Math 2.0

Real functions Proposition

Reversing a path changes the sign of a line integral

Builds on Line integral of a vector fieldLine integral of a scalar fieldLine integral as an integral over the parameter interval

Statement

Let γ ⁣:[a,b]→ℝn\gamma \colon [a, b] \to \mathbb{R}^n be a piecewise C1C^1 curve with image CC, and let γ− ⁣:[a,b]→ℝn\gamma^- \colon [a, b] \to \mathbb{R}^n, γ−(t)=γ(a+b−t)\gamma^-(t) = \gamma(a + b - t), be the reversed curve. For each continuous vector field FF on CC and each continuous function ff on CC, ∫γ−F⋅d𝐫=−∫γF⋅d𝐫,∫γ−f ds=∫γf ds.\int_{\gamma^-} F \cdot d\mathbf{r} = -\int_\gamma F \cdot d\mathbf{r}, \qquad \int_{\gamma^-} f \, ds = \int_\gamma f \, ds.

Remarks

Proof: by the parametric formula, (γ−)′(t)=−γ′(a+b−t)(\gamma^-)'(t) = -\gamma'(a + b - t). The substitution u=a+b−tu = a + b - t then gives both identities. The line integral of a vector field depends on the orientation of the curve. The line integral of a scalar field does not, because it uses only ∥γ′∥\|\gamma'\|. More generally, under a C1C^1 change of parameter φ\varphi with φ′>0\varphi' > 0, the vector line integral does not change. With φ′<0\varphi' < 0 it changes sign.

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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…conceptContinuity 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 …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…statementParametrised curve, definition, no evidence levelParametrised curvedefinitionVector field, definition, no evidence levelVector fielddefinitionRectifiable curve, definition, no evidence levelRectifiable curvedefinitionTangent 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…statementReversing a path changes the sign of a line integral, statement, no evidence levelReversing a path chan…statement
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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. Continuity in R^n definition
  8. Differentiable function of several variables definition
  9. Partial derivative definition
  10. Differentiable on an open set definition
  11. Higher partial derivatives definition
  12. Iterated derivative definition
  13. C^k function characterised by continuous derivatives statement
  14. Parametrised curve definition
  15. Vector field definition
  16. Rectifiable curve definition
  17. Tangent vector of a curve definition
  18. Line integral of a scalar field definition
  19. Line integral of a vector field definition
  20. Regular curve definition
  21. Arc length parametrisation definition
  22. Line integral as an integral over the parameter interval statement
  23. Reversing a path changes the sign of a line integral statement

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Further reading

  1. Web https://en.wikipedia.org/wiki/Line_integral, Line integral of a vector field
  2. Web https://en.wikipedia.org/wiki/Line_integral, Line integral of a scalar field
About this page
Tag0FSCXYG856
URIm2://0FSCXYG856
Aliasesorientation-reversal-of-line-integrals
Kindstatement, proposition
WritingW0 stub
MSC26B12
StatusThis copy of the site has no build data. Run m2 status 0FSCXYG856.
UsesLine integral of a vector field, Line integral of a scalar field, Line integral as an integral over the parameter interval

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