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Verify that the Divergence Theorem is true for the vector field F on the region E. F(x, y, z) = z, y, x , E is the solid ball x2 + y2 + z2 ≤ 64 For your answer, put in the flux across the boundary of E with respect to the outward normal.

Respuesta :

LammettHash

[tex]\vec F(x,y,z)=\langle z,y,x\rangle[/tex]

has divergence 1, so by the divergence theorem, the flux of [tex]\vec F[/tex] across the boundary of [tex]E[/tex] is exactly the volume of [tex]E[/tex],

[tex]\displaystyle\iiint_E\mathrm dV=\int_0^\pi\int_0^{2\pi}\int_0^8\rho^2\sin\varphi\,\mathrm d\rho\,\mathrm d\theta\,\mathrm d\varphi[/tex]

or simply [tex]\dfrac43\pi8^3=\boxed{\frac{2048\pi}3}[/tex].

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