Considering a vector field in two dimensions, $\vec V(x,y)$, I know that the derivative matrix (Jacobian matrix) is given by:
$\nabla \vec V(x,y) = \begin {bmatrix} \partial V_x/\partial x &\partial V_x/\partial y \\ \partial V_y/\partial x &\partial V_y/\partial y \\ \end{bmatrix}$
What is the proper formula to find the derivative matrix of a polar vector field in two dimensions $\vec P(r,\theta)$? I assume it is different as the $\nabla$ operator is defined in polar as:
$\nabla = \partial_r\ \vec e_r +(1/r)\partial_\theta \ \vec e_\theta $