Line, surface and volume integrals

[LucasGB]

Please help me check if the following reasoning is correct:

When considering line and surface integrals, one must integrate over a scalar or vector field. The infinitesimal line (dl) or surface (dA) segments can be treated either as vectors or scalars. Therefore, the only types of line and surface integrals one can run into are:

Vector x vector
Vector . vector
Scalar . vector
Scalar . scalar
Vector . scalar

Volume integrals, on the other hand, are simpler, since the infinitesimal volume segment (dV) can only be treated as a scalar. Therefore, we can only run into the following types of volume integrals:

Scalar . scalar
Vector . scalar

Does this check out? Tell me what you think.

 

[ExtravagantDreams]

In theory one can also integrate a scalar or vector field within volume with some directional vector. This would come up if you extended the divergence theorem to a four-dimensional space.

 

[LucasGB]

I see. But are the line and surface integals complete?