Find a potential function for the vector field?

Question

Find a potential function for the vector field f(x, y)=2x/y i+(1-x^2)/y^2 j.

Answer 1

Potential function U(x,y) defines the vector field f(x,y) =(2x/y) i + [(1-x²)/y ²]j. The "potentiality" (curl = 0) of this vector field can be noticed from the fact that

                                                                ∂U _y/∂x  = ∂U _x/∂y   = -2x/y² 

where

                     U_y = ∂U/∂y = (1-x²)/y²      and                     U_x = ∂U/∂x = 2x/y

Potential function is defined as the following integral

                                                        U(x,y) = ∫U_xdx + ∫U_ydy + const

Take a look at the first integral'

                                     ∫U_xdx  =     2 ∫xdx/y = x²/y    + p(y)

For the second one

                                     ∫U_ydy  =   (1-x²)  ∫dy/y²  = -(1-x²)/y + q(x) = (x²-1)/y   + q(x)

p(y) is an arbitrary function of y, and q(x) is an arbitrary function of x. To make both integrals reflecting the same function we can put p(y) = -1/y and q(x) = constant (c). Thus, we have

                                                              U(x,y) = (x²-1)/y + constant

Answer 2

Let ∇P = f(x, y)

∂P/∂x = 2x/y => P = x²/y + k(y)

∂P/∂y = (1-x²)/y² => P = (x²-1)/y + g(x)

Let k(y) = -1/y,  and g(x) = c, where c is a constant.

P(x, y) = (x²-1)/y + c

Check:

∂P/∂x = 2x/y,

∂P/∂y = (1-x²)/y²

f(x,y) = ∇P = 2x/y i+(1-x²)/y² j