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Stewart
Calculate the flux of the vector field
\(\bm{F}(x,y,z) = z\mathbf{i}+y\mathbf{j}+x\mathbf{k}\)
across the unit sphere \(x^2+y^2+z^2=1.\)
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Stewart
Evaluate \(\iint_S \bm{F}\cdot\mathrm{d}\bm{S},\) where
\[ \bm{F}(x,y,z) = xy\mathbf{i} + \Big(y^2+\mathrm{e}^{xz^2}\Big)\mathbf{j} + \sin(xy)\mathbf{k} \]
and \(S\) is the surface of the region \(E\) bounded by
the parabolic cylinder \(z=1-x^2\)
and the planes \(z=0\) and \(y=0\) and \(y+z=2.\)
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Use the divergence theorem to evaluate
\(\iint_S \bm{F}\cdot\mathrm{d}\bm{S},\) where
\[ \bm{F}(x,y,z) = xy\mathbf{i} - \frac{1}{2}y^2\mathbf{j} + z\mathbf{k} \]
and \(S\) consists of the three surfaces:
\(z=4-3^2-3y^2\) for \(1\leq z \leq 4\) on the top,
\(x^2+y^2=1\) for \(0\leq z \leq 1\) on the sides,
and \(z=0\) on the bottom.