Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: 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.

10
The categories of this paper are: q-bio
30
The categories of this paper are: q-alg
18
The categories of this paper are: econ
27
The categories of this paper are: comp-gas
3
The categories of this paper are: physics
34
The categories of this paper are: chem-ph
5
The categories of this paper are: gr-qc
32
The categories of this paper are: atom-ph
24
The categories of this paper are: dg-ga
9
The categories of this paper are: nlin
14
The categories of this paper are: math-ph
20
The categories of this paper are: acc-phys
12
The categories of this paper are: nucl-th
35
The categories of this paper are: mtrl-th
28
The categories of this paper are: alg-geom
13
The categories of this paper are: quant-ph