# 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.

#### Solution Preview

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.