Divergence Theorem
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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 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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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.
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 ...
Education
- BSc, Manipur University
- MSc, Kanpur University
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