Flux of vector field

• Feb 13th 2010, 02:19 PM
Flux of vector field
How do you calculate the flux of the vector field $F=xi+yj+zk$ across a disk $x^2+z^2\leq 1$ in the plane $y=3$ with the normal vector pointing in the $y$ direction
• Feb 13th 2010, 04:50 PM
TheEmptySet
Quote:

How do you calculate the flux of the vector field $F=xi+yj+zk$ across a disk $x^2+z^2\leq 1$ in the plane $y=3$ with the normal vector pointing in the $y$ direction

The unit normal vector to the disk is $\vec{n} = \hat{j}$

So by definition

$\iint \vec{F}\cdot d\vec{S}=\iint \vec{F} \cdot \vec{n}dS$

Where $dS$ is the projection of the disk into the $x-z$ plane

$\iint_DydA$ but on this disk y is constant and $y=3$

$3\iint_DdA$ since the integrand is constant the value of the integral is

$3\pi$