Path independence of line integrals

Statement

Let $U \subseteq \mathbb{R}^n$ be open and connected, and let $F \colon U \to \mathbb{R}^n$ be a continuous vector field. The following are equivalent.

1. $F$ is conservative on $U$.
2. For all piecewise $C^1$ curves $\gamma_1, \gamma_2$ in $U$ with the same initial point and the same end point, $\int_{\gamma_1} F \cdot d\mathbf{r} = \int_{\gamma_2} F \cdot d\mathbf{r}$.
3. For each closed piecewise $C^1$ curve $\gamma$ in $U$, $\int_\gamma F \cdot d\mathbf{r} = 0$.

Remarks

Statement 1 implies statement 2 by the gradient theorem. Statements 2 and 3 are equivalent: a closed curve splits into two curves with the same end points, and one uses reversal and additivity. Statement 2 implies statement 1: fix $p \in U$ and let $\phi(x) = \int_\gamma F \cdot d\mathbf{r}$ for a path $\gamma$ from $p$ to $x$. A short straight last segment in direction $e_i$ shows that $\partial \phi / \partial x_i = F_i$. Connectedness is used to join $p$ to each point. On a disconnected set, apply the result on each component.

Explanations

L1 human

For a conservative field, the work from one point to another does not depend on the route. Conversely, if the work never depends on the route, then the work from a fixed base point defines a potential for the field.