Math 2.0

Real functions Proposition

Divergence theorem on a rectangle in the plane

Builds on DivergenceFubini's theoremDouble integralGreen's theorem

Statement

Let a<ba < b and c<dc < d, and let R=[a,b]×[c,d]R = [a, b] \times [c, d]. Let F=(P,Q)F = (P, Q) be a vector field of class C1C^1 on an open set that contains RR. Then ∬R(∂P∂x+∂Q∂y)dA=∫cd(P(b,y)−P(a,y)) dy+∫ab(Q(x,d)−Q(x,c)) dx.\iint_R \left( \frac{\partial P}{\partial x} + \frac{\partial Q}{\partial y} \right) dA = \int_c^d \big( P(b, y) - P(a, y) \big) \, dy + \int_a^b \big( Q(x, d) - Q(x, c) \big) \, dx.

Remarks

The right side is the outward flux ∮∂RF⋅n ds\oint_{\partial R} F \cdot n \, ds. Here the outward unit normal nn is (1,0)(1, 0), (−1,0)(-1, 0), (0,1)(0, 1) and (0,−1)(0, -1) on the right, left, top and bottom sides. Proof: by Fubini's theorem, ∬R∂P/∂x dA=∫cd∫ab∂P/∂x dx dy\iint_R \partial P / \partial x \, dA = \int_c^d \int_a^b \partial P / \partial x \, dx \, dy. The fundamental theorem of calculus turns the inner integral into P(b,y)−P(a,y)P(b, y) - P(a, y). The QQ term is similar. This is the simplest case of the divergence theorem. In the plane, the divergence theorem is equivalent to Green's theorem.

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Euclidean space, object, no evidence levelEuclidean spaceobjectFunction, concept, no evidence levelFunctionconceptFréchet derivative, definition, no evidence levelFréchet derivativedefinitionFunction of several variables, concept, no evidence levelFunction of several v…conceptVector-valued function, concept, no evidence levelVector-valued functionconceptLimit 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 …definitionJacobian matrix, definition, no evidence levelJacobian matrixdefinitionHigher partial derivatives, definition, no evidence levelHigher partial deriva…definitionFubini's theorem, statement, no evidence levelFubini's theoremstatementIterated 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 curvedefinitionDivergence, definition, no evidence levelDivergencedefinitionLine integral of a vector field, definition, no evidence levelLine integral of a ve…definitionGreen's theorem, statement, no evidence levelGreen's theoremstatementDivergence theorem on a rectangle in the plane, statement, no evidence levelDivergence theorem on…statement
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.
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  1. Euclidean space object
  2. Function concept
  3. Fréchet derivative definition
  4. Function of several variables concept
  5. Vector-valued function concept
  6. Limit of a function of several variables concept
  7. Iterated integral definition
  8. Partial derivative concept
  9. Continuity in R^n definition
  10. Differentiable function of several variables definition
  11. Partial derivative definition
  12. Double integral definition
  13. Differentiable on an open set definition
  14. Jacobian matrix definition
  15. Higher partial derivatives definition
  16. Fubini's theorem statement
  17. Iterated derivative definition
  18. C^k function characterised by continuous derivatives statement
  19. Parametrised curve definition
  20. Vector field definition
  21. Closed curve definition
  22. Rectifiable curve definition
  23. Divergence definition
  24. Line integral of a vector field definition
  25. Green's theorem statement
  26. Divergence theorem on a rectangle in the plane statement

Where this leads

17 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/Green%27s_theorem, Relationship to the divergence theorem
  3. Web https://en.wikipedia.org/wiki/Green%27s_theorem, Proof when D is a simple region
About this page
Tag0GZ3D8F921
URIm2://0GZ3D8F921
Aliasesplanar-divergence-theorem-rectangle
Kindstatement, proposition
WritingW0 stub
MSC26B20
StatusThis copy of the site has no build data. Run m2 status 0GZ3D8F921.
UsesDivergence, Fubini's theorem, Double integral, Green's theorem

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