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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.

Choose a vector field \(\boldsymbol{\vec{F}}\) from the first column below. Choose a small loop \(C\) (that is, a simple, closed, positively-oriented curve) which does not go around the origin.

• Is \(\oint\boldsymbol{\vec{F}}\cdot d\boldsymbol{\hat{r}}\) positive, negative, or zero?
  
• Will a paddlewheel spin if placed inside your loop, and, if so, which way?
  

• Do you think \(\nabla\times\boldsymbol{\vec{F}}\) is zero or nonzero inside your loop?

  Explain.

• Compute \(\nabla\times\boldsymbol{\vec{F}}\). Did you guess right? Explain.
• Is \(\oint\boldsymbol{\vec{F}}\cdot\boldsymbol{\hat{n}}\,ds\) positive, negative, or zero? (\(\boldsymbol{\hat{n}}\) is the outward pointing normal vector to \(C\).)
• Is the net flow outwards across your loop positive, negative, or zero?
• Do you think \(\nabla\cdot\boldsymbol{\vec{F}}\) is zero or nonzero inside your loop? Explain.
• Compute \(\nabla\cdot\boldsymbol{\vec{F}}\). Did you guess right? Explain.
• Repeat the above steps for vector fields \(\boldsymbol{\vec{G}}\) and \(\boldsymbol{\vec{H}}\) chosen from the second and third columns.

\(-y\,\boldsymbol{\hat{x}}+x\,\boldsymbol{\hat{y}}\)	\((x+y)\,\boldsymbol{\hat{x}}+(y-x)\,\boldsymbol{\hat{y}}\)	\(e^{-y^2}\,\boldsymbol{\hat{y}}\)
\(x\,\boldsymbol{\hat{x}}+y\,\boldsymbol{\hat{y}}\)	\((y-x)\,\boldsymbol{\hat{x}}-(x+y)\,\boldsymbol{\hat{y}}\)	\(e^{-x^2}\,\boldsymbol{\hat{y}}\)

Main ideas

• Visualization of divergence and curl.

Prerequisites

• Definition of divergence and curl.
• Geometry of divergence and curl, either through a geometric definition or through Stokes' Theorem and the Divergence Theorem.

Warmup

• Students may need to be reminded what circulation is.
• Students may not have seen flux in 2 dimensions.
• Students may only have seen \(\boldsymbol{\hat{n}}\) for surfaces, not curves. Some students will set \(\boldsymbol{\hat{n}}=\boldsymbol{\hat{z}}\)! Emphasize that \(\boldsymbol{\hat{n}}\) is horizontal (and that \(ds\ne\boldsymbol{d\vec{S}}\)).

Props

• whiteboards and pens
• formula sheet for div and curl in spherical and cylindrical coordinates (Each group may need its own copy.)
• divergence and curl transparency
• blank transparencies and pens

Wrapup

• Discuss the effect of choosing loops of different shapes, especially those adapted to the given vector field.
• Talk about the geometry of sinks and sources (for divergence) and paddlewheels (for curl).

Details

In the Classroom

• While students are working on this activity, draw the vector fields on the board to use during the wrapup. Alternatively, bring an overhead transparency showing the vector fields (and blank transparencies for students to write on).
• Students like this lab; it should flow smoothly and quickly.
• Students may need to be reminded what \(\oint\) means, and that the positive orientation in the plane is counterclockwise.
• Yes, two pairs of questions are really the same.
• Make sure the paths do not go around the origin.
• Encourage each group to work on at least two vector fields, which are in different rows and columns. Include one vector field from the third column if time permits.
• Encourage each group to consider, for a single vector field, moving their loop to another location. This is especially effective (and in fact essential) for the two vector fields in the third column.
• See the discussion of using transparencies for Group Activity The Hill.
• Students may eventually realize that the vector fields in the middle column are linear combinations of the vector fields in the first column, which are in turn “pure curl” and “pure divergence”, respectively.

Subsidiary ideas

• Divergence and curl are not just about the behavior near the origin. Derivatives are about change --- the difference between nearby vectors.

Homework

(MHG refers to McCallum, Hughes Hallett, Gleason, et al.

• MHG 19.1:20
• MHG 20.2:16
• MHG 20.3:10,12,20
• MHG 20.4:22

Essay questions

• Which operation, curl or divergence is easier to understand?
• Which is more useful?
• Do you prefer to gauge curl from a plot or from a calculation? What about divergence?

