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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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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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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 ...

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