- Vectors:
But

**Note Well:**is now a function of ! Similarly:with a function of the angles that define the direction of . Specifically:

- Unit vectors (relative to Cartesian
:
- Direct Length
- Directed Area
- Volume Element
- Gradient:
- Divergence;
The divergence is constructed by the same argument that proves the
divergence theorem in a general curvilinear coordinate system, or
alternatively picks up pieces from
, etc, hence its
complexity:
Note that this follows from:

with , , , and , , . Take the contribution from :

because does not depend on , similarly for the other two pieces.

- Curl
The curl is evaluated in exactly the same way from the expression above,
but it ends up being much more complex:
- Laplacian
The Laplacian follows by applying the divergence rule to the gradient
rule and simplifying: