# Divergence theorem

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A formula in the integral calculus of functions in several variables that establishes a link between an -fold integral over a domain and an -fold integral over its boundary. Let the functions and their partial derivatives , , be Lebesgue integrable in a bounded domain whose boundary is the union of a finite set of piecewise-smooth -dimensional hypersurfaces oriented using the exterior normal . The Ostrogradski formula then takes the form

 (1)

If , , are the direction cosines of the exterior normals of the hypersurfaces forming the boundary of , then formula (1) can be expressed in the form

 (2)

where is the -dimensional volume element in while is the -dimensional volume element on .

In terms of the vector field , the formulas (1) and (2) signify the equality of the integral of the divergence of this field over the domain to its flux (see Flux of a vector field) over the boundary of :

For smooth functions, the formula was first obtained for the -dimensional case by M.V. Ostrogradski in 1828 (published in 1831, see [1]). He later extended it (1834) to cover -fold integrals for an arbitrary natural (published in 1838, see [2]). Using this formula, Ostrogradski found an expression for the derivative with respect to a parameter of an -fold integral with variable limits, and obtained a formula for the variation of an -fold integral; in one particular case, where , the formula was obtained by C.F. Gauss in 1813, for this reason it is also sometimes called the Ostrogradski–Gauss formula. A generalization of this formula is the Stokes formula for manifolds with boundary.

#### References

 [1] M.V. Ostrogradski, Mém. Acad. Sci. St. Petersbourg. Sér. 6. Sci. Math. Phys. et Naturelles , 1 (1831) pp. 117–122 [2] M.V. Ostrogradski, Mém. Acad. Sci. St. Petersbourg. Sér. 6. Sci. Math. Phys. et Naturelles , 1 (1838) pp. 35–58