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Inga [223]
3 years ago
12

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

Mathematics
1 answer:
schepotkina [342]3 years ago
4 0
\mathbf f(x,y,z)=5x\,\mathbf i+xy\,\mathbf j+2xz\,\mathbf k

\mathrm{div}(\mathbf f)=\dfrac{\partial(5x)}{\partial x}+\dfrac{\partial(xy)}{\partial y}+\dfrac{\partial(2xz)}{\partial z}=5+x+2x=5+3x


By the divergence theorem, the flux of \mathbf f across the boundary of E is given by


\displaystyle\iint_{\mathcal S}\mathbf f\cdot\mathrm d\mathbf S=\iiint_E\mathrm{div}(\mathbf f)\,\mathrm dV

=\displaystyle\int_{z=0}^{z=2}\int_{y=0}^{y=2}\int_{x=0}^{x=2}(5+3x)\,\mathrm dx\,\mathrm dy\,\mathrm dz=64
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