Relationships between the free energy change due to shear, the first normal stress coefficient and the density dependence of the coefficient of shear rate dependence of the pressure are derived for a linear viscoelastic fluid in steady shear flow. One of these relationships is a generalization of the Maxwell relations that are so useful in equilibrium thermodynamics, and another is the classical expression for the limiting zero shear rate value of the first normal stress coefficient obtained from the theory of non-linear rheology. A third relationship links the first normal stress coefficient to the stored elastic energy of a linear viscoelastic fluid in steady shear. The stored elastic energy, the first normal stress coefficient and the shear dilatancy are all independently evaluated for a simple fluid using non-equilibrium molecular dynamics simulations and the relationships between them are tested. It is found that the relationships are not satisfied for the conditions investigated. Possible reasons for the discrepancies are discussed. (C) 2007 Elsevier B.V. All rights reserved.