Enrichment

• Emphasize the importance of divergence and curl in applications.
• Ask students how to determine which vector fields are conservative! (A single closed path with nonzero circulation suffices to show that a vector field is not conservative. The best geometric way we know to show that a vector field is conservative is to try to draw the level curves for which the given vector field would be the gradient.)
• Discuss the fact that \(\boldsymbol{\hat{r}}\over r\) and \(\boldsymbol{\hat{\phi}}\over r\) are both curl-free and divergence-free; this is counterintuitive, but crucial for electromagnetism. (These are, respectively, the electric/magnetic field of a charged/current-carrying wire along the \(z\)-axis.)
• Discuss the behavior of \(\boldsymbol{\hat{r}}\over r^n\) and \(\boldsymbol{\hat{\phi}}\over r^n\), emphasizing that both the divergence and curl vanish when \(n=1\).
• Relate these examples to the magnetic field of a wire (\(\boldsymbol{\vec{B}}={\boldsymbol{\hat{\phi}}\over r}\)) and the electric field of a point charge (\(\boldsymbol{\vec{E}}={\boldsymbol{\hat{r}}\over r^2}\); this is the spherical \(r\)).
• Show students how to compute divergence and curl of these vector fields in cylindrical coordinates.
• Trying to estimate divergence and curl from a single plot of a vector field confronts students with the need to zoom in. Technology can be useful here.
• Point students to our paper on Electromagnetic Conic Sections, which appeared in Am. J. Phys. 70, 1129--1135 (2002), and which is also available on the Bridge Project website.
• Most physical applications of the divergence are 3-dimensional, rather than 2-dimensional. Each vector field in this activity could be regarded as a horizontal 3-dimensional vector field by assuming that there is no \(z\)-dependence, in which case the flux can be computed through a 3-dimensional box whose cross-section is the loop, and whose horizontal top and bottom do not contribute.

Basic algebraic and geometric properties of the dot product.

In this activity students use the known speed of earthquake waves to estimate the Young's modulus of the Earth's crust.

Students integrate numerically to find the electric field due to a cone of surface charge, and then visualize the result. This integral can be done in either spherical or cylindrical coordinates, giving students a chance to reason about which coordinate system would be more convenient.

Students write python programs to compute the potential due to a square of surface charge, and then to visualize the result. This activity can be used to introduce students to the process of integrating numerically.

• The superposition principle for the electrostatic potential;
• How to calculate the distance formula \(\frac{1}{|\vec{r} - \vec{r}'|}\) for a simple specific geometric situation;
• How to calculate the first few terms of a (binomial) power series expansion by factoring out the dimensionful quantity which is large;
• How the symmetries of a physical situation are reflected in the symmetries of the power series expansion.

• The superposition principle for the electrostatic potential;
• How to calculate the distance formula \(\frac{1}{|\vec{r} - \vec{r}'|}\) for a simple specific geometric situation;

Students work in small groups to use the superposition principle \[V(\vec{r}) =\frac{1}{4\pi\epsilon_0}\int\frac{\rho(\vec{r}^{\,\prime})}{\vert \vec{r}-\vec{r}^{\,\prime}\vert} \, d\tau^{\prime}\] to find an integral expression for the electrostatic potential, \(V(\vec{r})\), everywhere in space, due to a ring of charge.

In an optional extension, students find a series expansion for \(V(\vec{r})\) either on the axis or in the plane of the ring, for either small or large values of the relevant geometric variable. Add an extra half hour or more to the time estimate for the optional extension.

This is a great first programming activity.

Consider a system consisting of four point charges arranged on the corners of a square in 3D Cartesian space of coordinates \((x,y,z)\).

1. Write a python function that returns the potential at any point in space caused by four equal point charges forming a square. Make the sides of the square parallel to the \(x\) and \(y\) axes and on the \(z=0\) plane.

   To do this you will need the expression for a the potential due to a single point charge \(V= \frac{k_Cq}{r}\) where \(r\) is the distance from the point charge. You will also need to use the fact that the total potential is the sum of the potentials due to each individual point charge.

   It is important that we ask students first to create a function for the potential, and only then try to visualize the potential. This allows students to reason about the computation for a single point in space (defined in their choice of coordinate systems).

