If a potential function exists for a certain vector field, is it then automatically conservative? I know that if a vector field is conservative, a potential function exists, but does this relation also hold the other way around? In other words, does a potential function exist for a certain vector field if and only if the field in question is conservative?
 A: Say a vector field $\vec A$ has the potential $\phi$.
So we can write that $\vec A=\nabla \phi$
Now, line integral around the field from $M$ to $N$
$= \int_M^N \vec A \cdot \vec{dl}$ 
$= \int_M^N \nabla \phi \cdot \vec{dl}$ 
$= \int_M^N \left(\frac{\partial \phi}{\partial x}\hat i+ \frac{\partial \phi}{\partial y}\hat j+ \frac{\partial \phi}{\partial z}\hat k\right)  \cdot (dx\hat i+dy\hat j+dz\hat k)$ 
$= \int_M^N \left(\frac{\partial \phi}{\partial x}dx+ \frac{\partial \phi}{\partial y}dy+ \frac{\partial \phi}{\partial z}dz\right)$ 
$= \int_M^N d\phi$ 
$=\phi(M)-\phi(N)$
which is independent of the path joining $M$ and $N$.
So such vector fields having scalar potential are conservative.
A: EDIT : it seems you are looking for the "easy" implication that if a vector field is derived from a potential then it's conservative, which is always true. My answer actually says that what you are taking for granted (the reverse implication) is false in general. I guess you are working in a case where it's true, but it may interest you that there are restrictions.

It depends on the geometry of the space upon which the vector space is defined. If it's defined on $\mathbb{R}^n$, then yes. If it's (for instance) $\mathbb{R}^2\setminus \{0\}$, then no.
This is because a conservative vector field is basically a closed $1$-form, and a field that comes from a potential is basically an exact $1$-form, and closed $1$-forms are exact precisely when the first de Rham cohomology group is trivial. This happens for instance on a contractible space (such as $\mathbb{R}^n$), but not on any space.
It's actually a purely topological condition : it's equivalent to the fact that $\pi_1(X)^{ab}$ is a torsion group, where $X$ is the space on which the vector field is defined.
