The Del Operator in Other Coordinates

To convert the familiar del operator

$$\vec{\nabla} = \frac{\partial}{\partial x} \hat{i} + \frac{\partial}{\partial y} \hat{j} + \frac{\partial}{\partial z}\hat{k}$$

into a new coordinate system, we need to compute two things:

1.

The change in the partial derivatives: We can use the chain rule to determine what the partial derivatives look like in the new uvw coordinate system:

$$\frac{\partial}{\partial x} = \frac{\partial}{\partial u}\fr... ... x} + \frac{\partial}{\partial w}\frac{\partial w}{\partial x},$$

and so forth for the other partial derivative operators.

2.

The basis vectors in the new coordinates:

The basis vectors of the new coordinate system are not going to be $\hat{i}, \hat{j}, \hat{k}$. They'll be some vectors $\vec{e}_u, \vec{e}_v, \vec{e}_w$ that may not even be unit vectors, or have a constant direction.

All in all, the computations involved are quite complex. For a good description of this, check out section 8.11 (pp. 498-504) in Kreyszig's Advanced Engineering Mathematics, sixth edition.