Mathematical Complements: df or δf

This lecture covers mathematical tools for differentials of functions of multiple variables, including partial derivatives, second derivatives, and exact differentials. It explores the concept of conservative quantities in thermodynamics and the significance of different paths in integrals. Examples illustrate the practical applications of exact differentials and the conditions for their existence, such as the Poincaré lemma. Real-life examples, like mountain hiking, demonstrate the independence of elevation changes on the chosen path. Exercises challenge students to identify non-exact differential forms, emphasizing the importance of understanding the behavior of multivariable functions.