Divergence theorem

The divergence theorem equates the outward flux of a vector field through a closed surface with the volume integral of its divergence over the enclosed region.

Divergence: A scalar measure of a vector field’s net outward flow per unit volume at a point. The theorem integrates this local measure throughout the enclosed volume.
Vector field: A function assigning a vector to each point in a region of space. The theorem applies to a field whose values define flux and divergence.
Gauss's law: A law relating electric flux through a closed surface to the electric charge enclosed. The theorem turns its flux statement into a local equation involving charge density.
Carl Friedrich Gauss: A German mathematician and physicist whose work shaped number theory, geometry, and mathematical physics. The theorem is commonly associated with his work on flux and gravitational fields.
Flux: The signed amount of a vector field crossing an oriented surface. The theorem equates total outward surface flux with integrated divergence.
Surface integral: An integral that accumulates a scalar or vector quantity over a surface. The boundary side of the theorem is a surface integral of normal flux.
Continuity equation: A differential equation expressing local conservation of a quantity through its density and current. The theorem converts integral conservation over a region into a local balance law.
George Green: An English mathematical physicist known for foundational work on potential theory and integral theorems. His planar integral theorem belongs to the same family of boundary-to-interior results.
Gauss's theorem in electrostatics: An electrostatic law relating electric flux through a closed surface to the charge enclosed. It applies the divergence theorem to electric fields and charge density.
Volume integral: An integral that accumulates a quantity throughout a three-dimensional region. The interior side integrates divergence over the enclosed volume.

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