Math 2.0

Real functions Functions of several real variables Theorem

Divergence theorem

Builds on DivergenceFlux through a surfaceOrientable surfaceTriple integralDivergence theorem on a rectangle in the plane

Statement

Let V⊆ℝ3V \subseteq \mathbb{R}^3 be compact with a piecewise smooth boundary surface S=∂VS = \partial V, oriented by the outward unit normal nn. If FF is a vector field of class C1C^1 on an open set that contains VV, then ∭V∇⋅F dV=∬SF⋅n dS.\iiint_V \nabla \cdot F \, dV = \iint_S F \cdot n \, dS.

Remarks

The left side is the triple integral of the divergence, the total source strength in VV. The right side is the outward flux through the boundary. Example: for F(x,y,z)=(x,y,z)F(x, y, z) = (x, y, z) and the unit ball, ∇⋅F=3\nabla \cdot F = 3. So both sides equal 3⋅43π=4π3 \cdot \tfrac{4}{3} \pi = 4\pi. With gFg F in place of FF, for a C1C^1 function gg, one gets ∭V(F⋅∇g+g ∇⋅F) dV=∬Sg F⋅n dS\iiint_V (F \cdot \nabla g + g \, \nabla \cdot F) \, dV = \iint_S g \, F \cdot n \, dS. This is a form of integration by parts. The same statement holds in ℝn\mathbb{R}^n. For a rectangle in the plane, the proof is a direct use of the fundamental theorem of calculus.

History

Lagrange discovered the theorem in 1762, and Gauss proved special cases from 1813. Ostrogradsky gave the first proof of the general theorem in 1826, in his work on heat flow.

Reading path

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Euclidean space, object, no evidence levelEuclidean spaceobjectFunction, concept, no evidence levelFunctionconceptInfimum, concept, no evidence levelInfimumconceptSupremum, concept, no evidence levelSupremumconceptFréchet derivative, definition, no evidence levelFréchet derivativedefinitionFunction of several variables, concept, no evidence levelFunction of several v…conceptVector-valued function, concept, no evidence levelVector-valued functionconceptJordan measure, definition, no evidence levelJordan measuredefinitionLimit of a function of several variables, concept, no evidence levelLimit of a function o…conceptIterated integral, definition, no evidence levelIterated integraldefinitionPartial derivative, concept, no evidence levelPartial derivativeconceptContinuity in R^n, definition, no evidence levelContinuity in R^ndefinitionDifferentiable function of several variables, definition, no evidence levelDifferentiable functi…definitionPartial derivative, definition, no evidence levelPartial derivativedefinitionDouble integral, definition, no evidence levelDouble integraldefinitionDifferentiable on an open set, definition, no evidence levelDifferentiable on an …definitionGradient, definition, no evidence levelGradientdefinitionJacobian matrix, definition, no evidence levelJacobian matrixdefinitionHigher partial derivatives, definition, no evidence levelHigher partial deriva…definitionFubini's theorem, statement, no evidence levelFubini's theoremstatementTriple integral, definition, no evidence levelTriple integraldefinitionIterated derivative, definition, no evidence levelIterated derivativedefinitionC^k function characterised by continuous derivatives, statement, no evidence levelC^k function characte…statementParametrised curve, definition, no evidence levelParametrised curvedefinitionVector field, definition, no evidence levelVector fielddefinitionClosed curve, definition, no evidence levelClosed curvedefinitionRectifiable curve, definition, no evidence levelRectifiable curvedefinitionParametrised surface, definition, no evidence levelParametrised surfacedefinitionDivergence, definition, no evidence levelDivergencedefinitionLine integral of a vector field, definition, no evidence levelLine integral of a ve…definitionTangent plane and normal vector of a surface, definition, no evidence levelTangent plane and nor…definitionGreen's theorem, statement, no evidence levelGreen's theoremstatementSurface area, definition, no evidence levelSurface areadefinitionDivergence theorem on a rectangle in the plane, statement, no evidence levelDivergence theorem on…statementSurface integral, definition, no evidence levelSurface integraldefinitionOrientable surface, definition, no evidence levelOrientable surfacedefinitionFlux through a surface, definition, no evidence levelFlux through a surfacedefinitionDivergence theorem, statement, no evidence levelDivergence theoremstatement
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Euclidean space object
  2. Function concept
  3. Infimum concept
  4. Supremum concept
  5. Fréchet derivative definition
  6. Function of several variables concept
  7. Vector-valued function concept
  8. Jordan measure definition
  9. Limit of a function of several variables concept
  10. Iterated integral definition
  11. Partial derivative concept
  12. Continuity in R^n definition
  13. Differentiable function of several variables definition
  14. Partial derivative definition
  15. Double integral definition
  16. Differentiable on an open set definition
  17. Gradient definition
  18. Jacobian matrix definition
  19. Higher partial derivatives definition
  20. Fubini's theorem statement
  21. Triple integral definition
  22. Iterated derivative definition
  23. C^k function characterised by continuous derivatives statement
  24. Parametrised curve definition
  25. Vector field definition
  26. Closed curve definition
  27. Rectifiable curve definition
  28. Parametrised surface definition
  29. Divergence definition
  30. Line integral of a vector field definition
  31. Tangent plane and normal vector of a surface definition
  32. Green's theorem statement
  33. Surface area definition
  34. Divergence theorem on a rectangle in the plane statement
  35. Surface integral definition
  36. Orientable surface definition
  37. Flux through a surface definition
  38. Divergence theorem statement

Where this leads

16 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Divergence_theorem, Mathematical statement
  2. Web https://en.wikipedia.org/wiki/Divergence_theorem, History
  3. Web https://en.wikipedia.org/wiki/Divergence_theorem, Corollaries
About this page
Tag0JX86PSSA9
URIm2://0JX86PSSA9
Aliasesgauss-theorem, gauss-ostrogradsky-theorem
Kindstatement, theorem
WritingW1 stated
MSC26B20
StatusThis copy of the site has no build data. Run m2 status 0JX86PSSA9.
UsesDivergence, Flux through a surface, Orientable surface, Triple integral, Divergence theorem on a rectangle in the plane
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