Multivariable Calculus

Conservative Vector Fields

If $\mathbf{F}$ is conservative, there exists an $f$ such that $\mathbf{F}=\nabla f$.

Note

In order to calculate the line integral of a given curve, we need to determine whether $\mathbf{F}$ is conservative.

• Conservative Vector Field Tests

Properties of Conservative Vector Fields

If $\mathbf{F}$ is conservative and $C=\mathbf{r}(t):\alpha \le t\le \beta$:

$${\int }_C\mathbf{F}\cdot d\mathbf{r}=f(\mathbf{r}(\beta ))-f(\mathbf{r}(\alpha ))$$

Fundamental Theorem of Line Integrals

Additionally, ${\int }_C\mathbf{F}\cdot d\mathbf{r}$ is path-independent and the path traversed on $C$ is irrelevant in which only the endpoints a.k.a. the potential matters.

If we assume $C$ is a closed loop within $\mathbf{F}$, then the line integral of a closed loop is $0$.

$${\oint }_C\mathbf{F}\cdot d\mathbf{r}=0$$