arXiv:0712.2133v1 [math.FA] 13 Dec 2007 The Div-Curl Lemma Revisited Dan Poli ˇ sevski † Abstract. The Div-Curl Lemma, which is the basic result of the com- pensated compactness theory in Sobolev spaces(see [2]-[6]), was introduced by [1] with distinct proofs for the L 2 (Ω) and L p (Ω),p =2, cases. In this note we present a slightly different proof, relying only on a Green-Gauss integral formula and on the usual Rellich-Kondrachov compactness properties. Keywords: Compensated compactness; Weak convergence; Sobolev spaces. 2000 Mathematics Subject Classifications: 49J45; 46E40; 47B07. 1 An integral formula From now on, for any vector distribution w in R N , N ≥ 2, we denote: D(w)= n ∑ i=1 ∂w i ∂x i and C ij (w)= ∂w i ∂x j − ∂w j ∂x i ,1 ≤ i, j ≤ N . Theorem 1.1. Let Ω be an open bounded set of R N , N ≥ 2. Let p, q > 1 such that 1/p +1/q =1. If u ∈ [W 2,p loc (Ω)] N and v ∈ [W 2,q loc (Ω)] N , then for any ϕ ∈D(Ω) we have: 1