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.
Comments: 10 pages, 15 figures