(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Find the Flux of the Vector Field <-1, -1, -y> where the surface is the part of the plane region z + x = 1 that is on the ellipsoid [tex]{x}^{2}+2\,{y}^{2}+{z}^{2}=1[/tex]

(oriented in the +ve z direction)

2. Relevant equations

Surface Integral

3. The attempt at a solution

Parametrize the Surface:

<u, v, 1 - u>

The intersection of the plane and the ellipsoid is:

[tex]{u}^{2}+2\,{v}^{2}+ \left( 1-u \right) ^{2}=1[/tex]

[tex]{u}^{2}+{v}^{2}=u[/tex]

Which is a circle of radius 1/2 centered at (1/2,0)

Or the polar region [tex]0\leq r\leq \cos \left( \theta \right) [/tex] and [tex]0\leq \theta\leq 2\,\pi [/tex]

Then, r_{u}x r_{v}= <1, 0, 1>

Then dotting the vector field with the above vector = -1 - v

So the integral becomes:

[tex]\int \!\!\!\int \!-1-v{dv}\,{du}[/tex]

After converting to polar and limits for the circle:

[tex]\int _{0}^{2\,\pi }\!\int _{0}^{\cos \left( \theta \right) }\!-r-{r}^{

2}\sin \left( \theta \right) {dr}\,{d\theta}

[/tex]

Which gives me [tex]-1/2\,\pi [/tex]

But, when I try to find the flux with maple by using the Flux command, it gives me -pi/4

Am I doing it wrong? Could someone point out where I went wrong please?

Thank you!

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# Homework Help: Flux of a vector field over an elliptical region

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