Stokes’s theorem gives an alternative expression for the surface integral of the curl of a vector field. This is analogous to the divergence theorem, so Stokes’s theorem could be referred to as the ‘curl theorem’. The proof of the theorem is very similar to that for the divergence theorem, being based on the definition of curl in terms of a line integral

Let \( C \) be a closed curve which forms the boundary of a surface \( S \). Then for a continuously differentiable vector field \( \mathbf{u} \), Stokes’s theorem states that