— 1. Vector Integration —
When dealing with vectors and the mathematical operation we have three basic options:
- Volume integration
- Surface integration
- Line (contour) integration
The result of integrating a vector, , over a volume is also a vector and the result is given by the following expression:
Hence, the result of vector integration is just three separate integration operations (one of each spatial dimension).
The result of integrating the projection of a vector over an area is what is called surface integration.
Surface integration is always done with the normal component of over the surface in question. Thus what we need to define first is the normal of a surface at a given point. will be this normal. We still have the ambiguity of having two possible directions for the normal at any given point, but this is taken care of by defining the normal to be on the outward direction of a closed surface.
Hence the quantity of interest is ( for example) with
As for the line integral it is define along the path between two points and . Again we have to consider the normal of , but this time the quantity of interest is:
The quantity is an element of length along and is taken to be positive along the direction in which the path is defined.