Since ? is a dummy variable of integration, and since the change in the boundary ? is infinitesimal by assumption, the two integrals may be combined using the four-dimensional version of the divergence theorem into the following form
32.
The Gauss s divergence theorem shows that the diffusion equation is valid in the solid, liquid and gas states in every material as a material conservation law, if there is no sink and source in the given diffusion system.
33.
The divergence theorem is employed in any conservation law which states that the volume total of all sinks and sources, that is the volume integral of the divergence, is equal to the net flow across the volume's boundary.
34.
Then, as first demonstrated by Newton, and can easily be demonstrated using the divergence theorem, the acceleration of gravity at any given distance R from the center of the sphere depends only upon the total mass contained within R.
35.
The exterior derivative plays the role of the gradient and curl of vector calculus, and Stokes'theorem simultaneously generalizes the three theorems of vector calculus : the divergence theorem, Green's theorem, and the Kelvin-Stokes theorem.
36.
If u and \ vec f are sufficiently smooth functions, we can use the divergence theorem and change the order of the integration and \ partial / \ partial t to get a conservation law for the quantity u in the general form
37.
The divergence theorem states that the net outflux through a closed surface, in other words the net outflux from a 3D region, is found by adding the local net outflow from each point in the region ( which is expressed by the divergence ).
38.
The fundamental theorems of integral calculus in several variables namely Green's theorem, the divergence theorem, and Stokes'theorem & mdash; generalize to a theorem ( also called Stokes'theorem ) relating the exterior derivative and integration over submanifolds.
39.
As a result of the divergence theorem, a host of physical laws can be written in both a differential form ( where one quantity is the divergence of another ) and an integral form ( where the flux of one quantity through a closed surface is equal to another quantity ).
40.
In particular, the key fact is that, for vector fields \ mathbf { F } and \ mathbf { G }, integration by parts ( or the divergence theorem ) over a volume " V " enclosed by a surface " S " gives the identity:
