Table of Contents

Differentials

Prerequisites

Students should be able to:

  • Determine an ordinary derivative from a symbolic expression.
  • Hold variables constant when finding a derivative.
  • Sketch graphs and identify points and changes between points on those graphs.

In-class Content

Lecture: Differentials of 1D Functions (Lec - 20 min)

Main ideas

Lecture (30 minutes)

Ask students to make a sketch of f = 7x2 on the big white board.

What are the variables of interest? One idea here is to note that x is not the only variable here: f is also a variable from a physics perspective.

What representations do we have for this relationship? The symbolic equation is one, and the graph is another.

Introduce the differential quantities df and dx as small changes in f and x, respectively. Ask students to add df and dx to their graphs.

Then, ask students how df and dx are related to each other. Students should be able to articulate that this is a derivative.

Is the relationship the same if we choose a different point on the graph (a different initial x)?

Then relate df and dx using the symbolic representation: df = 14xdx.

The following two steps can be replaced by other activities on the relevant Hour page.
  • Give the students f = 5x2y3. What are the variables? What representations do we have (if time, hand out surfaces and have students make sketches similar to the above)? What does a differential relationship look like? For this last one, use the generic form df = A dx + B dy, and talk about the fact that this is another representation.
  • Do some specific examples of using the zapping with d strategy (see Zapping with d).

Activity: Total Differentials on a Surface

Link to Total Differentials on a Surface Activity

Activity Highlights

  1. This small group activity is designed to help students understand differentials in multivariable functions.
  2. Students work in small groups to generalize symbolic and graphical representations of differentials for a single variable function to a multivariable case using a plastic surface as a test case.
  3. The whole class wrap-up discussion emphasizes that differentials are a way of linearizing the relationship between variables by expressing how small changes in those variables are related.

Activity: Evaluating Total Differentials

Link to Evaluating Total Differentials Activity

Activity Highlights

  1. This small group activity is designed to help students understand differentials in multivariable functions.
  2. Students work in small groups to generalize symbolic and graphical representations of differentials for a single variable function to a multivariable case using a plastic surface as a test case.
  3. The whole class wrap-up discussion emphasizes that differentials are a way of linearizing the relationship between variables by expressing how small changes in those variables are related.

Activity: Covariation in Thermal Systems

Link to Covariation in Thermal Systems Activity

Activity Highlights

  1. This small group activity is designed to help students understand relationships between thermodynamic state variables.
  2. Students work in small groups to investigate states on a raising physics surface, and consider cycles.
  3. The whole class wrap-up discussion emphasizes the meaning of a “state” variable.

Homework for Energy and Entropy

  1. (mbZapWithD) Find the differential of special functions.

    Find the differential of each of the following expressions, ie. zap each of the following with $d$:

    1. $$f=3x-5z^2+2xy$$

    2. $$g=\sin^2(\omega t)$$

    3. $$h=\frac{\mu B}{k_B T}$$

    4. $$j=\exp\left(\frac{\mu B}{k_B T}\right)$$

    5. $$k=\ln\left(e^{\frac{\mu B}{k_B T}} +e^{-\frac{\mu B}{k_B T}}\right)$$

  2. (mbDifferentials) Finding the differential abstractly.

    Find the differential of $R$ where\ldots

    1. $R(B,C) = B^2 + C^2$

    2. $R(B,C) = BC$

    3. $R(B,C) = e^{B^2+C^2}$

    4. $R(B,C) = e^{S(B,C)}$

    5. $R(B,C) = ST$, where $S = S(B,C)$ and $T = T(B,C)$

  3. (mbDerivVSDiff) Find the differential of special functions.

    Consider $V=-A\cos(kx)$, where $A$ is a constant with dimensions of electrostatic potential.

    1. Find $dV$ and $\frac{dV}{dx}$. Discuss the differences and similarities between the two.

    2. What are the dimensions of $k$, $V$, $dx$, $dV$, and $\frac{dV}{dx}$?

  4. (mbMathematicaContours) Plot contours, Change of Variable.

    On this problem, you will use Mathematica to graph and interpret some functions following the steps below. Mathematica can be found in the computers on campus, or you can download a student license for free, or even use online tools like sandbox.open.wolframcloud.com. Please include both your graphs and your source code to make grading easier!

    1. Create a contour plot of the function $G = A^4 + B^2$. Play around with the options until you have a graph that has all of the following features:

      The axes are labeled. The contours are numbered. The plot is titled. There are 8 contours. The colored shading on the contours is different from the default.

    2. Describe how your contour plot changes if you modify the plot range (i.e., zooming in or out)?

    3. Suppose that $5A = 4U + 3V$ and $5B = 4V - 3U$.

      Make a contour plot of $G$ in terms of the new variables $U$ and $V$. Label and present your new graph as described above.

    4. Make a contour plot of $G$ in terms of the variables $A$ and $V$. Label and present your new graph as described above.

    5. Explain why all three of the contour plots you have made are valid representations of $G$. What is different about the three graphs?

  5. (CoffeeAndBagels) A thermo-like system. Requires Clairaut's theorem.

    \newcommand\juice{\ensuremath{\mathcal{O}}} \newcommand\cocoa{\ensuremath{\mathcal{C}}} \newcommand\priceorange{\ensuremath{\mathcal{P_O}}} \newcommand\pricecocoa{\ensuremath{\mathcal{P_C}}}

    In economics, the term utility is roughly related to overall happiness. Many things affect your happiness, including the amount of money you have and the amount of coffee you drink. We cannot directly measure your happiness, but we can measure how much money you are willing to give up in order to obtain coffee or bagels. If we assume you choose wisely, we can thus determine that your happiness increases when you decrease your amount of money by that amount in exchange for increasing your coffee consumption. Thus money is a (poor) measure of happiness or utility.

    Money is also a nice quantity because it is conserved—just like energy! You may gain or lose money, but you always do so by a transaction. (There are some exceptions to the conservation of money, but they involve either the Fed, counterfeiters, or destruction of cash money, and we will ignore those issues.)

    In this problem, we will assume that you have bought all the coffee and bagels you want (and no more), so that your happiness has been maximized. Thus you are in equilibrium with the coffee shop. We will assume further that you remain in equilibrium with the coffee shop at all times, and that you can sell coffee and bagels back to the coffee shop at cost.\footnote{Yes, this is ridiculous. It would be slightly less ridiculous if we were talking about nations and commodities, but also far less humorous.}

    Thus your savings $S$ can be considered to be a function of your bagels $B$ and coffee $C$. In this problem we will also discuss the prices $P_B$ and $P_C$, which you may not assume are independent of $B$ and $C$. It may help to imagine that you have

    1. The prices of bagels and coffee $P_B$ and $P_C$ have derivative relationships between your savings and the quantity of coffee and bagels that you have. What are the units of these prices? What is the mathematical definition of $P_C$ and $P_B$?

    2. Write down the total differential of your savings, in terms of $B$, $C$, $P_B$ and $P_C$.

    3. Use the equality of mixed partial derivatives (Clairut's theorem) to find a relationship between $P_B$, $P_C$, $B$ and $C$. Write this relationship mathematically, and also describe in words what it means.

    4. Solve for the total differential of your net worth. Once again use Clairut's theorem considering second derivatives of $W$ to find a different partial derivative relationship between $P_B$, $P_C$, $B$ and $C$.