Answer
We show that any vector field of the form ${\bf{F}} = \left( {f\left( x \right),g\left( y \right),h\left( z \right)} \right)$ has a potential function given by $V\left( {x,y,z} \right) = F\left( x \right) + G\left( y \right) + H\left( z \right)$, where $F\left( x \right)$, $G\left( y \right)$, and $H\left( z \right)$ are the antiderivatives of $f\left( x \right)$, $g\left( y \right)$, and $h\left( z \right)$, respectively.
Work Step by Step
We have a vector field: ${\bf{F}} = \left( {f\left( x \right),g\left( y \right),h\left( z \right)} \right)$.
Assume that $f$, $g$, and $h$ are continuous and there are antiderivatives of $f$, $g$, and $h$.
Let $F\left( x \right)$, $G\left( y \right)$, and $H\left( z \right)$ be the antiderivatives of $f$, $g$, and $h$, respectively such that
$\frac{{dF\left( x \right)}}{{dx}} = f\left( x \right)$, ${\ \ \ \ }$ $\frac{{dG\left( y \right)}}{{dy}} = g\left( y \right)$, ${\ \ \ \ }$ $\frac{{dH\left( z \right)}}{{dz}} = h\left( z \right)$
We can always choose a function defined by
$V\left( {x,y,z} \right) = F\left( x \right) + G\left( y \right) + H\left( z \right)$
Now, we evaluate the gradient of $V\left( {x,y,z} \right)$:
$\nabla V\left( {x,y,z} \right) = \left( {\frac{{\partial V}}{{\partial x}},\frac{{\partial V}}{{\partial y}},\frac{{\partial V}}{{\partial z}}} \right) = \left( {\frac{{dF\left( x \right)}}{{dx}},\frac{{dG\left( y \right)}}{{dy}},\frac{{dH\left( z \right)}}{{dz}}} \right)$
$\nabla V\left( {x,y,z} \right) = \left( {\left( {f\left( x \right),g\left( y \right),h\left( z \right)} \right)} \right)$
Since ${\bf{F}} = \left( {f\left( x \right),g\left( y \right),h\left( z \right)} \right)$, so $\nabla V\left( {x,y,z} \right) = {\bf{F}}$.
By definition, $V\left( {x,y,z} \right)$ is a potential function for ${\bf{F}}$. Hence, any vector field of the form ${\bf{F}} = \left( {f\left( x \right),g\left( y \right),h\left( z \right)} \right)$ has a potential function given by $V\left( {x,y,z} \right) = F\left( x \right) + G\left( y \right) + H\left( z \right)$, where $F\left( x \right)$, $G\left( y \right)$, and $H\left( z \right)$ are the antiderivatives of $f\left( x \right)$, $g\left( y \right)$, and $h\left( z \right)$, respectively.
