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jianjunzhou2008 · Dec 14, 2008, 07:27 PM

Use Stokes' Theorem to evaluate

F(x,y,z)=(x^2)(z^2)i+(y^2)(z^2)j+xyzk, S is the part of the paraboloid z=x^2+y^2 that lies inside the cylinder x^2+y^2=4, oriented upward:confused:

galactus · Dec 15, 2008, 09:18 AM

Because it is oriented upward, we can parametrize S:

$x=rcos(t), \;\ y=rsin(t), \;\ z=r^{2}$

$dx=-rsin(t), \;\ dy=rcos(t), \;\ dz=2r$

Now, use those in $\oint_{C} F\cdot dr=\oint_{C}\left[x^{2}z^{2}dx+y^{2}z^{2}dy+xyzdz\right]$

Find your curl F by taking the cross product

$\begin{vmatrix}i&j&k\\\frac{{\partial}}{{\partial} x}&\frac{{\partial}}{{\partial}y}&\frac{{\partial} }{{\partial}z}\\x^{2}z^{2}&y^{2}z^{2}&xyz\end{vmat rix}$

There's a start.