I am given that F=2xyi + (x^2 + z + 1)j + yk
How do I calculate the scalar potential for F by calculating an appropriate work integral?
You can calculate the potential of one point in the force field, relative to another point, by integrating along any path from the second point to the first. If the force is conservative, the path doesn't matter so you can take the simplest path, say, three lines parallel to the axes. That is
integrate from (0,0,0) to (x,0,0), then from (x,0,0) to (x,y,0), then from (x,y,0) to (x,y,z). That is the potential of any point relative to (0,0,0). The potential relative to some other point would be that plus a constant. It would be really good idea to check that this is a conservative force field so that it has a potential.
Solve the following simultaneously:
$\displaystyle \frac{\partial \phi}{\partial x} = 2xy$ .... (1)
$\displaystyle \frac{\partial \phi}{\partial y} = x^2 + z + 1$ .... (2)
$\displaystyle \frac{\partial \phi}{\partial z} = y$ .... (3)
From (1): $\displaystyle \phi = x^2 y + f(y, z)$. Substitute into (2).
From (2): $\displaystyle f(y, z) = (z + 1)y + g(z)$. Therefore $\displaystyle \phi = x^2 y + (z + 1)y + g(z)$. Substitute into (3).
From (3): $\displaystyle g(z) = C$.
Therefore $\displaystyle \phi = x^2 y + (z + 1)y + C$. It's convenient to define the scalar potential so that C = 0.