surface integral, flux

[mathstresser]

\(\displaystyle \L\\ F(x,y,z)= <zxe^y, -xze^y, z>\)

S: x+y+z=1

in the first octant (and has downward orientation)

z=1-x-y


\(\displaystyle \L\\\int_{S}\int_{S} F dot ds = \int_{S}\int_{S} F dot (z-sub-x cross z-sub-x\)

I tried to work the problem but I don't know what to do when I get to z-sub-x cross z-sub-x.

 

[daon]

\(\displaystyle \L

\int _S \int F \cdot N dS = \int _R \int <zxe^y, -xze^y, z> \cdot \frac{<(-1), (-1), (-1)>}{\sqrt{1+(-1)^2+(-1)^2}}(\sqrt{(-1)^2 + (-1)^2 + 1})dA \\ = \int _R \int <(1-x-y)xe^y, \,\, -x(1-x-y)e^y, \,\, (1-x-y)> \cdot <-1,-1,-1>dA\)

Its been a while for me, but I believe this is right. The rest shouldn't be too bad.

Hope that helps,
-Daon