In the previous section, we discussed vector differentiation. We can extend our notion of integration of scalar functions to that of vector functions.
Let be a vector field defined over a region and let be a curve in this region. At each point will have a value. We consider an arc element and a unit tangent vector at a point on the curve. If we fix 2 points and on the curve and allow to slide along the curve from to , we get the path of integration.
Above integral is known as the line integral.
Note that as , . Hence, above integral can also be written as
If the definite integral does not depend on the path , the field is known as conservative field. It can be shown that this is true, when there exists a scalar point function , such that .
The closed path integral, where coincides with , , is in a conservative field.
Gauss’ Divergence Theorem
The surface integral of a vector point function over a surface is defined as the component of normal to the surface , taken over the entire surface. In other words,
is the surface integral. Since the elementary area can be expressed as , the surface integral is actually a double integral.
Gauss’ Divergence Theorem states that the surface integral of the normal component of the function taken over a closed surface is equal to the volume integral of taken over the volume enclosed by the surface . In other words,
The surface integral of curl of normal component of a vector point function taken over an open surface bounded by a closed curve is equal to the line integral of the tangential component of over . Thus,
The Green’s lemma (or theorem) is a special case of Stokes’ theorem, when the surface is in plane and thus .