Given a vector field determined by equation in polar coordinates how do I find the equivalent equation in cartesian coordinates?

Question

I have the expression of a vector field in polar coordinates, how do I find the expression in Cartesian one ?

$$X = \frac{\partial }{\partial \rho }+\frac{\partial }{\partial \theta }$$

Answer 1

If you are familiar with the transformation rule for vector, it's become easy to solve. That is, if you have two different coordinate charts $\tilde{x}^i$ and $x^i$, then the basis vector transform as (with summation convention) $$ \frac{\partial }{\partial \tilde{x}^i} = \frac{\partial x^j}{\partial \tilde{x}^i} \frac{\partial }{\partial {x}^j} $$ In your case, the coordinates are $(\rho, \theta)$ and $(x,y)$. So for example one of them transform as $$ \frac{\partial}{\partial \rho} = \frac{\partial x}{\partial \rho} \frac{\partial }{\partial x} + \frac{\partial y}{\partial \rho} \frac{\partial }{\partial y} $$ And you can find the explicit expression by finding the coefficient of $\frac{\partial }{\partial x}$ and $\frac{\partial}{\partial y}$ by the relation of the two coordinates, $$ x = \rho \cos \theta $$ $$ y = \rho \sin \theta $$