Taking a contour integral ∮ ∮ γ γ d ϕ ϕ = ∮ ∮ γ γ d ϕ ϕ d x d x over any contractible closed path γ γ , we would expect it to be zero (for example, by the fundamental theorem of calculus. However, this is not the case due to the singular nature of vortices (which give singularities in ϕ ϕ ).