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djyliett [7]
3 years ago
12

Use the divergence theorem to find the outward flux of the vector field F(xyz)= 4x^2 i + 4y^2 j + 3z^2 k across the boundary of

the rectangular prism: 0
Mathematics
1 answer:
Pavel [41]3 years ago
5 0

\vec F(x,y,z)=4x^2\,\vec\imath+4y^2\,\vec\jmath+3z^2\,\vec k\implies\nabla\cdot\vec F=8x+8y+6z

Let S be the surface of the rectangular prism bounded by the planes x=0, x=a, y=0, y=b, z=0, and z=c. By the divergence theorem, the integral of \vec F over S is given by the integral of \nabla\cdot\vec F over the interior of S (call it R):

\displaystyle\iint_S\vec F\cdot\mathrm d\vec S=\iiint_R(\nabla\cdot\vec F)\,\mathrm dV

=\displaystyle\int_0^c\int_0^b\int_0^a(8x+8y+6z)\,\mathrm dx\,\mathrm dy\,\mathrm dz=\boxed{abc(4a+4b+3c)}

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