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So I'm trying to solve this by using Stokes' :$F(x,y,z)=2y\cos(z)i+e^x\sin(z)j+xe^yk$, where $S$ is the hemisphere $x^2+y^2+z^2=9$ oriented upward, $z$ larger than or equal to $0$. I do this by letting $r(t)=3\cos ti+3\sin tj+0k$ , and then evaluate integral $F\cdot dr$ from $0$ to $2\pi$. Lo and behold and I get $-18\pi$. Now, that ain't right. What am I doing wrong?

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When you write, "I'm trying to solve this..." I think what you mean is that you are trying to integrate a ceertain function over a certain set, but you never say this, so it's not entirely clear what it is that you are trying to solve. Please edit the question to clarify. –  Gerry Myerson May 1 '12 at 1:16
    
If you were trying to evaluate $\displaystyle \int_{\{x^2+y^2+z^2=9\}\cap\{z \geq 0\} } \nabla \times F \cdot dS = \oint_{x^2+y^2=9} F\cdot dr$, $-18\pi$ looks correct to me? –  Shuhao Cao May 1 '12 at 2:25
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