Summary of Conservative Vector Fields

The theorems in this section require curves that are closed, simple, or both, and regions that are connected or simply connected.
The line integral of a conservative vector field can be calculated using the Fundamental Theorem for Line Integrals. This theorem is a generalization of the Fundamental Theorem of Calculus in higher dimensions. Using this theorem usually makes the calculation of the line integral easier.
Conservative fields are independent of path. The line integral of a conservative field depends only on the value of the potential function at the endpoints of the domain curve.
Given vector field [latex]{\bf{F}}[/latex], we can test whether [latex]{\bf{F}}[/latex] is conservative by using the cross-partial property. If [latex]{\bf{F}}[/latex] has the cross-partial property and the domain is simply connected, then [latex]{\bf{F}}[/latex] is conservative (and thus has a potential function). If [latex]{\bf{F}}[/latex] is conservative, we can find a potential function by using the Problem-Solving Strategy.
The circulation of a conservative vector field on a simply connected domain over a closed curve is zero

Key Equations

Fundamental Theorem for Line Integrals
[latex]\displaystyle\int_{C}\nabla {f}\cdot {d{\bf{r}}}=f({\bf{r}}(b))-f({\bf{r}}(a))[/latex]
Circulation of a conservative field over curve [latex]C[/latex] that encloses a simply connected region
[latex]\displaystyle\oint_{C}\nabla {f}\cdot {d{\bf{r}}}=0[/latex]

closed curve: a curve for which there exists a parameterization [latex]{\bf{r}}(t),a\le{t}\le{b}[/latex], such that [latex]{\bf{r}}(a)={\bf{r}}(b)[/latex], and the curve is traversed exactly once

connected region: a region in which any two points can be connected by a path with a trace contained entirely inside the region

Fundamental Theorem for Line Integrals: the value of the line integral [latex]\displaystyle\int_{C} {\nabla}{f}\cdot{d{\bf{r}}}[/latex] depends only on the value of [latex]f[/latex] at the endpoints of [latex]C[/latex]: [latex]\displaystyle\int_{C} {\nabla}{f}\cdot{d{\bf{r}}}=f({\bf{r}}(b)))-f({\bf{r}}(a))[/latex]

independent of path (path independent): a vector field [latex]{\bf{F}}[/latex] has path independence if [latex]\displaystyle\int_{C_{1}} {\bf{F}}\cdot{d{\bf{r}}}=\displaystyle\int_{C_{2}} {\bf{F}}\cdot{d{\bf{r}}}[/latex] for any curves [latex]C_{1}[/latex] and [latex]C_{2}[/latex] in the domain of [latex]{\bf{F}}[/latex] with the same initial points and terminal points

simply connected region: a region that is connected and has the property that any closed curve that lies entirely inside the region encompasses points that are entirely inside the region