Pedro F. da Silva Júnior

arXiv2022

On the Integrability of Pfaffian Forms on ${\mathbb R}^{n}$

This paper details the lesser known conditions on ${\mathbb {R}}^{n}$ for the integrability of pfaffian forms, or 1-forms. Emphasis is given to locality of these conditions, and proofs in some additional detail are provided for theorems due to Clairaut and Carathéodory. Considering the importance of the integrability of pfaffian forms, in particular in mathematical-physics, this paper shows that: there is a hidden content in Carathéodory's theorem in the direction of a global integrability.

math-ph↗

arXiv2021

On the Deduction of the Carathéodory's Axiom of the Second Law of Thermodynamics from the Clausius and Kelvin Principles

Carathéodory's formalism for classical thermodynamics is a rich alternative approach to this theory, although unpopular with students and physics professors. This approach dispenses with the content of thermal machines for the presentation of the second law of thermodynamics. In this paper, we discuss Carathéodory's formalism historically, and show how Carathéodory's axiom of the second law of thermodynamics is derived, didactically, from the Clausius principle and the Kelvin principle. In addition, also providing an objective character for this paper, in the sense of seeking to popularize the teaching of Carathéodory's formalism in disciplines of classical thermodynamics at undergraduate level, we guide the reader to obtain the entropy and mathematical content of the second law of thermodynamics through this formalism. Finally, considering the wide reviewed literature, the proof we gave for deducing the Carathéodory's axiom of the second law of thermodynamics from the Clausius principle is new.

physics.class-ph↗