Consider some arbitrary circuit. Approximate the circuit with lumped elements, so that time-varying magnetic fields are contained to each component and the field in the region exterior to the circuit is negligible. Based on this assumption, the Maxwell–Faraday equation reveals that ∇ ∇ × × E = − − ∂ ∂ B ∂ ∂ t = 0 in the exterior region. If each of the components has a finite volume, then the exterior region is simply connected, and thus the electric field is conservative in that region. Therefore, for any loop in the circuit, we find that ∑ ∑ i V i = − − ∑ ∑ i ∫ ∫ P i E ⋅ ⋅ d l = ∮ ∮ E ⋅ ⋅ d l = 0 where P i are paths around the exterior of each of the components, from one terminal to another.