Math 2.0

Real functions Proposition

Curl of a gradient is zero

Builds on CurlGradientSchwarz's theorem for C^2 functionsConservative vector field

Statement

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

Remarks

Proof: the first component of the curl of ∇ϕ\nabla \phi is ∂y∂zϕ−∂z∂yϕ\partial_y \partial_z \phi - \partial_z \partial_y \phi, and this is zero by Schwarz's theorem. The other components are similar. So a conservative C1C^1 vector field is irrotational. This gives a quick test: if curl⁡F≠0\operatorname{curl} F \ne 0 at some point, then FF has no scalar potential. In ℝn\mathbb{R}^n, the same argument shows ∂iFj=∂jFi\partial_i F_j = \partial_j F_i for all i,ji, j when F=∇ϕF = \nabla \phi with ϕ\phi of class C2C^2. The converse needs a condition on UU. It holds on convex sets, but not on every open set.

Reading path

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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 …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…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…statementConservative vector field, definition, no evidence levelConservative vector f…definitionCurl, definition, no evidence levelCurldefinitionCurl of a gradient is zero, statement, no evidence levelCurl of a gradient is…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.
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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. 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. Conservative vector field definition
  20. Curl definition
  21. Curl of a gradient is zero statement

Where this leads

5 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Vector_calculus_identities, Curl of gradient is zero
About this page
Tag0M2JRW1J0N
URIm2://0M2JRW1J0N
Aliasescurl-grad-vanishes
Kindstatement, proposition
WritingW0 stub
MSC26B12
StatusThis copy of the site has no build data. Run m2 status 0M2JRW1J0N.
UsesCurl, Gradient, Schwarz's theorem for C^2 functions, Conservative vector field

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