Math 2.0

Real functions Definition

Line integral of a vector field

Builds on Vector fieldParametrised curveRectifiable curve

Definition

Let γ ⁣:[a,b]→ℝn\gamma \colon [a, b] \to \mathbb{R}^n be a rectifiable curve with image CC, and let F ⁣:C→ℝnF \colon C \to \mathbb{R}^n be a continuous vector field. For a partition a=t0<t1<⋯<tk=ba = t_0 < t_1 < \dots < t_k = b and points τi∈[ti−1,ti]\tau_i \in [t_{i-1}, t_i], consider the sum ∑i=1kF(γ(τi))⋅(γ(ti)−γ(ti−1))\sum_{i=1}^{k} F(\gamma(\tau_i)) \cdot \big( \gamma(t_i) - \gamma(t_{i-1}) \big). The line integral of FF along γ\gamma is the limit of these sums as max⁡i(ti−ti−1)→0\max_i (t_i - t_{i-1}) \to 0. It is written ∫γF⋅d𝐫or∫γF1 dx1+⋯+Fn dxn.\int_\gamma F \cdot d\mathbf{r} \quad \text{or} \quad \int_\gamma F_1 \, dx_1 + \dots + F_n \, dx_n.

Remarks

The limit exists, and it does not depend on the choice of the points τi\tau_i. For a C1C^1 curve, ∫γF⋅d𝐫=∫abF(γ(t))⋅γ′(t) dt\int_\gamma F \cdot d\mathbf{r} = \int_a^b F(\gamma(t)) \cdot \gamma'(t) \, dt, by the parametric formula. For a regular curve with unit tangent vector TT, ∫γF⋅d𝐫=∫γ(F⋅T) ds\int_\gamma F \cdot d\mathbf{r} = \int_\gamma (F \cdot T) \, ds. So it is the line integral of the scalar field F⋅TF \cdot T. The value depends on the direction of travel. Reversing the path changes its 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 curvedefinitionLine integral of a vector field, definition, no evidence levelLine integral of a ve…definition
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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. Line integral of a vector field definition

Where this leads

36 pages build on this one.

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, Derivation
About this page
Tag0YS8RX87F4
URIm2://0YS8RX87F4
Kinddefinition
WritingW1 stated
MSC26B12
DefinesLine integral of a vector field
UsesVector field, Parametrised curve, Rectifiable curve

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