Say I have a scalar function $V(x,y,z) :\mathbb{R^3}\to \mathbb{R}$ which is a scalar potential of some vector field $\vec{F}$. I can write:
$\nabla V=\frac{\partial V}{\partial x}\hat{x}+\frac{\partial V}{\partial y}\hat{y}+\frac{\partial V}{\partial z}\hat{z}$
I also know $\vec{F}=\nabla V$, so:
$\vec{F}=\frac{\partial V}{\partial x}\hat{x}+\frac{\partial V}{\partial y}\hat{y}+\frac{\partial V}{\partial z}\hat{z}$
Now, if V was only dependent upon one variable, say $x$, could I write the following connection ?
$V=\int \vec{F} \cdot \vec{dx}$
If so, what algebraic steps have I done here? It sure makes sense, but I fail at explaining it to myself. And what if V wasn't only a function of x, but of y and z too? Would it haven been right to write:
$V_x=\int \vec{F} \cdot \vec{dx}$
$V_y=\int \vec{F} \cdot \vec{dy}$
$V_z=\int \vec{F} \cdot \vec{dz}$
and then $V=V_x+V_y+V_z$ ?