Math 2.0

Global analysis, analysis on manifolds Definition

Closed and exact forms

Builds on Differential formExterior derivativeOne-formConservative vector fieldCurl

Definition

Let U⊆ℝnU \subseteq \mathbb{R}^n be open, and let ω\omega be a differential k-form on UU with C1C^1 coefficients, where k≥1k \ge 1. The form ω\omega is closed if dω=0d\omega = 0, where dd is the exterior derivative. The form ω\omega is exact if ω=dη\omega = d\eta for some (k−1)(k - 1)-form η\eta on UU.

Remarks

Each exact form dηd\eta with η\eta of class C2C^2 is closed, because d d = 0. A one-form ∑ifi dxi\sum_i f_i \, dx_i is closed if and only if ∂jfi=∂ifj\partial_j f_i = \partial_i f_j for all i,ji, j. It is exact if and only if (f1,…,fn)(f_1, \dots, f_n) is a conservative field. In ℝ3\mathbb{R}^3, closed means that the curl is zero. The angle form dθ=−y dx+x dyx2+y2d\theta = \frac{-y \, dx + x \, dy}{x^2 + y^2} on ℝ2∖{0}\mathbb{R}^2 \setminus \{0\} is closed but not exact. Its integral around the unit circle is 2π2\pi, not 00. On a convex open set, each closed form is exact. This is the Poincaré lemma.

Reading path

To understand this page from the ground up, read these 28 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 derivativedefinitionC^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 fielddefinitionWedge 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…definitionOne-form, definition, no evidence levelOne-formdefinitionExterior derivative, definition, no evidence levelExterior derivativedefinitionClosed and exact forms, definition, no evidence levelClosed and exact formsdefinition
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. 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. C^k function characterised by continuous derivatives statement
  17. Differential form definition
  18. Parametrised curve definition
  19. Vector field definition
  20. Wedge product of forms definition
  21. Conservative vector field definition
  22. Rectifiable curve definition
  23. Curl definition
  24. Divergence definition
  25. Line integral of a vector field definition
  26. One-form definition
  27. Exterior derivative definition
  28. Closed and exact forms definition

Where this leads

2 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Exterior_derivative, Closed and exact forms
  2. Web https://en.wikipedia.org/wiki/Closed_and_exact_differential_forms, Examples
  3. Web https://en.wikipedia.org/wiki/Closed_and_exact_differential_forms, Vector field analogies
About this page
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Kinddefinition
WritingW0 stub
MSC58A12
DefinesClosed and exact forms
UsesDifferential form, Exterior derivative, One-form, Conservative vector field, Curl

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