Explore BrainMass

Explore BrainMass

    Potential function of a vector field

    This content was COPIED from BrainMass.com - View the original, and get the already-completed solution here!

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

    thank you

    © BrainMass Inc. brainmass.com October 10, 2019, 2:04 am ad1c9bdddf

    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.