Conservative Vector Field

A conservative vector field is a category of vector fields. They are especially used in physics, to define conservative forces.

Definition §

Conservative Vector Field

A vector field $\vec{F}$ that is $C^1$ on either ${\mathbb{R}}^2$ or ${\mathbb{R}}^3$ is considered conservative iff it satisfies any of the four equivalent conditions:

• Zero on closed loops: Given any simple, closed oriented curve curve $C$, it holds that:

$${\oint }_C\vec{F}\cdot d\vec{c}=0$$

• Path Independance: For any two different oriented simple curves $C_1$ and $C_2$, having the same start and end points:

$${\int }_{C_1}\vec{F}\cdot \text{d}\vec{c}={\int }_{C_2}\vec{F}\cdot \text{d}\vec{c}$$

• Existence of scalar potential: There exists a scalar function $\varphi$ (with a constant vector) such that:

$$\vec{F}=\nabla (\varphi +\vec{c})$$

• Is irrotational: Has zero curl

$$\nabla \times \vec{F}=\vec{0}$$