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Activities

Small Group Activity

30 min.

Name the experiment (changing entropy)
Students are placed into small groups and asked to create an experimental setup they can use to measure the partial derivative they are given, in which entropy changes.
• Found in: Energy and Entropy course(s) Found in: Name the experiment sequence(s)

Small Group Activity

10 min.

Fourier Transform of a Derivative

The formula for the inverse Fourier transform shows that a function $f(x)$ can be written in terms of its Fourier transform via $$f(x)= \frac{1}{\sqrt{2\pi}} \int_{-\infty}^{\infty} \tilde{f}(k)\, e^{ikx}\, dk$$ Take the derivative of both sides of this equation with respect to $x$ and simplify. Interpret your expression as the inverse Fourier transform of something.

Instructor's Guide

Introduction

Students will need a short lecture giving the definition of the inverse Fourier Transform $${\cal{F}}^{-1}(\tilde{f}) =f(x)= \frac{1}{\sqrt{2\pi}} \int_{-\infty}^{\infty} f(k)\, e^{ikx}\, dk$$

Student Conversations

The logic of this problem may feel a little backwards to students. Be prepared to be more directive than normal in helping the groups that get stuck. Or consider doing this problem as a mini-lecture, rather than a group activity, especially if time is tight.

Wrap-up

The result if this calculation is an essential formula in solving differential equations with Fourier transforms.
• Found in: Periodic Systems course(s) Found in: Fourier Transforms and Wave Packets sequence(s)

Problem

5 min.

Directional Derivative

You are on a hike. The altitude nearby is described by the function $f(x, y)= k x^{2}y$, where $k=20 \mathrm{\frac{m}{km^3}}$ is a constant, $x$ and $y$ are east and north coordinates, respectively, with units of kilometers. You're standing at the spot $(3~\mathrm{km},2~\mathrm{km})$ and there is a cottage located at $(1~\mathrm{km}, 2~\mathrm{km})$. You drop your water bottle and the water spills out.

1. Plot the function $f(x, y)$ and also its level curves in your favorite plotting software. Include images of these graphs. Special note: If you use a computer program written by someone else, you must reference that appropriately.
2. In which direction in space does the water flow?
3. At the spot you're standing, what is the slope of the ground in the direction of the cottage?
4. Does your result to part (c) make sense from the graph?

• Found in: Static Fields, AIMS Maxwell, Problem-Solving course(s) Found in: Gradient Sequence sequence(s)

Problem

Derivative of Fermi-Dirac function
Derivative of Fermi-Dirac function Show that the magnitude of the slope of the Fermi-Direc function $f$ evaluated at the Fermi level $\varepsilon =\mu$ is inversely proportional to its temperature. This means that at lower temperatures the Fermi-Dirac function becomes dramatically steeper.
• Found in: Thermal and Statistical Physics course(s)

Small White Board Question

10 min.

Partial Derivatives from a Contour Map
In this sequence of small whiteboard questions, students are shown the contour graph of a function of two variables and asked to find the derivative. They discover that, without a function to differentiate, they must instead think of the derivative as a ratio of small changes. This requires them to pick two nearby points. Which two?
• Found in: AIMS Maxwell, Static Fields, Surfaces/Bridge Workshop, Problem-Solving course(s) Found in: Gradient Sequence sequence(s)

Small Group Activity

30 min.

Name the experiment
Student groups design an experiment that measures an assigned partial derivative. In a compare-and-contrast wrap-up, groups report on how they would measure their derivatives.
• Found in: Energy and Entropy course(s) Found in: Name the experiment sequence(s)

Small Group Activity

30 min.

Directional Derivatives
This small group activity using surfaces relates the geometric definition of directional derivatives to the components of the gradient vector. Students work in small groups to measure a directional derivative directly, then compare its components with measured partial derivatives in rectangular coordinates. The whole class wrap-up discussion emphasizes the relationship between the geometric gradient vector and directional derivatives.
• Found in: Vector Calculus I course(s) Found in: Gradient Sequence sequence(s)

Small Group Activity

30 min.

Quantifying Change
In this activity, students will explore how to calculate a derivative from measured data. Students should have prior exposure to differential calculus. At the start of the activity, orient the students to the contour plot - it's busy.

Small Group Activity

30 min.

Heat and Temperature of Water Vapor
In this introduction to heat capacity, students determine a derivative that indicates how much the internal energy changes as the temperature changes when volume is held constant.

Small Group Activity

30 min.

Chain Rule Measurement
This small group activity using surfaces combines practice with the multivariable chain rule while emphasizing numerical representations of derivatives. Students work in small groups to measure partial derivatives in both rectangular and polar coordinates, then verify their results using the chain rule. The whole class wrap-up discussion emphasizes the relationship between a directional derivative in the $r$-direction and derivatives in $x$- and $y$-directions using the chain rule.
• Found in: Vector Calculus I course(s)