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    Potential function of a vector field

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    Let F (x,y) = (4x^3y^3+1/x) i + (3x^4y^2- 1/y) j. find the potential function of F.

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    https://brainmass.com/math/computing-values-of-functions/potential-function-vector-field-361366

    Solution Preview

    We have:

    F (x,y) = (4x^3y^3+1/x) i + (3x^4y^2- 1/y) j

    Then a potential function V would have to satisfy:

    dV/dx = 4x^3y^3+1/x (1)

    dV/dy = 3x^4y^2- 1/y (2)

    We can check if this is possible by using the symmetry of second derivatives:

    d^2V/(dx dy) = d^2V/(dydx)

    This means that we should have:

    d/dy [4x^3y^3+1/x] = d/dx [3x^4y^2- 1/y]

    Working out the partial derivatives on both sides gives:

    12 x^3 y^2 = 12 x^3x^3 y^2

    So, we see that the potential function exists. Also note ...

    Solution Summary

    We first show that the given vector field has a potential function and then compute it step by step.

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