Math 2.0

Global analysis, analysis on manifolds Theorem

The exterior derivative squares to zero

Builds on Exterior derivativeWedge product of formsSchwarz's theorem for C^2 functionsCurl of a gradient is zeroDivergence of a curl is zeroClosed and exact forms

Statement

Let U⊆ℝnU \subseteq \mathbb{R}^n be open, and let ω\omega be a differential kk-form on UU with coefficients of class C2C^2. Then d(dω)=0.d(d\omega) = 0.

Remarks

Proof: for ω=f dxI\omega = f \, dx_I, the exterior derivative gives d(dω)=∑j,l∂l∂jf dxl∧dxj∧dxId(d\omega) = \sum_{j, l} \partial_l \partial_j f \, dx_l \wedge dx_j \wedge dx_I. By Schwarz's theorem, ∂l∂jf=∂j∂lf\partial_l \partial_j f = \partial_j \partial_l f, while dxl∧dxj=−dxj∧dxldx_l \wedge dx_j = -dx_j \wedge dx_l. So the terms cancel in pairs, and the terms with j=lj = l are zero. In ℝ3\mathbb{R}^3, the cases k=0k = 0 and k=1k = 1 are the identities curl of a gradient is zero and divergence of a curl is zero. It follows that each exact form with C2C^2 coefficients is closed.

Reading path

To understand this page from the ground up, read these 33 pages in order. The foundations come first.

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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…conceptPartial 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 derivativedefinitionGradient, definition, no evidence levelGradientdefinitionDifferentiable 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…definitionIterated derivative, definition, no evidence levelIterated derivativedefinitionSymmetry of second derivatives, statement, no evidence levelSymmetry of second de…statementC^k function characterised by continuous derivatives, statement, no evidence levelC^k function characte…statementDifferential form, definition, no evidence levelDifferential formdefinitionParametrised curve, definition, no evidence levelParametrised curvedefinitionVector field, definition, no evidence levelVector fielddefinitionSchwarz's theorem for C^2 functions, statement, no evidence levelSchwarz's theorem for…statementWedge product of forms, definition, no evidence levelWedge product of formsdefinitionConservative vector field, definition, no evidence levelConservative vector f…definitionRectifiable curve, definition, no evidence levelRectifiable curvedefinitionCurl, definition, no evidence levelCurldefinitionDivergence, definition, no evidence levelDivergencedefinitionLine integral of a vector field, definition, no evidence levelLine integral of a ve…definitionCurl of a gradient is zero, statement, no evidence levelCurl of a gradient is…statementDivergence of a curl is zero, statement, no evidence levelDivergence of a curl …statementOne-form, definition, no evidence levelOne-formdefinitionExterior derivative, definition, no evidence levelExterior derivativedefinitionClosed and exact forms, definition, no evidence levelClosed and exact formsdefinitionThe exterior derivative squares to zero, statement, no evidence levelThe exterior derivati…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.
  • no evidence level
  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. Partial derivative concept
  8. Continuity in R^n definition
  9. Differentiable function of several variables definition
  10. Partial derivative definition
  11. Gradient definition
  12. Differentiable on an open set definition
  13. Jacobian matrix definition
  14. Higher partial derivatives definition
  15. Iterated derivative definition
  16. Symmetry of second derivatives statement
  17. C^k function characterised by continuous derivatives statement
  18. Differential form definition
  19. Parametrised curve definition
  20. Vector field definition
  21. Schwarz's theorem for C^2 functions statement
  22. Wedge product of forms definition
  23. Conservative vector field definition
  24. Rectifiable curve definition
  25. Curl definition
  26. Divergence definition
  27. Line integral of a vector field definition
  28. Curl of a gradient is zero statement
  29. Divergence of a curl is zero statement
  30. One-form definition
  31. Exterior derivative definition
  32. Closed and exact forms definition
  33. The exterior derivative squares to zero statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Exterior_derivative, In terms of axioms
  2. Web https://en.wikipedia.org/wiki/Exterior_derivative, Closed and exact forms
  3. Web https://en.wikipedia.org/wiki/Vector_calculus_identities, Curl of gradient is zero
About this page
Tag0X308R2DBE
URIm2://0X308R2DBE
Aliasesd-squared-is-zero
Kindstatement, theorem
WritingW0 stub
MSC58A10
StatusThis copy of the site has no build data. Run m2 status 0X308R2DBE.
UsesExterior derivative, Wedge product of forms, Schwarz's theorem for C^2 functions, Curl of a gradient is zero, Divergence of a curl is zero, Closed and exact forms

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