We present a systematic treatment of the theory of Compensated Compactness
under Murat's constant rank assumption. We give a short proof of a sharp weak
lower semicontinuity result for signed integrands, extending the results of
Fonseca--M\"uller. The null Lagrangians are an important class of signed
integrands, since they are the weakly continuous functions. We show that they
are precisely the compensated compactness quantities with Hardy space
integrability, thus proposing an answer to a question raised by
Coifman-Lions-Meyer-Semmes. Finally we provide an effective way of computing
the null Lagrangians associated with a given operator.Comment: Corrected typos and added an example in Section