# Flux of vector field

• Feb 13th 2010, 01:19 PM
Flux of vector field
How do you calculate the flux of the vector field $\displaystyle F=xi+yj+zk$ across a disk $\displaystyle x^2+z^2\leq 1$ in the plane $\displaystyle y=3$ with the normal vector pointing in the $\displaystyle y$ direction
• Feb 13th 2010, 03:50 PM
TheEmptySet
Quote:

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

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

So by definition

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

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

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

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

$\displaystyle 3\pi$