Math 2.0

Real functions Theorem

Line integral as an integral over the parameter interval

Builds on Line integral of a scalar fieldLine integral of a vector fieldTangent vector of a curveArc length parametrisation

Statement

Let γ ⁣:[a,b]→ℝn\gamma \colon [a, b] \to \mathbb{R}^n be a C1C^1 curve with image CC. Let f ⁣:C→ℝf \colon C \to \mathbb{R} be continuous, and let F ⁣:C→ℝnF \colon C \to \mathbb{R}^n be a continuous vector field. Then ∫γf ds=∫abf(γ(t)) ∥γ′(t)∥ dt,∫γF⋅d𝐫=∫abF(γ(t))⋅γ′(t) dt.\int_\gamma f \, ds = \int_a^b f(\gamma(t)) \, \|\gamma'(t)\| \, dt, \qquad \int_\gamma F \cdot d\mathbf{r} = \int_a^b F(\gamma(t)) \cdot \gamma'(t) \, dt.

Remarks

The left sides are the line integral of a scalar field and the line integral of a vector field. The right sides are ordinary integrals in one variable. The formula follows from Δsi=∫ti−1ti∥γ′(t)∥ dt\Delta s_i = \int_{t_{i-1}}^{t_i} \|\gamma'(t)\| \, dt, the mean value theorem for each component of γ\gamma, and the uniform continuity of f∘γf \circ \gamma and γ′\gamma'. The formulas extend to piecewise C1C^1 curves, by summing over the pieces. Example: for F(x,y)=(−y,x)F(x, y) = (-y, x) and γ(t)=(cos⁡t,sin⁡t)\gamma(t) = (\cos t, \sin t), t∈[0,2π]t \in [0, 2\pi], the integrand is sin⁡2t+cos⁡2t=1\sin^2 t + \cos^2 t = 1, so ∫γF⋅d𝐫=2π\int_\gamma F \cdot d\mathbf{r} = 2\pi.

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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…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

Where this leads

10 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Line_integral, Derivation
  2. Web https://en.wikipedia.org/wiki/Line_integral, Line integral of a vector field
About this page
Tag0W3W1VYWBP
URIm2://0W3W1VYWBP
Aliasesparametric-formula-for-line-integrals
Kindstatement, theorem
WritingW0 stub
MSC26B12
StatusThis copy of the site has no build data. Run m2 status 0W3W1VYWBP.
UsesLine integral of a scalar field, Line integral of a vector field, Tangent vector of a curve, Arc length parametrisation

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