Legendre Transformation and Convex Functions

Let us consider two smooth, differentiable functions x) and ir y). If they are related by 

Then we take the limit A 0, and substitute the definition (3.83) to obtain 

If a function 4 is sufficiently smooth, the convexity of p is equivalent to the monotonicity of d4 /dx, as observed above. 

2 Thermodynamic Functions in a Thermo-mechanical Field Solids with Small Strain 

As described in Sect.3.2, the change of internal energy du(e, s) under the assumption of a small strain theory is given by 

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