Line Integral

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

$$\int_\boldsymbol{\sigma}{\bf F}\cdot d{\bf s} = \int^b_a {\bf F}(\boldsymbol{\sigma}(t))\cdot\boldsymbol{\sigma}'(t)\,dt$$ (1)

$$= \int_C F_1\,dx+F_2\,dy+F_3\,dz,$$ (2)

where

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

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

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

For $z$ Complex, $\gamma: z=z(t)$, and $t\in[a,b]$,

$$\int_\gamma f\,dz = \int^b_a f(z(t))z'(t)\,dt.$$ (5)

See also Contour Integral, Path Integral