Consider the polar coordinate transformation
$$ x = r\cos \theta $$ $$ y = r\sin \theta $$
I am trying to find the most direct way to compute the coordinate basis vectors
$$ \frac{\partial}{\partial x}, \frac{\partial}{\partial y} $$
The exterior derivatives of $x$ and $y$ are given by
$$ dx = \cos \theta dr - r\sin\theta d\theta $$ $$ dy = \sin \theta dr + r \cos \theta d\theta $$ Considering that $dx$ and $dy$ comprise a basis in the cotangent space dual to the tangent space with basis elements $\frac{\partial}{\partial x}$ and $\frac{\partial}{\partial y}$, in principle, one ought to be able to compute the latter by using the duality relations $$ dx \left(\frac{\partial}{\partial x}\right) = 1, dx \left(\frac{\partial}{\partial y}\right) = 0, dy \left(\frac{\partial}{\partial y}\right) = 1, dy \left(\frac{\partial}{\partial x}\right) = 0 $$
Unfortunately, I am unable to make the algebra for this work out as expected. Is this approach promising, and perhaps I'm simply missing some algebraic manipulations or is there a better approach to this problem?