2. Once you have written the above function, use it to plot the electrostatic potential versus position along the three cartesian axes.

   Since the students have already written a function for their potential, they can create a plot by creating an array for \(x\) (or \(y\), or \(z\)), and then passing that array to their function, along with scalars for the other two coordinates. Many students will discover this simply by modifying an example script they find on the web, replacing \(\sin(x)\) or similar with their function. It is well worth showing this easier approach to students who attempt who attempt to write a loop in order to compute the potential at each point in space.

   We ask students to explicitly plot the potential along axes because students seldom spontaneously think to create a 1D plot such as this.

3. Label your axes.
4. Work out the first non-zero term in a power series approximation for the potential at large \(x\), small \(x\), etc. Plot these approximations along with your computed potential, and verify that they agree in the range that you expect. Useful 1\(^{st}\) order Taylor expansions are: \begin{eqnarray} \sqrt{1+\epsilon} &\sim& 1+\frac{\epsilon}{2} \\ \frac{1}{1+\epsilon} &=& 1-\epsilon \end{eqnarray} where \(\epsilon\) is a small quantity.

   This may need to be omitted on the first Tuesday of class, since students probably will not yet have seen power series approximations. It may work in this case to at least talk about what is expected at large distance, since "it looks like a point charge" is reasoning students do make.

   Students struggle with the \(x\) approximations (assuming the square is in the xy plane). Each pair will probably need to have a little lecture on grouping terms according to the power of \(x\), and keeping only those terms for which they have every instance.

Extra fun

Create one or more different visualizations of the electrostatic potential. For example a 2D representation in the \(z=0\) plane.

More extra fun

Create a plot of the potential along a straight line that is not one of the axes. Hint: start from a line on the \(z=0\) plane, then try a random straight line. You can use your browser for help.

Even more extra fun

Move the charges around (e.g., off the \(z=0\) plane) and see what happens to your graphs

Dipole fun

Repeat the above (especially the limiting cases!) for four point charges in which half are positive and half negative, with the positive charges neighbors.

Common visualizations for 2D slices of space include contour plots, color plots, and "3D plots". Another option (less easy) would be to visualize an equipotential surface in 3 dimensions. It is worth reminding students to consider other planes than those at \(x=0\), \(y=0\), and \(z=0\).

Quadrupole fun

Repeat the above (especially the limiting cases!) for four point charges in which half are positive and half negative, with the positive charges diagonal from one another. It will help in this case to place the charges on the axes (rotating the square by 45 degrees), since otherwise the potential on each axis will be zero.

Students solve numerically for the potential due to a spherical shell of charge. Although this potential is straightforward to compute using Gauss's Law, it serves as a nice example for numerically integrating in spherical coordinates because the correct answer is easy to recognize.

You have a system that consists of two identical (fair) six-sided dice. Imagine that you will perform an experiment where you roll the pair of dice together and record the observable: the norm of the difference between the values displayed by the two dice.

1. What are the possible results of the observable for each roll?

   
   

2. What is the theoretical probability of measuring each of those results? Assume the results are fair.

   Plot a probability histogram. Use your histogram to make a guess about where the average value is and the standard deviation.

   
   

3. Use your theoretical probabilities to determine a theoretical average value of the observable (the expectation value)? Indicate the expectation value on your histogram.

   
   

4. Use your theoretical probabilities to determine the standard deviation (the uncertainty) of the distribution of possible results. Indicate the uncertainty on your histogram.

   
   

5. Challenge: Use

   1. Dirac bra-ket notation
   2. matrices

   to represent:

   • the possible states of the dice after a measurement is made;

   • the state of the dice when you're shaking them up in your hand;

   • an operator that represents the norm of the difference of the dice.

1. The diagonal of the rectangle on the left below shows (a blown-up picture of) an infinitesimal displacement from the point (\(x\), \(y\)) to the nearby point
   (\(x+dx\), \(y+dy\)).

   • Label the rectangle with the lengths of the sides.

   • Express the sides of the rectangle indicated by arrows as vectors.

     Use the unit vectors \(\boldsymbol{\hat{x}}\) and \(\boldsymbol{\hat{y}}\).

