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13 June, 21:52

Let r = x i + y j + z k and r = |r|. If F = r/r p, find div F. (Enter your answer in terms of r and p.) div F =

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  1. 13 June, 21:58
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    The answer in terms of r and p is div F = (3 - p) / r^p.

    Step-by-step explanation:

    Given:

    F = R/r^p

    F = / ||^p

    F =.

    Hence, For div F,

    We take partial derivative:

    div F = (∂/∂x) x / (x^2+y^2+z^2) ^ (p/2) + (∂/∂y) y / (x^2+y^2+z^2) ^ (p/2) + (∂/∂z) z / (x^2+y^2+z^2) ^ (p/2)

    Now, we use the rational derivative rule to find the derivatives:

    div F = [1 (x^2+y^2+z^2) ^ (p/2) - x * px (x^2+y^2+z^2) ^ (p/2 - 1) ] / (x^2+y^2+z^2) ^p + [1 (x^2+y^2+z^2) ^ (p/2) - y * py (x^2+y^2+z^2) ^ (p/2 - 1) ] / (x^2+y^2+z^2) ^p + [1 (x^2+y^2+z^2) ^ (p/2) - z * pz (x^2+y^2+z^2) ^ (p/2 - 1) ] / (x^2+y^2+z^2) ^p

    div F = (x^2+y^2+z^2) ^ (p/2 - 1) {[ (x^2+y^2+z^2) - px^2] + [ (x^2+y^2+z^2) - py^2] + [ (x^2+y^2+z^2) - pz^2]} / (x^2+y^2+z^2) ^p

    div F = [3 (x^2+y^2+z^2 - p (x^2+y^2+z^2) ] / (x^2+y^2+z^2) ^ (p/2 + 1)

    div F = (3 - p) (x^2+y^2+z^2) / (x^2+y^2+z^2) ^ (p/2 + 1)

    div F = (3 - p) / (x^2+y^2+z^2) ^ (p/2)

    Now it comes like,

    div F = (3 - p) / r^p.
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