Gauss` Theorem or Gauss` Divergence Theorem


What is the Gauss` Theorem or Gauss` Divergence Theorem?

According to Gauss theorem “If a volume V is enclosed by a surface S then the volume integral of the divergence of a vector field F over V is equal to the surface integral of

over the surface S. i.e.

Thus, the divergence theorem equates a surface integral with a triple integral over the volume inside the surface. The L.H.S. of the above equation represents the total flux of vector field F diverging through the volume V while R.H.S. represents the total flux of that vector field through the closed surface S. The divergence theorem finds its great significance in electrostatics and fluid dynamics.

Physical interpretation of Gauss` Theorem or Gauss` Divergence Theorem:

Let us consider that a vector field F that represents the velocity field of a fluid flow in a three dimensional region of volume V and which is enclosed by a surface S. Then the volume integral of divergence of  ( ) over the volume V represents the expansion (if  of the fluid inside the region. On the other hand, the surface integral of  over the surface S represents the net flow of fluid through the surface S. For example, imagine that a gas is produced inside a container by some chemical reactions. As the volume of the gas increases then it starts to expand. Let the gas is somehow allow to leak out through the walls of the container. Then according to Gauss`s theorem the total expansion of the gas inside the container equals the total outward flow of the gas through the walls.

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