Math 2.0

Real functions Functions of several real variables Theorem

Green's theorem

Builds on Closed curveLine integral of a vector fieldDouble integralPartial derivative

Statement

Let CC be a positively oriented, piecewise smooth, simple closed curve in ℝ2\mathbb{R}^2, and let DD be the bounded region enclosed by CC. Let LL and MM be real functions with continuous partial derivatives on an open set that contains D∪CD \cup C. Then ∮C(L dx+M dy)=∬D(∂M∂x−∂L∂y)dA.\oint_C (L \, dx + M \, dy) = \iint_D \left( \frac{\partial M}{\partial x} - \frac{\partial L}{\partial y} \right) dA.

Remarks

Positive orientation means that CC goes counterclockwise, so DD is on the left. Proof sketch: first let DD be a region between the graphs of two functions of xx. Write the double integral of ∂L/∂y\partial L / \partial y as an iterated integral, and use the fundamental theorem of calculus. This gives ∮CL dx=−∬D∂L/∂y dA\oint_C L \, dx = -\iint_D \partial L / \partial y \, dA. The MM term is similar, and general regions are cut into such pieces. Example: with L=−yL = -y and M=xM = x, the right side is 22 times the area of D. Green's theorem is the planar case of Stokes' theorem. George Green stated a similar result in 1828, and Riemann gave a proof in 1851.

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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…conceptContinuity 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 …definitionHigher partial derivatives, definition, no evidence levelHigher partial deriva…definitionIterated 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 curvedefinitionLine integral of a vector field, definition, no evidence levelLine integral of a ve…definitionGreen's theorem, statement, no evidence levelGreen's 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.
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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. Continuity in R^n definition
  8. Differentiable function of several variables definition
  9. Partial derivative definition
  10. Double integral definition
  11. Differentiable on an open set definition
  12. Higher partial derivatives definition
  13. Iterated derivative definition
  14. C^k function characterised by continuous derivatives statement
  15. Parametrised curve definition
  16. Vector field definition
  17. Closed curve definition
  18. Rectifiable curve definition
  19. Line integral of a vector field definition
  20. Green's theorem statement

Where this leads

20 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Green%27s_theorem, Theorem
  2. Web https://en.wikipedia.org/wiki/Green%27s_theorem, Proof when D is a simple region
  3. Web https://en.wikipedia.org/wiki/Green%27s_theorem, History
About this page
Tag0HR3N02317
URIm2://0HR3N02317
Kindstatement, theorem
WritingW1 stated
MSC26B20
StatusThis copy of the site has no build data. Run m2 status 0HR3N02317.
UsesClosed curve, Line integral of a vector field, Double integral, Partial derivative
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