   • The diagonal of this rectangle is the vector differential \(d\vec{r}\). Express \(d\vec{r}\) in terms of \(\boldsymbol{\hat{x}}\) and \(\boldsymbol{\hat{y}}\).
   • Find the length \(ds=|d\vec{r}|\) of the diagonal.
                        

2. The diagonal of the “rectangle” on the right above shows (a blown-up picture of) the same infinitesimal displacement, now expressed in polar coordinates, from the point (\(r\), \(\phi\)) to the nearby point (\(r+dr\), \(\phi+d\phi\)).
   • Label the rectangle with the lengths of the sides. Careful!
   • Express the sides of the rectangle indicated by arrows as vectors.
     Use the natural orthonormal basis defined by the picture, that is, let \(\hat{r}\) be the unit vector which points in the direction of increasing \(\vec{r}\), and let \(\hat{\phi}\) be the unit vector which points in the direction of increasing \(\phi\). Do not attempt to express these vectors in terms of \(\boldsymbol{\hat{x}}\) and \(\boldsymbol{\hat{y}}\)! You do not need to worry about the fact that some sides of the rectangle aren't straight; the rectangle is so small that this error is negligible.
   • The diagonal of this rectangle is again the vector differential \(d\boldsymbol{\vec{r}}\). Express \(d\boldsymbol{\vec{r}}\) in terms of \(\hat{r}\) and \(\hat{\phi}\)
   • Find the length \(ds=|d\vec{r}|\) of the diagonal.

Essentials

Main ideas

• Introduces \(d\boldsymbol{\vec{r}}\), the key to vector calculus, as a geometric object.

Don't skip this activity if you use nonrectangular basis vectors! ^*

Prerequisites

• Familiarity with \(\boldsymbol{\hat{r}}\) and \(\boldsymbol{\hat{\phi}}\). The Acceleration activity is a good introduction to those vectors.

Warmup

Draw a picture on the board showing \(d\boldsymbol{\vec{r}}\) as the infinitesimal change in the position vector \(\boldsymbol{\vec{r}}\) between two infinitesimally close points.

Props

• whiteboards and pens
• Big arrows, perhaps made of straws, which can represent an orthonormal basis, and which can be moved around a curve on the board.

Wrapup

• Emphasize that \(d\boldsymbol{\vec{r}}\) is the same geometric object regardless of how it is expressed.
• Discuss the geometry of \(ds\) as the magnitude of \(d\boldsymbol{\vec{r}}\), that is, \(ds=|d\boldsymbol{\vec{r}}|\).
• This is a good place to introduce the idea of “what sort of a beast is it”. The vector differential \(d\boldsymbol{\vec{r}}\) is an infinitesimal differential having both direction and (infinitesimal) length. When writing an expression for \(d\boldsymbol{\vec{r}}\), students should make sure that each term has these same properties.

Details

In the Classroom

Most groups will miss the factor of \(r\) in the \(\boldsymbol{\hat{\phi}}\) component of \(d\boldsymbol{\vec{r}}\). Watch for this as you walk around the classroom. A good thing to point out is that \(d\phi\) is not a length.

Some groups will then remember the formula for arclength and be able to figure out the rest on their own. Other groups will need to be reminded about the relationship between arclength and radius on a circle. A good way to do this is to ask them for the formula for the circumference of a circle, then half a circle, a quarter, etc. Make sure to give the angles in radians! Eventually, they get the point.

Some students may wonder whether the top of the (Cartesian) rectangle is \(\pm dx\,\boldsymbol{\hat{x}}\). This question is ill-posed, since the sign of \(dx\) itself depends on which way you're going; you can't change your mind in the middle of a problem. The safest way to resolve such problems is to anchor all vectors to the same point, as shown in the figures.

For the polar rectangle, many students will realize that that there are second-order differences between the two arcs, but few will realize that there are also second-order differences in the radial sides, due to changes in \(\boldsymbol{\hat{r}}\).

Plan to spend some extra time addressing the nature of \(d\vec{r}\). Basis vectors, arc length, dot product and magnitude; there's a great deal to take in and it's easy to lose sight of the forest for the trees. People will benefit from a deeper understanding at this stage.

Subsidiary ideas

• This is a good place to emphasize the relationship between the dot product and the Pythagorean Theorem.

