• Total differentials
• Cyclic chain rule
• Legendre transformations
  • Chain rule diagrams
  • Differential substitution

Estimated Time: 35 minutes

• Students evaluate two given partial derivatives
• Students must use a Legendre transformation for one of the given partial derivatives
• Students may find it helpful to use a chain rule diagram

• Familiarity with partial derivatives and their interpretations
• Familiarity with total differentials
• Familiarity with Legendre transformations
• Familiarity with the cyclical chain rule

• Tabletop Whiteboard with markers
• A handout for each student

• Students may recognize one of the equations as a hyperbolic tangent. Credit them for recognizing it.
• Students may find the physical truth of the equations to be inspiring, so do not hesitate to state that the equations are valid physical relations.
• Students may ask what magnetic susceptibility, $\chi_B$, is. Dr. Round provides an eloquent response of sufficient detail: “It's the thing that's defined to be that.”
• Students may be confused as to how to apply the cyclic chain rule.