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