**Theorem 0.0.1 **  The **curl theorem** or **Stokes' theorem** states that the loop integral of a continuously differentiable vector field $\vec{F}$ along the boundary path $P$ of a smooth oriented surface $S$ is the surface integral of the curl of $\vec{F}$ over the surface $S$.
\[\int_P\vec{F}\cdot d\vec{P} = \iint_S(\nabla\times\vec{F})\cdot d\vec{S}\]