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Math Help - Find potential Function

  1. #1
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    Find potential Function

    Let S be the set of all x =/= O in R^n. Let r = abs(x), and let f be the vector field defined on S by the equation:

    f(x) = (r^p)(x),

    where p is a real constant. Find a potential function for f on S.

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  2. #2
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    Sorry, r = to the norm of x
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  3. #3
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    So what you really have is ||\vec{x}||^p \vec{x}?

    In 3 dimensions, that is x(x^2+ y^2+ z^2)^{p/2}\vec{i}+ y(x^2+ y^2+ z^2)^{p/2}\vec{j}+ z(x^2+ y^2+ z^2)^{p/2}\vec{k} and you want a function f(x, y, z) such that
    \frac{\partial f}{\partial x}= x(x^2+ y^2+ z^2)^{p/2}
    \frac{\partial f}{\partial y}= y(x^2+ y^2+ z^2)^{p/2}
    \frac{\partial f}{\partial z}= z(x^2+ y^2+ z^2)^{p/2}

    Okay, what is an anti-derivative of x(x^2+ a^2)^{p/2}?
    I recommend the substitution u= x^2+ a^2.
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