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Mathematics for Physics

207 Line Integral

A scalar obtained by integrating a field along a curve that is used to measure accumulated value along a path.

definition [d] (Line Integral = Path Integral = Curve Integral) The scalar obtained by integrating a continuous vector field \(\mathbf{F}\) along a smooth curve \(C\) given by \(\mathbf{r}(t)\), \(a \leq t \leq b\):

  • \(\displaystyle \int_{C} \mathbf{F} \cdot d\mathbf{r} = \int_{a}^{b} \mathbf{F}\!\bigl(\mathbf{r}(t)\bigr) \cdot \dfrac{d\mathbf{r}}{dt}(t)\, dt\) .

where

  • \(C\) is the path of integration.
  • \(\mathbf{F}\) is a vector field.
  • \(\mathbf{r}(t)\) is a parametrization of \(C\).
  • \(t\) is the parameter.
  • \(a\) and \(b\) are the parameter endpoints.
  • \(d\mathbf{r} = \dfrac{d\mathbf{r}}{dt}(t)\, dt\) is the infinitesimal displacement along \(C\).
  • \(\dfrac{d\mathbf{r}}{dt}(t)\) is the tangent vector along \(C\).

definition [d] (Line Integral = Path Integral) The scalar obtained by integrating a continuous scalar function \(f\) with respect to arc length along a smooth curve \(C\) given by \(\mathbf{r}(t)\), \(a \leq t \leq b\):

  • \(\displaystyle \int_{C} f\, ds = \int_{a}^{b} f\!\bigl(\mathbf{r}(t)\bigr)\, \left\lvert \dfrac{d\mathbf{r}}{dt}(t) \right\rvert\, dt\) .

where

  • \(C\) is the path of integration.
  • \(f\) is a scalar function.
  • \(\mathbf{r}(t)\) is a parametrization of \(C\).
  • \(t\) is the parameter.
  • \(a\) and \(b\) are the parameter endpoints.
  • \(ds = \left\lvert \dfrac{d\mathbf{r}}{dt}(t) \right\rvert\, dt\) is the arc-length element.
  • \(\dfrac{d\mathbf{r}}{dt}(t)\) is the tangent vector along \(C\).

207.1 References

  1. Stewart, J. Calculus. — arc length \(L = \int_{a}^{b} \left\lvert \dfrac{d\mathbf{r}}{dt}(t) \right\rvert\, dt\); line / path integrals (closest indexed material; dedicated line-integral chapter not fully excerpted in the notebook).
  2. Kreyszig, E. Advanced Engineering Mathematics, 10th ed. Wiley, 2011. — line / curve integral.
  3. Riley, K. F., Hobson, M. P., & Bence, S. J. Mathematical Methods for Physics and Engineering. Cambridge University Press, 2006. — path integral / work integral.