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Divergence theorem - Intuition | | Divergence theorem - Intuition: Encyclopedia II - Divergence theorem - Intuition | | The intuitive content is simple. 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 water 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 ...
See also:Divergence theorem, Divergence theorem - Intuition, Divergence theorem - Mathematical statement, Divergence theorem - Example, Divergence theorem - Applications, Divergence theorem - Electrostatics, Divergence theorem - Gravity, Divergence theorem - History | | | Divergence theorem, Divergence theorem - Applications, Divergence theorem - Electrostatics, Divergence theorem - Example, Divergence theorem - Gravity, Divergence theorem - History, Divergence theorem - Intuition, Divergence theorem - Mathematical statement | | |
| | | Divergence theorem: Encyclopedia II - Divergence theorem - Intuition
Divergence theorem - Intuition
The intuitive content is simple. 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 water 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.
Other related archives1762, 1813, 1825, 1831, Bouguer plate, Carl Friedrich Gauss, G, GFDL, Gauss's law, George Green, Green's theorem, Joseph Louis Lagrange, Mathematical theorems, Mikhail Vasilievich Ostrogradsky, PlanetMath, Stokes theorem, Vector calculus, boundary, compact, conservation law, continuously differentiable, divergence, electrostatics, flow, fluid dynamics, fundamental theorem of calculus, gravitational field, gravity anomalies, normals, physics, piecewise, shell theorem, smooth, surface integrals, unit sphere, vector calculus, vector field, volume
 Adapted from the Wikipedia article "Intuition", under the G.N U Free Docmentation License. Please also see http://en.wikipedia.org/wiki |
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