Divergensteoremet

Divergensteoremet, commonly called the divergence theorem or Gauss’s theorem, is a central result in vector calculus. It states that the flux of a smooth vector field through the closed boundary of a region equals the total divergence of the field inside the region. In its standard form for a solid V in three-dimensional space with boundary surface S, oriented outward, and for a vector field F with continuous first partial derivatives on a region containing V, the theorem asserts that the surface flux equals the volume integral of the divergence: ∬_S F · n dS = ∭_V div F dV, where n is the outward unit normal to S.

Equivalently, using differential form notation, ∬_S F · dS = ∭_V div F dV. The divergence is div F = ∂F1/∂x + ∂F2/∂y + ∂F3/∂z, describing the rate at which F spreads out from a point.

Conditions typically required are that V is bounded with a piecewise smooth boundary S, and F is continuously differentiable on an open set containing V. The theorem provides a bridge between a local property of a field (its divergence) and a global property (net flux across the boundary).

Applications span physics and engineering, notably Gauss’s law in electromagnetism and the analysis of fluid flow, where it converts volume sources into boundary flux. The divergence theorem is a special case of more general results in vector calculus, and it relates to Stokes’ theorem and Green’s theorem in higher dimensions.