Math 2.0

Real functions Theorem

Path independence of line integrals

Builds on Conservative vector fieldLine integral of a vector fieldGradient theoremReversing a path changes the sign of a line integralAdditivity of line integrals under concatenationPartial derivative

Statement

Let U⊆ℝnU \subseteq \mathbb{R}^n be open and connected, and let F ⁣:U→ℝnF \colon U \to \mathbb{R}^n be a continuous vector field. The following are equivalent.

  1. FF is conservative on UU.
  2. For all piecewise C1C^1 curves γ1,γ2\gamma_1, \gamma_2 in UU with the same initial point and the same end point, ∫γ1F⋅d𝐫=∫γ2F⋅d𝐫\int_{\gamma_1} F \cdot d\mathbf{r} = \int_{\gamma_2} F \cdot d\mathbf{r}.
  3. For each closed piecewise C1C^1 curve γ\gamma in UU, ∫γF⋅d𝐫=0\int_\gamma F \cdot d\mathbf{r} = 0.

Remarks

Statement 1 implies statement 2 by the gradient theorem. Statements 2 and 3 are equivalent: a closed curve splits into two curves with the same end points, and one uses reversal and additivity. Statement 2 implies statement 1: fix p∈Up \in U and let ϕ(x)=∫γF⋅d𝐫\phi(x) = \int_\gamma F \cdot d\mathbf{r} for a path γ\gamma from pp to xx. A short straight last segment in direction eie_i shows that ∂ϕ/∂xi=Fi\partial \phi / \partial x_i = F_i. Connectedness is used to join pp to each point. On a disconnected set, apply the result on each component.

Reading path

To understand this page from the ground up, read these 36 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 derivativedefinitionTotal derivative, concept, no evidence levelTotal derivativeconceptFunction 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…conceptDifferentiable function of several variables, concept, no evidence levelDifferentiable functi…conceptParametrised curve, concept, no evidence levelParametrised curveconceptPartial 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, concept, no evidence levelGradientconceptChain rule in several variables, statement, no evidence levelChain rule in several…statementDifferentiable on an open set, definition, no evidence levelDifferentiable on an …definitionGradient, definition, no evidence levelGradientdefinitionHigher partial derivatives, definition, no evidence levelHigher partial deriva…definitionIterated derivative, definition, no evidence levelIterated derivativedefinitionChain rule for a composition with a curve, statement, no evidence levelChain rule for a comp…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 fielddefinitionRectifiable curve, definition, no evidence levelRectifiable curvedefinitionConservative vector field, definition, no evidence levelConservative vector f…definitionTangent vector of a curve, definition, no evidence levelTangent vector of a c…definitionLine integral of a scalar field, definition, no evidence levelLine integral of a sc…definitionLine integral of a vector field, definition, no evidence levelLine integral of a ve…definitionRegular curve, definition, no evidence levelRegular curvedefinitionArc length parametrisation, definition, no evidence levelArc length parametris…definitionLine integral as an integral over the parameter interval, statement, no evidence levelLine integral as an i…statementAdditivity of line integrals under concatenation, statement, no evidence levelAdditivity of line in…statementReversing a path changes the sign of a line integral, statement, no evidence levelReversing a path chan…statementGradient theorem, statement, no evidence levelGradient theoremstatementPath independence of line integrals, statement, no evidence levelPath independence of …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. Function composition concept
  4. Fréchet derivative definition
  5. Total derivative concept
  6. Function of several variables concept
  7. Vector-valued function concept
  8. Limit of a function of several variables concept
  9. Differentiable function of several variables concept
  10. Parametrised curve concept
  11. Partial derivative concept
  12. Continuity in R^n definition
  13. Differentiable function of several variables definition
  14. Partial derivative definition
  15. Gradient concept
  16. Chain rule in several variables statement
  17. Differentiable on an open set definition
  18. Gradient definition
  19. Higher partial derivatives definition
  20. Iterated derivative definition
  21. Chain rule for a composition with a curve statement
  22. C^k function characterised by continuous derivatives statement
  23. Parametrised curve definition
  24. Vector field definition
  25. Rectifiable curve definition
  26. Conservative vector field definition
  27. Tangent vector of a curve definition
  28. Line integral of a scalar field definition
  29. Line integral of a vector field definition
  30. Regular curve definition
  31. Arc length parametrisation definition
  32. Line integral as an integral over the parameter interval statement
  33. Additivity of line integrals under concatenation statement
  34. Reversing a path changes the sign of a line integral statement
  35. Gradient theorem statement
  36. Path independence of line integrals statement

Where this leads

2 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Conservative_vector_field, Path independence
  2. Web https://en.wikipedia.org/wiki/Conservative_vector_field, Path independence and conservative vector field
  3. Web https://en.wikipedia.org/wiki/Gradient_theorem, Converse of the gradient theorem
About this page
Tag0BFN21TF5F
URIm2://0BFN21TF5F
Aliasesconservative-iff-path-independent
Kindstatement, theorem
WritingW0 stub
MSC26B20
StatusThis copy of the site has no build data. Run m2 status 0BFN21TF5F.
UsesConservative vector field, Line integral of a vector field, Gradient theorem, Reversing a path changes the sign of a line integral, Additivity of line integrals under concatenation, Partial derivative

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