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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.
The integral, 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'.