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Divergence Theorem for a Function

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1. Check the divergence theorem for the function:

V(vector)= r^2*cos(theta)(r(hat))+r^2*cos(phi(Theta(hat))

-r^2*cos(theta)*sin (phi)(phi(hat))

Use as the volume one octant of the sphere of radius R.

https://brainmass.com/physics/scalar-and-vector-operations/divergence-theorem-function-162419

Solution Summary

See attached file for step by step calculations for checking the divergence theorem for a function.

\$2.19

Let V be a region in &#8299;3complying with the hypotheses of the divergence theorem, and denote by S its boundary surface. Let also &#966;: &#8594; &#8299; be a scalar function, and c an arbitrary constant vector. By applying the divergence theorem to the vector field &#966;c (1) show that: (&#8747;&#8747;&#8747;v &#9660;&#966;dV - &#8747;&#8747;s &#966;ndS).c = 0 with the understanding that the integral of a vector is the vector of the integrals of the components. (2) Use the above result to deduce carefully that: &#8747;&#8747;&#8747;v &#9660;&#966;dV = &#8747;&#8747;s &#966;ndS.

Real Analysis
Divergence Theorem

Let V be a region in &#8299;3complying with the hypotheses of the divergence theorem,
and denote by S its boundary surface. Let also &#966;: &#8594; &#8299; be a scalar function, and c an arbitrary constant vector.

By applying the divergence theorem to the vector field &#966;c
(1) show that:

(&#8747;&#8747;&#8747;v &#9660;&#966;dV - &#8747;&#8747;s &#966;ndS).c = 0
with the understanding that the integral of a vector is the vector of the integrals of the components.

(2) Use the above result to deduce carefully that:

&#8747;&#8747;&#8747;v &#9660;&#966;dV = &#8747;&#8747;s &#966;ndS.

See the attached file.

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