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## Evaluating Total Differentials: Instructor's Guide

### Main Ideas

- Students gain experience finding total differentials of non-trivial functions
- Students are introduced to variables and physical quantities relevant to thermodynamics
- Students discuss their answers to identify difficulties and tricks they found useful when approaching this, and similar, problems

### Students' Task

*Estimated Time: 10 minutes*

In groups, students are asked to find the total differential of various functions, example below, to gain practice with taking and interpreting total differentials.

Example function: $U(x,y)=-\frac{L^2 U_0}{L^2+x^2+y^2}$ where $L$ and $U_0$ are constants.

### Prerequisite Knowledge

- Basic understanding of differentials
- Familiarity with partial derivatives

### Props/Equipment

- Tabletop Whiteboard with markers
- A handout for each student

### Activity: Introduction

### Activity: Student Conversations

Students sometimes:

- Do not show what variables are held constant in partial derivatives
- Multiply terms in a total differential instead of adding them
- Do not include the differentials multiplied to the terms on the right side of the total differential
- Do not show a differential on the left side of the total differential

### Activity: Wrap-up

Once the class has had a few minutes find the total differentials of the given functions, go through the given functions one by one, openly asking (to the class) a group to report their result. Once a group reports their result, ask (to the class) the other groups if they got the same solution. If there are other solutions, ask for groups to justify their solution. If this does not result in a unanimous agreement upon a correct answer, take the total differential of the function in question on the whiteboard before moving on to the next function.