By the Kelvin–Stokes theorem we can rewrite the line integrals of the fields around the closed boundary curve ∂Σ to an integral of the "circulation of the fields" (i.e. their curls) over a surface it bounds, i.e. ∮ ∮ ∂ ∂ Σ Σ B ⋅ ⋅ d ℓ ℓ = ∬ ∬ Σ Σ ( ∇ ∇ × × B ) ⋅ ⋅ d S , Hence the modified Ampere law in integral form can be rewritten as ∬ ∬ Σ Σ ( ∇ ∇ × × B − − μ μ 0 ( J + ε ε 0 ∂ ∂ E ∂ ∂ t ) ) ⋅ ⋅ d S = 0. Since Σ can be chosen arbitrarily, e.g. as an arbitrary small, arbitrary oriented, and arbitrary centered disk, we conclude that the integrand is zero if and only if Ampere's modified law in differential equations form is satisfied. The equivalence of Faraday's law in differential and integral form follows likewise.