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Vesna [10]
2 years ago
8

Calculate the flux of the vector field f⃗ (x,y,z)=cos(x2+y2)k⃗ through the disk x2+y2≤9 oriented upward in the plane z=4.

Mathematics
1 answer:
garik1379 [7]2 years ago
3 0
Parameterize the disk S by

\mathbf r(s,t)=\begin{cases}x(s,t)=s\cos t\\y(s,t)=s\sin t\\z(s,t)=4\end{cases}

where 0\le s\le3 and 0\le t\le2\pi. Call this parameterized region T. Then

\mathbf r_s\times\mathbf r_t=(\cos t\,\mathbf i+\sin t\,\mathbf j)\times(-s\sin t\,\mathbf i+s\cos t\,\mathbf j)=s\,\mathbf k

The flux over the disk is given by the surface integral

\displaystyle\iint_S\mathbf f(x,y,z)\cdot\mathrm dS=\iint_T\mathbf f(x(s,t),y(s,t),z(s,t))\cdot(\mathbf r_s(s,t)\times\mathbf r_t(s,t))\,\mathrm ds\,\mathrm dt
=\displaystyle\int_{t=0}^{t=2\pi}\int_{s=0}^{s=3}(\cos(s^2)\,\mathbf k)\cdot(s\,\mathbf k)\,\mathrm ds\,\mathrm dt
=\displaystyle2\pi\int_{s=0}^{s=3}s\cos(s^2)\,\mathrm ds
=\displaystyle\pi\int_{\sigma=0}^{\sigma=9}\cos\sigma\,\mathrm d\sigma

(where we take \sigma=s^2)

=\pi\sin 9
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