Conservative Fields

Conditions

1. Clairaut’s Conditions are satisfied

$$\begin{array}{rl}\frac{\partial F_1}{\partial y}& =\frac{\partial F_2}{\partial x}\\ \frac{\partial F_1}{\partial z}& =\frac{\partial F_3}{\partial x}\\ \frac{\partial F_2}{\partial z}& =\frac{\partial F_3}{\partial y}\end{array}$$

2. $E$ is simply connected (no holes)

Properties

• $⟺\vec{F}=\vec{\nabla }f\text{ in }E$
• $⟺\text{curl}\vec{F}=\vec{0}$ in simply connected $E$
• $(⟹\text{curl}\vec{F}=\vec{0}\text{ in }E)$
• $⟺$
  • Path Independent Line Integrals: ${\int }_C\vec{F}\cdot d\vec{r}={\int }_{C^{\prime }}\vec{F}\cdot d\vec{r}$
  • Circulation of $\vec{F}=0$: ${\oint }_C\vec{F}\cdot d\vec{r}=0\text{ for }C\in E$

Potential

$$f(x,y,z)={\int }_{x_0}^x{F}_1(t,y_0,z_0)dt+{\int }_{y_0}^y{F}_1(x,t,z_0)dt+{\int }_{z_0}^z{F}_1(x,y,t)dt+C$$