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Line Integral

The line integral on a curve $\boldsymbol{\sigma}$ is defined by

$\displaystyle \int_\boldsymbol{\sigma}{\bf F}\cdot d{\bf s}$ $\textstyle =$ $\displaystyle \int^b_a {\bf F}(\boldsymbol{\sigma}(t))\cdot\boldsymbol{\sigma}'(t)\,dt$ (1)
  $\textstyle =$ $\displaystyle \int_C F_1\,dx+F_2\,dy+F_3\,dz,$ (2)

{\bf F} \equiv \left[{\matrix{F_1\cr F_2\cr F_3\cr}}\right].
\end{displaymath} (3)

If $\nabla\cdot{\bf F} = 0$ (i.e., it is a Divergenceless Field), then the line integral is path independent and

\int^{(x,y,z)}_{(a,b,c)} F_1\,dx+F_2\,dy+F_3\,dz = \int^{(x,...
...(x,y,c)}_{(x,b,c)} F_2\,dy + \int^{(x,y,z)}_{(x,y,c)} F_3\,dz.
\end{displaymath} (4)

For $z$ Complex, $\gamma: z=z(t)$, and $t\in[a,b]$,
\int_\gamma f\,dz = \int^b_a f(z(t))z'(t)\,dt.
\end{displaymath} (5)

See also Contour Integral, Path Integral

© 1996-9 Eric W. Weisstein