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Thermodynamic identities with sunray diagrams
(2019)cite arxiv:1912.11485Comment: 10 pages, 15 figures; submitted to American Journal of Physics.
Аннотация
One of the hurdles in learning thermodynamics is a plethora of complicated
partial derivative identities. Students suffer from difficulties in deriving,
justifying, memorizing, or interpreting the identities, misconceptions about
partial derivatives, and a lack of deeper understandings about the meaning of
the identities. Here, we propose a diagrammatic method, the "sunray diagram,"
for the calculus of differentials and partial derivatives that resolves all of
the aforementioned difficulties. With the sunray diagram, partial derivative
identities can be instantly obtained in an intuitive manner by sliding arrows.
Furthermore, the sunray diagram is more than an ad hoc machinery but based on
the geometric structure of thermodynamics and admits direct physical
interpretation on the P-V (or T-S) plane. Employing the language of
differential forms and symplectic geometry, we show that the sunray diagram and
Maxwell's previous work utilizing equal-area sliding of parallelograms are
different visualizations of the same mathematical syntax, while the sunray
diagram being more convenient in practice. We anticipate that our discussion
introduces the geometry of thermodynamics to learners and enriches the
graphical pedagogy in physics education.
