Math 2.0

Global analysis, analysis on manifolds Definition

Exterior derivative

Builds on Differential formWedge product of formsOne-formPartial derivativeGradientCurlDivergence

Definition

Let U⊆ℝnU \subseteq \mathbb{R}^n be open, and let ω=∑IfI dxI\omega = \sum_I f_I \, dx_I be a differential k-form on UU whose coefficients fIf_I are of class C1C^1. The exterior derivative of ω\omega is the (k+1)(k + 1)-form dω=∑I∑j=1n∂fI∂xj dxj∧dxI,d\omega = \sum_I \sum_{j=1}^n \frac{\partial f_I}{\partial x_j} \, dx_j \wedge dx_I, where ∧\wedge is the wedge product.

Remarks

For a function ff, df=∑j∂jf dxjdf = \sum_j \partial_j f \, dx_j is its differential, the one-form that corresponds to the gradient. The operator dd is linear and satisfies d(α∧β)=dα∧β+(−1)kα∧dβd(\alpha \wedge \beta) = d\alpha \wedge \beta + (-1)^k \alpha \wedge d\beta for a kk-form α\alpha. For C2C^2 forms, d d = 0. In the plane, d(L dx+M dy)=(∂xM−∂yL) dx∧dyd(L \, dx + M \, dy) = (\partial_x M - \partial_y L) \, dx \wedge dy. In ℝ3\mathbb{R}^3, d(F1 dx+F2 dy+F3 dz)d(F_1 \, dx + F_2 \, dy + F_3 \, dz) has the components of the curl of FF as coefficients of dy∧dzdy \wedge dz, dz∧dxdz \wedge dx and dx∧dydx \wedge dy. Also d(F1 dy∧dz+F2 dz∧dx+F3 dx∧dy)=(∇⋅F) dx∧dy∧dzd(F_1 \, dy \wedge dz + F_2 \, dz \wedge dx + F_3 \, dx \wedge dy) = (\nabla \cdot F) \, dx \wedge dy \wedge dz, which is the divergence.

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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 derivativedefinition
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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

Where this leads

10 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Exterior_derivative, In terms of local coordinates
  2. Web https://en.wikipedia.org/wiki/Exterior_derivative, In terms of axioms
  3. Web https://en.wikipedia.org/wiki/Exterior_derivative, Exterior derivative in vector calculus
About this page
Tag004FRCXH6M
URIm2://004FRCXH6M
Kinddefinition
WritingW1 stated
MSC58A10
DefinesExterior derivative
UsesDifferential form, Wedge product of forms, One-form, Partial derivative, Gradient, Curl, Divergence

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