Integralsatz

Integralsatz, or integral theorem, is a term used in mathematics to denote a family of theorems that relate integrals over a region to integrals over its boundary or to other related integrals. In particular, in vector calculus the term commonly refers to theorems that connect volume or surface integrals with boundary integrals, providing powerful tools for computation and interpretation.

The most well-known Integralsätze are the divergence theorem, Green’s theorem and Stokes’ theorem. The divergence theorem (Gauss’s theorem) states that the integral of the divergence of a vector field over a volume equals the flux of the field through the boundary surface: ∭_V (∇·F) dV = ∬_S F·n dS, where S is the boundary of V with outward orientation. Green’s theorem is the two-dimensional version that relates a line integral around a simple closed curve to a double integral over the region it encloses: ∬_D (∂Q/∂x − ∂P/∂y) dA = ∮_C (P dx + Q dy). Stokes’ theorem generalizes these ideas to higher dimensions, equating the surface integral of the curl of a vector field over a surface to a line integral around its boundary: ∬_S (∇×F)·n dS = ∮_∂S F·dr.

Beyond these classical theorems, the term also encompasses the generalized Stokes’ theorem for differential forms, which unifies many such results as ∫_∂M ω = ∫_M dω on oriented manifolds. Applications span physics (electromagnetism, fluid dynamics), engineering, and computer science, where converting volume or surface integrals to boundary terms can simplify analysis and provide physical insight.