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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.

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Solution Summary

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

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Let V be a region in 3complying with the hypotheses of the divergence theorem,
and denote by S its boundary surface. Let also φ: →  be a scalar function, and c an arbitrary constant vector.
By applying the divergence theorem to the vector field φc
(1) show that:

(∫∫∫v ▼φdV - ∫∫s φndS).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:

∫∫∫v ▼φdV = ∫∫s φndS.

Real Analysis
Divergence Theorem

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

By applying the divergence theorem to the vector field φc
(1) show that:

(∫∫∫v ▼φdV - ∫∫s φndS).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:

∫∫∫v ▼φdV = ∫∫s φndS.

See the attached file.

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