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More precisely, the divergence theorem states that the surface integral of a vector field over a closed surface, which is called the "flux" through the surface, is equal to the volume integral of the divergence over the region enclosed by the surface. Intuitively, it states that "the sum of all sources of the field in a region (with sinks ...
In physics (specifically electromagnetism), Gauss's law, also known as Gauss's flux theorem (or sometimes Gauss's theorem), is one of Maxwell's equations. It is an application of the divergence theorem , and it relates the distribution of electric charge to the resulting electric field .
Gauss's law for magnetism thus states that the net magnetic flux through a closed surface equals zero. The integral and differential forms of Gauss's law for magnetism are mathematically equivalent, due to the divergence theorem. That said, one or the other might be more convenient to use in a particular computation.
This identity is derived from the divergence theorem applied to the vector field F = ψ ∇φ while using an extension of the product rule that ∇ ⋅ (ψ X) = ∇ψ ⋅X + ψ ∇⋅X: Let φ and ψ be scalar functions defined on some region U ⊂ R d, and suppose that φ is twice continuously differentiable, and ψ is once continuously differentiable.
As a component of the 4D Gauss' Theorem / Stokes' Theorem / Divergence Theorem [ edit ] In vector calculus , the divergence theorem , also known as Gauss's theorem or Ostrogradsky's 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.
By the divergence theorem, Gauss's law for the field P can be stated in differential form as: =, where ∇ · P is the divergence of the field P through a given surface containing the bound charge density .
Mathematically, we can state the law of charge conservation as a continuity equation: = ˙ ˙ (). where / is the electric charge accumulation rate in a specific volume at time t, ˙ is the amount of charge flowing into the volume and ˙ is the amount of charge flowing out of the volume; both amounts are regarded as generic functions of time.
The following is a proof of half of the theorem for the simplified area D, a type I region where C 1 and C 3 are curves connected by vertical lines (possibly of zero length). A similar proof exists for the other half of the theorem when D is a type II region where C 2 and C 4 are curves connected by horizontal lines (again, possibly of zero ...