Math 2.0

Real functions Proposition

Divergence of a curl is zero

Builds on DivergenceCurlSchwarz's theorem for C^2 functions

Statement

Let U⊆ℝ3U \subseteq \mathbb{R}^3 be open, and let A ⁣:U→ℝ3A \colon U \to \mathbb{R}^3 be a vector field of class C2C^2. Then ∇⋅(∇×A)=0on U.\nabla \cdot (\nabla \times A) = 0 \quad \text{on } U.

Remarks

Proof: with A=(A1,A2,A3)A = (A_1, A_2, A_3), the divergence of the curl is ∂x(∂yA3−∂zA2)+∂y(∂zA1−∂xA3)+∂z(∂xA2−∂yA1)\partial_x (\partial_y A_3 - \partial_z A_2) + \partial_y (\partial_z A_1 - \partial_x A_3) + \partial_z (\partial_x A_2 - \partial_y A_1). The six terms cancel in pairs by Schwarz's theorem. So a field that has a vector potential is divergence-free. On an open ball, the converse is also true: a C1C^1 divergence-free field is the curl of some field. The identity is a special case of the rule d d = 0 for the exterior derivative.

Reading path

To understand this page from the ground up, read these 21 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 derivativedefinitionDifferentiable 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…statementSymmetry of second derivatives, statement, no evidence levelSymmetry of second de…statementVector field, definition, no evidence levelVector fielddefinitionSchwarz's theorem for C^2 functions, statement, no evidence levelSchwarz's theorem for…statementCurl, definition, no evidence levelCurldefinitionDivergence, definition, no evidence levelDivergencedefinitionDivergence of a curl is zero, statement, no evidence levelDivergence of a curl …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. Differentiable on an open set definition
  12. Jacobian matrix definition
  13. Higher partial derivatives definition
  14. Iterated derivative definition
  15. C^k function characterised by continuous derivatives statement
  16. Symmetry of second derivatives statement
  17. Vector field definition
  18. Schwarz's theorem for C^2 functions statement
  19. Curl definition
  20. Divergence definition
  21. Divergence of a curl is zero statement

Where this leads

2 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Vector_calculus_identities, Divergence of curl is zero
  2. Web https://en.wikipedia.org/wiki/Divergence, Properties
About this page
Tag0F56CMMC5P
URIm2://0F56CMMC5P
Aliasesdiv-curl-vanishes
Kindstatement, proposition
WritingW0 stub
MSC26B12
StatusThis copy of the site has no build data. Run m2 status 0F56CMMC5P.
UsesDivergence, Curl, Schwarz's theorem for C^2 functions

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