Let $X: \mathbb R^3 \rightarrow \mathbb R^3$ be a smooth vector field, and $f: \mathbb R^3 \rightarrow \mathbb R$ be its potential (i.e. $X= grad\ f$).
When $f$ has a minimum point $p_{min}$ and all points in some neighborhood $U$ of $p_{min}$ are regular points of $f$, is $p_{min}$ a point of all integral curves in some neighborhood $V (\subset U)$ of $p_{min}$? I think this is true, but I don't know how I prove this.
For example, the minimum point of $f(x,y,z) := x^2 + y^2 + z^2$ is $(0,0,0)$, and this satisfies above.