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A different technique, which goes back to Laplace (1812), [3] is the following. Let = =. Since the limits on s as y → ±∞ depend on the sign of x, it simplifies the calculation to use the fact that e −x 2 is an even function, and, therefore, the integral over all real numbers is just twice the integral from zero to infinity.
Since the flux is defined as an integral of the electric field, this expression of Gauss's law is called the integral form. A tiny Gauss's box whose sides are perpendicular to a conductor's surface is used to find the local surface charge once the electric potential and the electric field are calculated by solving Laplace's equation.
By the Kelvin–Stokes theorem we can rewrite the line integrals of the fields around the closed boundary curve ∂Σ to an integral of the "circulation of the fields" (i.e. their curls) over a surface it bounds, i.e. = (), Hence the Ampère–Maxwell law, the modified version of Ampère's circuital law, in integral form can be rewritten as ((+)) =
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). [12]
The source free equations can be written by the action of the exterior derivative on this 2-form. But for the equations with source terms (Gauss's law and the Ampère-Maxwell equation), the Hodge dual of this 2-form is needed. The Hodge star operator takes a p-form to a (n − p)-form, where n is the number of dimensions.
Gauss's law for gravity is often more convenient to work from than Newton's law. [1] The form of Gauss's law for gravity is mathematically similar to Gauss's law for electrostatics, one of Maxwell's equations. Gauss's law for gravity has the same mathematical relation to Newton's law that Gauss's law for electrostatics bears to Coulomb's law.
It is an arbitrary closed surface S = ∂V (the boundary of a 3-dimensional region V) used in conjunction with Gauss's law for the corresponding field (Gauss's law, Gauss's law for magnetism, or Gauss's law for gravity) by performing a surface integral, in order to calculate the total amount of the source quantity enclosed; e.g., amount of ...
This relation is known as Gauss's law for electric fields in its integral form and it is one of Maxwell's equations. While the electric flux is not affected by charges that are not within the closed surface, the net electric field, E can be affected by charges that lie outside the closed surface. While Gauss's law holds for all situations, it ...