Difference between revisions of "Divergence theorem"
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Revision as of 09:10, 3 February 2014
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
:
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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 |
Comments
The result embodied in formula (2) above is most often known as the divergence theorem. It is (finally) equivalent to the Gauss formula (Gauss integral formula)
![]() |
where is the surface element for
,
is the outward pointing unit normal at
, and
is the
-th standard unit vector.
References
[a1] | H. Triebel, "Analysis and mathematical physics" , Reidel (1986) pp. Sect. 9.3.1 |
[a2] | A.M. Krall, "Applied analysis" , Reidel (1986) pp. 380 |
[a3] | A.P. Wills, "Vector analysis with an introduction to tensor analysis" , Dover, reprint (1958) pp. 97ff |
[a4] | C. von Westenholz, "Differential forms in mathematical physics" , North-Holland (1981) pp. 286ff |
Divergence theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Divergence_theorem&oldid=12667