I missing? Theta always being zero makes zero sense to me... Even if the x and y in the vector are supposed to be different from the ones in the matrix theta would be zero in lots of weird situations.
You are right, the transformations at the very end of the linked article, as they are written, do not make sense. They should be intended as the transformations between the infinitesimals $(dr,d\theta)$ and $(dx,dy)$. For instance, the last one should be read as $$dr=\frac{xdx}{\sqrt{x^2+y^2}}+\frac{ydy}{\sqrt{x^2+y^2}}=\frac{xdx+ydy}{r}$$ and $$d\theta=-\frac{ydx}{x^2+y^2}+\frac{xdy}{x^2+y^2}=\frac{-ydx+xdy}{r^2}.$$ This webpage is much more clear in my opinion. Also the link given by Anne Bauval is very helpful: How do you show that $d\theta = \frac{x dy - y dx }{x^2 + y^2}$?