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In vector calculus, the divergence theorem, also known as Gauss' theorem or Ostrogradsky-Gauss theorem is a result that links the divergence of a vector field to the value of surface integrals of the flow defined by the field. The divergence theorem is an important result for the mathematics of physics, in particular in electrostatics and fluid dynamics.
The intuitive content is simple: if there's water flowing in some area, and you are interested in how much water flows out of a certain region within that area, then you 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 in fact the case.
The divergence theorem is thus a conservation law, stating that the volume total of all sinks and sources, i.e. the volume integral of the divergence, is equal to the net flow across the volume's boundary.
Suppose V is a subset of Rn (think of the case n=3 for now) 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
where S = ∂V is the boundary of V oriented by outward-pointing normals, and dS is shorthand for NdS, the outward pointing normal of the boundary ∂V.
We note that Gauss' theorem follows from the more general Stokes' theorem, which itself generalizes the fundamental theorem of calculus.
Applied to an electrostatic field we get Gauss's law: the divergence is a constant times the volume charge density.
Applied to a gravitational field we get that the surface integral is -4π GAccording to the law of universal gravitation, the attractive force between two bodies is proportional to the product of their masses and inversely proportional to the square of the distance between them. The constant of proportionality is called , the gr times the mass inside, regardless of how the mass is distributed, and regardless of any masses outside.
In the case of a spherically symmetric mass distribution we can conclude from this that the field strength at a distance r from the center is inward with a magnitude of G/rē times the total mass at a smaller distance, regardless of any masses at a larger distance.
For example, a hollow sphere does not produce any gravity inside. The gravitational field inside is the same as if the hollow sphere were not there (i.e. the field is that of any masses inside and outside the sphere only).
In the case of an infinite cylindrically symmetric mass distribution we can conclude that the field strength at a distance r from the center is inward with a magnitude of 2G/r times the total mass per unit length at a smaller distance, regardless of any masses at a larger distance.
For example, an infinite hollow cylinder does not produce any gravity inside.
We can conclude that for an infinite, flat plate (Bouguer plate) of thickness H gravity outside the plate is perpendicular to the plate, towards it, with magnitude 2πG times the mass per unit area, independent of the distance to the plate (see also gravity anomaliesDefinition Physical geodesy is the study of the physical properties of the gravity field of the Earth, the geopotential, with a view to their application in geodesy. Traditional geodetic instruments such as theodolites rely on the gravity field for orient).
More generally, for a mass distribution with the density depending on one Cartesian coordinate z only, gravity for any z is 2πG times the difference in mass per unit area on either side of this z value.
In particular, a combination of two equal parallel infinite plates does not produce any gravity inside.
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