The flux through a surface (or the integral of the vector flux density g over a closed surface) equals the divergence of the flux density integrated over the volume contained by the surface:
Another way of writing this is
where is the area element on the bounding surface of volume , with volume element .
Commonly called Gauss's theorem or the Gauss-Ostrogradski theorem .
Generalization to arbitrary dimensions
The divergence theorem is a general mathematical result that may be applied in arbitrary dimensions.
If we have an integral over the volume in bounded by in then the following relation holds
Here, is a vector field and the integrand of the (hyper-)volume integral is an exact divergence of that vector field. The quantity is the outward-pointing
normal vector to the (hyper-) surface Each is the respective direction cosine
obtained by forming the inner product of the normal to the boundary with the respective coordinate axis unit vector.
Because we are in arbitrary dimensions and .
Consider the case when and is a three-dimensional volume bounded by the two-dimensional
boundary . The volume integral may be written in three terms, one for each coordinate direction
If we concentrate on the volume integral and apply the fundamental theorem of calculus we obtain
Here, the functions and define points on of greater
and lesser respectively.
Now, if we consider the surface integral
and concentrate on the term
On the part of that is of greater the values of
and . On the part of the surface that of lesser values of the values of and . Here on the respective portion of the surface.
Thus, the surface integral for the is the same as the volume integral of the
Similar results may be found for the and , proving the divergence theorem
for 3 dimensions.
This same method generalizes for the case of dimensions, proving the theorem for the arbitrary case.
- ↑ Greenspan, H. P., Benney, D. J., & Turner, J. E. (1986). Calculus: an introduction to applied mathematics. McGraw-Hill Ryerson Ltd.