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Mathematical methods of diagonalization of quadratic forms applied to the study of stability of thermodynamic systems

In this paper, we use quadratic forms diagonalization methods applied to the function thermodynamic energy to analyze the stability of physical systems. Taylor's expansion was useful to write a quadratic expression for the energy function. We consider the same methodology to expanding the thermodynamic entropy and investigate the signs of the second-order derivatives of the entropy as well as previously to the thermodynamic energy function. The signs of the second-order derivatives to the Helmholtz, enthalpy and Gibbs functions are also analysed. We show the immediate consequences on the stability of physical systems due to the signs or curvatures of the second-order derivatives of these thermodynamic functions. The thermodynamic potentials are presented and constructed pedagogically as well as demonstrated the main mathematical aspects these surfaces. We demonstrate the power of superposition of mathematical and physical aspects to understand the stability of thermodynamic systems. Besides, we provide a consistent mathematical demonstration of the minimum, maximum, and saddle conditions of the potentials. We present here a detailed approach on aspects related to the curvature of the thermodynamic functions of physical interest with consequences on stability. This work can be useful as a part or supplement material of the traditional physics curriculum that requires a solid formation in thermodynamics, particularly about formal aspects on the stability.