Conservative Vector Fields

Conservative vector fields

A vector field $\mathbf {F}$ for which $\nabla \times \mathbf {F} =\mathbf {0}$ at all points is an "conservative" vector field. $\mathbf {F}$ can also be referred to as being "irrotational" since the gain around any closed curve is always 0.

A key property of a conservative vector field $\mathbf {F}$ is that the gain of $\mathbf {F}$ along a continuous curve is purely a function of the curve's end points. If $C_{1}$ and $C_{2}$ are two continuous curves which share the same starting point $\mathbf {q} _{0}$ and end point $\mathbf {q} _{1}$ , then $\int _{\mathbf {q} \in C_{1}}\mathbf {F} (\mathbf {q} )\cdot d\mathbf {q} =\int _{\mathbf {q} \in C_{2}}\mathbf {F} (\mathbf {q} )\cdot d\mathbf {q}$ . In other words, the gain is purely a function of $\mathbf {q} _{0}$ and $\mathbf {q} _{1}$ . This property can be derived from Stokes' theorem as follows:

Invert the orientation of $C_{2}$ to get $-C_{2}$ and combine $C_{1}$ and $-C_{2}$ to get a continuous closed curve $C_{3}=C_{1}-C_{2}$ , linking the curves together at the endpoints $\mathbf {q} _{0}$ and $\mathbf {q} _{1}$ . Let $\sigma$ denote a surface for which $C_{3}$ is the counterclockwise oriented boundary.

Stokes' theorem states that $\int _{\mathbf {q} \in C_{3}}\mathbf {F} (\mathbf {q} )\cdot d\mathbf {q} =\iint _{\mathbf {q} \in \sigma }(\nabla \times \mathbf {F} )|_{\mathbf {q} }\cdot \mathbf {dS} =0$ . The gain around $C_{3}$ is the gain along $C_{1}$ minus the gain along $C_{2}$ : $\int _{\mathbf {q} \in C_{3}}\mathbf {F} (\mathbf {q} )\cdot d\mathbf {q} =\int _{\mathbf {q} \in C_{1}}\mathbf {F} (\mathbf {q} )\cdot d\mathbf {q} +\int _{\mathbf {q} \in -C_{2}}\mathbf {F} (\mathbf {q} )\cdot d\mathbf {q}$ $=\int _{\mathbf {q} \in C_{1}}\mathbf {F} (\mathbf {q} )\cdot d\mathbf {q} -\int _{\mathbf {q} \in C_{2}}\mathbf {F} (\mathbf {q} )\cdot d\mathbf {q}$ . Therefore:

$\int _{\mathbf {q} \in C_{3}}\mathbf {F} (\mathbf {q} )\cdot d\mathbf {q} =0$ $\implies \int _{\mathbf {q} \in C_{1}}\mathbf {F} (\mathbf {q} )\cdot d\mathbf {q} =\int _{\mathbf {q} \in C_{2}}\mathbf {F} (\mathbf {q} )\cdot d\mathbf {q}$