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Gelneren [198K]
3 years ago
9

Verify that the divergence theorem is true for the vector field f on the region

Mathematics
1 answer:
Leni [432]3 years ago
6 0
Surface integral: Parameterize the closed surface by

\mathbf s_1(u,v)=u\cos v\,\mathbf i+u\sin v\,\mathbf j+(25-u^2)\,\mathbf k
\mathbf s_2(u,v)=u\cos v\,\mathbf i+u\sin v\,\mathbf j

with 0\le u\le5 and 0\le v\le2\pi, where \mathbf s_1 defines the paraboloid part and \mathbf s_2 the planar part of the total surface \mathcal S.

We have

{\mathbf s_1}_u\times{\mathbf s_1}_v=2u^2\cos v\,\mathbf i+2u^2\sin v\,\mathbf j+u\,\mathbf k
{\mathbf s_2}_u\times{\mathbf s_2}_v=u\,\mathbf k

so we get

\displaystyle\iint_{\mathcal S}\mathbf f(x,y,z)\cdot\mathrm d\mathbf S=
\displaystyle\iint_{\mathcal S_1}\mathbf f(\mathbf s_1(u,v))\cdot(2u^2\cos v\,\mathbf i+2u^2\sin v\,\mathbf j+u\,\mathbf k)\,\mathrm du\,\mathrm dv+\iint_{\mathcal S_2}\mathbf f(\mathbf s_2(u,v))\cdot(u\,\mathbf k)\,\mathrm du\,\mathrm dv

The second integral vanishes when computing the dot product, so we're left with the first integral which reduces to

\displaystyle\int_{v=0}^{v=2\pi}\int_{u=0}^{u=5}(25u-u^3(2u\cos v-1))\,\mathrm du\,\mathrm dv=\frac{625\pi}2

Volume integral (divergence theorem): We have divergence

\nabla\times\mathbf f(x,y,z)=\dfrac{\partial(x^2)}{\partial x}+\dfrac{\partial(xy)}{\partial y}+\dfrac{\partial z}{\partial z}=2x+x+1=3x+1

By the divergence theorem, the flux is equivalent to the volume integral

\displaystyle\iiint_{\mathcal V}\nabla\times\mathbf f(x,y,z)\,\mathrm dV=\iiint_{\mathcal V}(3x+1)\,\mathrm dV

where \mathcal V denotes the space enclosed by the surface \mathcal S. Converting to cylindrical coordinates lets us write the integral as

\displaystyle\int_{\theta=0}^{\theta=2\pi}\int_{r=0}^{r=5}\int_{z=0}^{z=25-r^2}(3r\cos\theta+1)r\,\mathrm dz\,\mathrm dr\,\mathrm d\theta=\frac{625\pi}2

as desired.
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