An inexact differential δ δ u is a differential for which the integral over some two paths with the same end points is different. Specifically, there exist integrable paths γ γ 1 , γ γ 2 : [ 0 , 1 ] → → R such that γ γ 1 ( 0 ) = γ γ 2 ( 0 ) , γ γ 1 ( 1 ) = γ γ 2 ( 1 ) and ∫ ∫ γ γ 1 δ δ u ≠ ∫ ∫ γ γ 2 δ δ u In this case, we denote the integrals as Δ Δ u | γ γ 1 and Δ Δ u | γ γ 2 respectively to make explicit the path dependence of the change of the quantity we are considering as u .