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Divergence Theorem (aka Gauss's Theorem and different from Gauss's Law)
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The divergence of a vector field, F, is defined as:
_ _
∇.F = ∂F_{x}/∂x + ∂F_{y}/∂y + ∂F_{z}/∂z --- a SCALAR
The divergence can be regarded as the spreading of a field away from center.
Consider a region bounded by a 3D surface (i.e. a sphere). Then, the Divergence
Theorem can be stated as:
^{ } _ _ _
∫∇.F dV = ∫(F.n) dσ
where dσ is a small part of the surface area.
The divergence theorem is a mathematical statement of the physical fact that, in the
absence of the creation or destruction of matter, the density within a region of
space can change only by having it flow into or away from the region through its
boundary. For example, imagine a lake with constant depth fed by a source. The
water has to 'overflow' through its boundary for the lake to stay at the same
depth and keep M/V = ρ constant.
Example: A sphere
^{ } _
∫∇.F dV = F4πR^{2}
Let LHS = M (mass) so that,
_ _
=> F = (M/4πR^{2})(R/R)
We have considered a point mass but it doesn't matter how big the source is as
long as it doesn't exceed the size of the sphere. A real gravitational field
has negative divergence (convergence)