The total volumetric flux through the test surface is
where  is an arc-length parameter defined on the curve , with  at the point  and  at the point .
Here  is the unit vector perpendicular to the test surface, i.e.,
where  is the  rotation matrix corresponding to a  anticlockwise rotation about the positive  axis:
The integrand in the expression for  is independent of , so the outer integral can be evaluated to yield