The $\nabla$-operator is simple in cartesian coordinates, $[\partial_x,\partial_y,\partial_z]$, but in spherical coordinates, it becomes $[\partial_r, \frac{1}{r}\partial_\theta, \frac{1}{r\sin\theta}\partial_\varphi]$ and in cylindrical coordinates $[\partial_\rho, \frac{1}{\rho}\partial_\varphi, \partial_z]$; is there a general formula for converting into a different coordinate system, perhaps in terms of a Jacobian?
(Sub-question: Is there any reason why similar operators like $\hat{\nabla} = [\partial_r, \partial_\theta, \partial_\varphi]$ aren't in use?)
Best Answer
There is a large body of literature on this subject. I found a section in an old calculus book on Orthogonal Curvilinear Coordinates in the chapter on Vector Analysis. It's too complex to present here. (The book is Advanced Calculus for Applications by Hildebrand.) However, you can get started here: Curvilinear coordinates.