Everything about Gauss S Theorem totally explained
In
vector calculus, the
divergence theorem, also known as
Gauss's theorem (
Carl Friedrich Gauss),
Ostrogradsky's theorem (
Mikhail Vasilievich Ostrogradsky), or
Gauss-Ostrogradsky theorem is a result that relates the flow (that is,
flux) of a
vector field through a
surface to the behavior of the vector field inside the surface.
More precisely, the divergence theorem states that the outward
flux of a vector field through a surface is equal to the triple integral of the
divergence on the region inside the surface. Intuitively, it states that
the sum of all sources minus the sum of all sinks gives the net flow out of a region.
The divergence theorem is an important result for the mathematics of
engineering, in particular in
electrostatics and
fluid dynamics.
The theorem is a special case of the more general
Stokes' theorem, which generalizes the
fundamental theorem of calculus.
Intuition
If a fluid is flowing in some area, and we wish to know how much fluid flows out of a certain region within that area, then we need to add up the sources inside the region and subtract the sinks. The fluid flow is represented by a vector field, and the vector field's divergence at a given point describes the strength of the source or sink there. So, integrating the field's divergence over the interior of the region should equal the integral of the vector field over the region's boundary. The divergence theorem says that this is true.
The divergence theorem is thus a
conservation law which states that the volume total of all sinks and sources, the volume integral of the divergence, is equal to the net flow across the volume's boundary.
Mathematical statement
Suppose
V is a subset of
Rn (in the case of
n = 3,
V represents a volume in 3D space) which is
compact and has a
piecewise smooth boundary. If
F is a
continuously differentiable vector field defined on a neighborhood of
V, then we have
»
because the unit ball W has
volume 4π/3.
Applications
"Differential form" and "integral form" of physical laws
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). Three examples are
Gauss's law (in
electrostatics),
Gauss's law for magnetism, and
Gauss's law for gravity.
Continuity equations
Continuity equations offer more examples of laws with both differential and integral forms, related to each other by the divergence theorem. In
fluid dynamics,
electromagnetism,
quantum mechanics, and a number of other fields, there are
continuity equations that describe the conservation of mass, momentum, energy, probability, or other quantities. Generically, these equations state that the divergence of the flow of the conserved quantity is equal to the distribution of "sources" or "sinks" of that quantity. The divergence theorem states that any such continuity equation can be written in a differential form (in terms of a divergence) and an integral form (in terms of a flux).
Inverse-square laws
Any "inverse-square law" can instead be written in a "Gauss's law"-type form (with a differential and integral form, as described above). Two examples are
Gauss's law (in electrostatics), which follows from the inverse-square
Coulomb's law, and
Gauss's law for gravity, which follows from the inverse-square
Newton's law of universal gravitation. The derivation of the Gauss's law-type equation from the inverse-square formulation (or vice-versa) is exactly the same in both cases; see either of those articles for details.
History
The
theorem was first discovered by
Joseph Louis Lagrange in
1762, then later independently rediscovered by
Carl Friedrich Gauss in
1813, by
George Green in
1825 and in
1831 by
Mikhail Vasilievich Ostrogradsky, who also gave the first proof of the theorem. Subsequently, variations on the Divergence theorem are called Gauss's Theorem,
Green's theorem, and Ostrogradsky's theorem.
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This article was originally based on the GFDL article from PlanetMath at http://planetmath.org/encyclopedia/Divergence.html Further Information
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