Math 2.0

Real functions Theorem

Curl-free vector field on a convex set is conservative

Builds on CurlConservative vector fieldScalar potentialCurl of a gradient is zeroDifferentiation under the integral signLine integral of a vector field

Statement

Let U⊆ℝ3U \subseteq \mathbb{R}^3 be open and convex, and let F ⁣:U→ℝ3F \colon U \to \mathbb{R}^3 be a vector field of class C1C^1 with ∇×F=0\nabla \times F = 0 on UU. Then FF is conservative on UU. Explicitly, for a fixed p∈Up \in U, the function ϕ(x)=∫01F(p+t(x−p))⋅(x−p) dt\phi(x) = \int_0^1 F\big(p + t (x - p)\big) \cdot (x - p) \, dt satisfies ∇ϕ=F\nabla \phi = F on UU.

Remarks

The integral ϕ(x)\phi(x) is the line integral of FF along the segment from pp to xx. Convexity makes sure that this segment lies in UU. To prove ∇ϕ=F\nabla \phi = F, differentiate under the integral sign and use ∂jFi=∂iFj\partial_j F_i = \partial_i F_j, which is the condition ∇×F=0\nabla \times F = 0. The same proof works in ℝn\mathbb{R}^n under the condition ∂jFi=∂iFj\partial_j F_i = \partial_i F_j for all i,ji, j, and on star-shaped open sets. The result holds more generally on simply connected open sets. It fails on some other open sets. With the converse, it gives: on a convex open set, a C1C^1 field is conservative if and only if its curl is zero.

Reading path

To understand this page from the ground up, read these 29 pages in order. The foundations come first.

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Euclidean space, object, no evidence levelEuclidean spaceobjectFunction, concept, no evidence levelFunctionconceptFunction composition, concept, no evidence levelFunction compositionconceptFréchet 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 levelGradientdefinitionChain rule in several variables, statement, no evidence levelChain rule in several…statementDifferentiable on an open set, definition, no evidence levelDifferentiable on an …definitionHigher partial derivatives, definition, no evidence levelHigher partial deriva…definitionDifferentiation under the integral sign, statement, no evidence levelDifferentiation under…statementIterated 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…statementParametrised 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…statementRectifiable curve, definition, no evidence levelRectifiable curvedefinitionConservative vector field, definition, no evidence levelConservative vector f…definitionCurl, definition, no evidence levelCurldefinitionLine integral of a vector field, definition, no evidence levelLine integral of a ve…definitionScalar potential, definition, no evidence levelScalar potentialdefinitionCurl of a gradient is zero, statement, no evidence levelCurl of a gradient is…statementCurl-free vector field on a convex set is conservative, statement, no evidence levelCurl-free vector fiel…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. Function composition concept
  4. Fréchet derivative definition
  5. Function of several variables concept
  6. Vector-valued function concept
  7. Limit of a function of several variables concept
  8. Partial derivative concept
  9. Continuity in R^n definition
  10. Differentiable function of several variables definition
  11. Partial derivative definition
  12. Gradient definition
  13. Chain rule in several variables statement
  14. Differentiable on an open set definition
  15. Higher partial derivatives definition
  16. Differentiation under the integral sign statement
  17. Iterated derivative definition
  18. Symmetry of second derivatives statement
  19. C^k function characterised by continuous derivatives statement
  20. Parametrised curve definition
  21. Vector field definition
  22. Schwarz's theorem for C^2 functions statement
  23. Rectifiable curve definition
  24. Conservative vector field definition
  25. Curl definition
  26. Line integral of a vector field definition
  27. Scalar potential definition
  28. Curl of a gradient is zero statement
  29. Curl-free vector field on a convex set is conservative statement

Where this leads

2 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Conservative_vector_field, Irrotational vector fields
  2. Web https://en.wikipedia.org/wiki/Poincar%C3%A9_lemma, Direct proof
About this page
Tag0B22P010D6
URIm2://0B22P010D6
Aliasespoincare-lemma-for-vector-fields
Kindstatement, theorem
WritingW0 stub
MSC26B12
StatusThis copy of the site has no build data. Run m2 status 0B22P010D6.
UsesCurl, Conservative vector field, Scalar potential, Curl of a gradient is zero, Differentiation under the integral sign, Line integral of a vector field

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