The divergence theorem is valid for piecewise smooth domains " D ", and hence " D " needs to be piecewise smooth.
22.
This theorem is a special case of the divergence theorem, and is essentially the higher dimensional equivalent of integration by parts with and the gradient of replacing and.
23.
This classical statement, along with the classical divergence theorem, the fundamental theorem of calculus, and Green's theorem are simply special cases of the general formulation stated above.
24.
Only the " differential form " of the equations is given, not the " integral form "; to get the integral forms apply the divergence theorem or Kelvin Stokes theorem.
25.
The third equality follows by the divergence theorem and shows, again, a sum ( or, in this case, an integral ) of outward normal derivatives over all boundary locations.
26.
More precisely, the divergence theorem states that the outward flux of a vector field through a closed surface is equal to the volume integral of the divergence over the region inside the surface.
27.
The divergence theorem states that any such continuity equation can be written in a differential form ( in terms of a divergence ) and an integral form ( in terms of a flux ).
28.
The direct computation of this integral is quite difficult, but we can simplify the derivation of the result using the divergence theorem, because the divergence theorem says that the integral is equal to:
29.
The direct computation of this integral is quite difficult, but we can simplify the derivation of the result using the divergence theorem, because the divergence theorem says that the integral is equal to:
30.
The divergence at a point represents the degree to which a small volume around the point is a source or a sink for the vector flow, a result which is made precise by the divergence theorem.
