Teaching Differentials in Thermodynamics using Spatial Visualization
  The greatest difficulty that is encountered by students in Thermodynamics classes is to find relationships between variables and solve a total differential equation that relates one thermodynamic state variable to two mutually-independent state variables. Rules of differentiation, including the total differential and the cyclic rule, are fundamental to solving thermodynamics problems. However, many students do not understand such basic mathematics principles, because ordinary calculus and thermodynamics textbooks do not provide specific geometric explanations for the differential rules. Spatial visualization allows students to develop a new line of reasoning. In this investigation, the total differential rule and the cyclic rule are interpreted geometrically. 
Chih-Yueh Wang and Jin-Han Hou
physics.ed-ph