10.4 Application: Meaning of Divergence and Curl
The Divergence Theorem and Stokes's Theorem provide the interpretation of the
divergence and curl that we have given above.
over a surface S, measures the flux of v through the surface, which is
proportional to the number of arrows of v that cross S.
By the divergence theorem if we take a tiny region V, the integral of div
v over this region (which is the average value of div v in it
times the volume of V), is the net outflux of v over the surface
of V. Thus this outflux, which for V centered on the point r' is a measure
of the number of v arrows originating from around the point r'
is directly proportional to the average divergence of v around r'.
An exactly analogous interpretation of Stokes's Theorem on a surface S including
the point r' provides our interpretation of the curl. The circulation
integral of v around a small cycle encircling r can be interpreted as
the difference between the path integral of v going around r'
on one side and the other. By Stokes' Theorem, this is proportional to the area
of the region between the paths times the average value of the component of
curl v normal to S in that area. Thus Stokes' Theorem means that
the average component of curl v normal to S around r' is directly
proportional to the amount of path dependence of v in S produced in the
neighborhood of r'.