Line, surface and volume integrals

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Line and surface integrals involve integrating over scalar or vector fields, with infinitesimal segments treated as either vectors or scalars. The types of line and surface integrals include combinations of vector and scalar fields, such as vector x vector and scalar . vector. Volume integrals are more straightforward, as the infinitesimal volume segment is treated as a scalar, leading to fewer types of integrals, primarily scalar . scalar and vector . scalar. The discussion also touches on the potential for integrating within a volume using directional vectors, particularly in the context of extending the divergence theorem to four-dimensional space. The completeness of the identified line and surface integrals is questioned, indicating a need for further clarification.
LucasGB
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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.
 
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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.
 
I see. But are the line and surface integals complete?
 

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