Path Integral

Let $\gamma$ be a Path given parametrically by ${\bf\sigma}(t)$. Let $s$ denote Arc Length from the initial point. Then

$$\int_\gamma f(s)\,ds = \int_\gamma f({\bf\sigma} (t))\, \vert\sigma'(t)\vert\,dt$$

$$= \int_\gamma f(x(t),y(t),z(t))\, \vert{\bf\sigma}'(t)\vert\,dt.$$

See also Line Integral

References

Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetterling, W. T. ``Evaluation of Functions by Path Integration.'' §5.14 in Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England: Cambridge University Press, pp. 201-204, 1992.