List of formulas related to the circulation of a vector field and the Stokes formula:
- Circulation of a vector field ( \mathbf{F} ) along a closed contour ( C ):
[
\Gamma = \oint_C \mathbf{F} \cdot d\mathbf{r}
]
where ( \Gamma ) is the circulation, ( \mathbf{F} ) is the vector field, ( d\mathbf{r} ) is the length element along the contour ( C ). - Stokes formula:
[
\oint_C \mathbf{F} \cdot d\mathbf{r} = \iint_S (\nabla \times \mathbf{F}) \cdot d\mathbf{S}
]
where:
- ( C ) — closed loop,
- ( S ) — a surface bounded by a contour ( C ),
- ( \nabla \times \mathbf{F} ) is the curl of the vector field ( \mathbf{F} ),
- ( d\mathbf{S} ) is the vector element of the surface area ( S ).