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Divergence Theorem

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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Real Analysis
Divergence Theorem

Let V be a region in 3complying with the hypotheses of the divergence theorem,
and denote by S its boundary ...

Solution Summary

This solution is comprised of a detailed explanation of the Divergence Theorem.
It contains step-by-step explanation for the following problem:
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.

Solution contains detailed step-by-step explanation.

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