List of formulas related to the circulation of a vector field and the Stokes formula:

  1. 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 ).
  2. 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 ).

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