Homework

• Have students determine \(d\boldsymbol{\vec{r}}\) in 3 dimensions in rectangular, cylindrical and spherical coordinates. (Spherical coordinates are tricky; most students miss the factor of \(\sin\theta\) in the \(\boldsymbol{\vec{\phi}}\) component.)
• Find \(d\boldsymbol{\vec{r}}\) along the diagonal of a square.

Enrichment

• Emphasize that \(d\boldsymbol{\vec{r}}\) is the concept which unifies most of vector calculus.
• It may be helpful to some students to be asked to orient the arrows (see Props) themselves at various points in the plane.

Students use chain rule diagrams to construct a multivariable chain rule in terms of differentials.

This is a small group activity for groups of 3-4. The students will be given one of 10 matrices. The students are then instructed to find the eigenvectors and eigenvalues for this matrix and record their calculations on their medium-sized whiteboards. In the class discussion that follows students report their finding and compare and contrast the properties of the eigenvalues and eigenvectors they find. Two topics that should specifically discussed are the case of repeated eigenvalues (degeneracy) and complex eigenvectors, e.g., in the case of some pure rotations, special properties of the eigenvectors and eigenvalues of hermitian matrices, common eigenvectors of commuting operators.

In this small group activity, students multiply a general 3x3 matrix with standard basis row/column vectors to pick out individual matrix elements. Students generate the expressions for the matrix elements in bra/ket notation.

Students calculate the flux from the vector field \(\vec{F} = C\, z\, \hat{z}\) through a right cone of height \(H\) and radius \(R\) .

The formula for the inverse Fourier transform shows that a function \(f(x)\) can be written in terms of its Fourier transform via \begin{equation} f(x)= \frac{1}{\sqrt{2\pi}} \int_{-\infty}^{\infty} \tilde{f}(k)\, e^{ikx}\, dk \end{equation} 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 \begin{equation} {\cal{F}}^{-1}(\tilde{f}) =f(x)= \frac{1}{\sqrt{2\pi}} \int_{-\infty}^{\infty} f(k)\, e^{ikx}\, dk \end{equation}

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.

Find the Fourier transform of the (simplified) Gaussian function \begin{equation} f(x)=e^{-x^2} \end{equation} You may want to use the value of the following integral \begin{equation} \int_{-\infty}^{\infty} e^{-x^2}\, dx = \sqrt{\pi} \end{equation}

Find the Fourier transform of a plane wave.

Instructor's Guide

Introduction

If students know about the Dirac delta function and its exponential representation, this is a great second example of the Fourier transform that students can work out in-class for themselves.

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

Student Conversations

Students may ask what is meant by a plane wave. Help them figure out what is meant, from the context or give them the formula if time is tight.

Keep the time dependence in or leave it out depending on how much time you have to deal with a little extra algebraic confusion.

Wrap-up

This example is (almost) the inverse of Fourier Transform of the Delta Function. If you really want the inverse problem, change the prompt to “Find the inverse Fourier transform of a plane wave.”

Suppose you have a definite function \(f(x)\) in mind and you already know its Fourier transform, i.e. you know how to do the integral \begin{equation} \tilde{f}(k)=\frac{1}{\sqrt{2\pi}} \int_{-\infty}^{\infty}e^{-ikx}\, f(x)\, dx \end{equation} Find the Fourier transform of the shifted function \(f(x-x_0)\).

Instructor's Guide

Introduction

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

Student Conversations

This example will feel very abstract to some students. It may be difficult for them to understand that the conditions of the problem state that the know both \(f(x)\) and \(\tilde{f}(k)\). This problem is about changing \(f\) slightly (by shifting its argument by \(x_0\)) and then asking how \(\tilde{f}\) changes, in response.

Wrap-up

The result from this calculation underlies why it is possible to factor out the time dependence in the Fourier transform of a plane wave, Fourier Transform of a Plane Wave. Even though the problem is somewhat abstract, it is super important in applications for this reason.

Students calculate the Fourier transform of the Dirac delta function.

This short small group activity introduces students to the Leibniz notation used for partial derivatives in thermodynamics, in which the variables being held constant are given explicitly. Students are guided to associate variables to their proper categories.

Students review using the Arms representation to represent states for discrete quantum systems and connecting the Arms representation to histogram and matrix representation. The student then extend the Arms representation to begin exploring the continuous position basis.