Here is the catch: We are given a 2-D velocity flow incompressible, irrotational of the form: --> ^ ^ ^ ^ V = u i + v j = [4y -x(1+x)] i + y(2x+1) j and we are asked to find the flow potential which obeys the Laplace Eq. for 2-D incompressible, irrotational flow: dΦ = u dx + v dy in other words: ∂Φ ---- = u ∂x ∂Φ ---- = v ∂y I integrated the first one and then the second one and compared the two functions and combined the terms, but at the end the Φ does not satisfy the first equation only the second one. Another technique, I integrated the first function with respect to x and Φ is expressed as 4xy - x^2/2 - x^3/3 + f(y) = Φ (x,y) now I differentiate with respect to y and equate it to v: 4x - f'(y) = y(2x+1) which solves to f(y) = xy^2 + y^2/2 -4xy + C Plug it in the above expression and get: Φ (x,y) = xy^2 + y^2/2 - x^2/2 - x^3/3 + C now the first partial diff. eq is not satisfied but the first is. Can someone explain what is wrong here?