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

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- Feb 13th 2010, 01:19 PMpkrleadsFlux 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 PMTheEmptySet
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

$\displaystyle 3\pi$