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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.

    © BrainMass Inc. brainmass.com March 4, 2021, 6:13 pm ad1c9bdddf
    https://brainmass.com/math/real-analysis/35020

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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.

    $2.49

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