what to do with a constant of a scalar potential if all you need is a constant

Question

if i construct a simple scalar potential say $F=[0,1,2z]$ then
fx=0
fy=0
fz=2
so $ f = 2z + g (x,y) $
now what? what function of x and y could equal 1?

Answer 1

Let's role with F = $0$i + j + $2z$k

Potential functions require that

F $=\nabla{f}$

Then

$\frac{\partial{f}}{dx}=0$

$\frac{\partial{f}}{dy}=1$

$\frac{\partial{f}}{dz}=2z$

We can start anywhere we would like. Let's start with the first

$f(x,y,z) = C_1(y,z)$

$f$ is indepedent of $x$. Now take the y partial

$\frac{\partial{f}}{dy}=\frac{\partial{C_1}}{dy} = 1$

Solve for $C_1$

$C_1(y,z) = y + C_2(z)$

$f(y,z) = y + C_2(z)$

$\frac{\partial{f}}{dz}=\frac{\partial{C_2}}{dz} = 2z$

$C_2(z) = z^2 + K$

So

$f(x,y,z) = y + z^2 + K$

For some constant K