Question

Can anybody explain divergence theorem to me?

Answer 1

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Let's say I have a rigid container filled with some gas. If the gas starts to expand but the container does not expand, what has to happen? Since we assume that the container does not expand (it is rigid) but that the gas is expanding, then gas has to somehow leak out of the container. (Or I suppose the container could burst, but that counts as both gas leaking out of the container and the container expanding.)


If I go to a gas station and pump air into one of my car's tires, what has to happen to the air inside the tire? (Assume the tire is rigid and does not expand as I put air inside it.) The air inside of the tire compresses.


These two examples illustrate the divergence theorem (also called Gauss's theorem). Recall that if a vector field F represents the flow of a fluid, then the divergence of F represents the expansion or compression of the fluid. The divergence theorem says that the total expansion of the fluid inside some three-dimensional region W equals the total flux of the fluid out of the boundary of W. In math terms, this means the triple integral of divF over the region W is equal to the flux integral (or surface integral) of F over the surface ∂W that is the boundary of W (with outward pointing normal):



"Math Insight." The Idea behind the Divergence Theorem -. NyKamp DQ, n.d. Web. 08 Jan. 2014.

Keywords: divergence, divergence theorem, surface integral, triple integral



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Answer 2

Above is a good explanation.

for the formula itself - see http://mathworld.wolfram.com/DivergenceTheorem.html

n u get an example on how to use it:
http://tutorial.math.lamar.edu/Classes/CalcIII/DivergenceTheorem.aspx

let me guess - Calculus 3 right? ;)

Answer 3

the divergence theorem just says you can find the total flux through a closed surface by summing up the "pressures" (divergences) in the volume bounded by said surface: