Activities and Homework Problems

Student explore the properties of an orthonormal basis using the Cartesian and \(S_z\) bases as examples.

Students generate a list of properties a glass of water might have. The class then discusses and categorizes those properties.

In rectangular coordinates, the natural unit vectors are \(\{\boldsymbol{\hat x},\boldsymbol{\hat y}\}\), which point in the direction of increasing \(x\) and \(y\), respectively. Similarly, in polar coordinates the natural unit vectors are \(\boldsymbol{\hat r}\), which points in the direction of increasing \(r\), and \(\boldsymbol{\hat\phi}\), which points in the direction of increasing \(\phi\).

The unit tangent vector to a parametric curve is the unit vector tangent to the curve which points in the direction of increasing parameter. The principal unit normal vector to a parametric curve is the unit vector perpendicular to the curve “in the direction of bending”, which is the direction of the derivative of the unit tangent vector.

1. Consider the parametric curve \(\boldsymbol{\vec r} = 3\cos\phi\,\boldsymbol{\hat x} + 3\sin\phi\,\boldsymbol{\hat y}\) with \(\phi\in[0,2\pi]\). Calculate the unit tangent vector \(\boldsymbol{\hat T}\) and the principal unit normal vector \(\boldsymbol{\hat N}\) for this curve in terms of \(\boldsymbol{\hat x}\) and \(\boldsymbol{\hat y}\).
2. Consider a circle of radius \(3\) centered at the origin. Determine the unit tangent vector \(\boldsymbol{\hat T}\) and the principal unit normal vector \(\boldsymbol{\hat N}\) for this curve in terms of \(\boldsymbol{\hat r}\) and \(\boldsymbol{\hat\phi}\).
3. Compare your answers.

Instructor's Guide

Main ideas

• Geometric introduction of \(\boldsymbol{\hat r}\) and \(\boldsymbol{\hat\phi}\).
• Geometric introduction of unit tangent and normal vectors.

Prerequisites

• The position vector \(\vec{r}\).
• The derivative of the position vector is tangent to the curve.

Warmup

See the prerequisites. It is possible to briefly introduce these ideas immediately preceding this activity.

Props

• whiteboards and pens

Wrapup

Emphasize that \(\boldsymbol{\hat r}\) and \(\boldsymbol{\hat\phi}\) do not live at the origin! Encourage students to use the figure provided, which may help alleviate this confusion.

Point out to the students that \(\boldsymbol{\hat r}\) and \(\boldsymbol{\hat\phi}\) are defined everywhere (except at the origin), whereas \(\boldsymbol{\hat{T}}\) and \(\boldsymbol{\hat{N}}\) are properties of the curve. It is only on circles that these two notions coincide; \(\boldsymbol{\hat r}\) and \(\boldsymbol{\hat\phi}\) are adapted to round problems, and circles are round! Symmetry is important.

Emphasize that \(\{\boldsymbol{\hat r},\boldsymbol{\hat\phi}\}\) can be used as a basis (except at the origin). Point out to the students that their answer to the last problem gives them a formula expressing \(\boldsymbol{\hat r}\) and \(\boldsymbol{\hat\phi}\) in terms of \(\boldsymbol{\hat{x}}\) and \(\boldsymbol{\hat{y}}\). When comparing these basis vectors, they should all be drawn with their tails at the same point.

Details

We have had success helping students master the idea of “direction of bending” by describing the curve as part of a pickle jar; the principal unit normal vector points at the pickles!

In the Classroom

The easiest way to find \(\boldsymbol{\hat{N}}\) is to use the dot product to find vectors orthogonal to \(\boldsymbol{\hat{T}}\), then normalize. Students must then use the “direction of bending” criterion to choose between the two possible orientations.

Finding \(\boldsymbol{\hat{N}}\) in this way requires the student to give names to the its unknown components. This is a nontrivial skill; many students will have trouble with this.

It may be important to draw some examples. Despite that, students still feel wary of embracing \(\boldsymbol{\hat{r}}\) and \(\boldsymbol{\hat{\phi}}\). People will feel more comfortable over the next few classes but emphasize the geometry: the circle is still the circle and the unit tangent remains the same regardless.

Some students are natural geometors and will realize what the desired vectors are. This is terrific. Certainly, it is worthwhile to explicitly demonstrate this, but the point is the geometry can often do the work for you. This will convince students of the value of smart coordinates.

Subsidiary ideas

• Dividing any vector by its length yields a unit vector.
• Using the dot product to find vectors perpendicular to a given vector.

Homework

• Some students will not be comfortable unless they work out the components of \(\boldsymbol{\hat r}\) and \(\boldsymbol{\hat\phi}\) with respect to \(\boldsymbol{\hat{x}}\) and \(\boldsymbol{\hat{y}}\). Let them.

Enrichment

• What units does a unit vector have? Do \(\boldsymbol{\hat r}\) and \(\boldsymbol{\hat\phi}\) have the same units?

Students hold rulers and meter sticks to represent a vector field. The instructor holds a hula hoop to represent a small area element. Students are asked to describe the flux of the vector field through the area element.

Students learn about the geometric meaning of the amplitude and period parameters in the sine function. They also practice sketching the sum of two functions by hand.

Students learn how to express Angular Momentum as a vector quantity in polar coordinates, and then in Cylindrical and Spherical Coordinates

If the microscopic world was classical, predict \(U_{\text{classical}}(T)\) for the following “toy molecules” in the gas phase.

(a) (b) (c) (d)

• Each ball is a point mass \(m\) with no moment of inertia.
• The zig-zag lines are springs which are freely jointed at the balls.
• Vibrational motion of the springs is very small (\(\ll\) the length of the spring).
• The springs can extend and compress, but cannot twist or flex.
• The straight lines are rigid rods.

In this activity, students apply the Stefan-Boltzmann equation and the principle of energy balance in steady state to find the steady state temperature of a black object in near-Earth orbit.

Students solve for the equations of motion of a box sliding down (frictionlessly) a wedge, which itself slides on a horizontal surface, in order to answer the question "how much time does it take for the box to slide a distance \(d\) down the wedge?". This activities highlights finding kinetic energies when the coordinate system is not orthonormal and checking special cases, functional behavior, and dimensions.

Students are placed into small groups and asked to calculate the total differential of a function of two variables, each of which is in turn expressed in terms of two other variables.

This activity starts with a brief lecture introduction to power series and a short derivation of the formula for calculating the power series coefficients.

\[c_n={1\over n!}\, f^{(n)}(z_0)\]

Students use this formula to compute the power series coefficients for a \(\sin\theta\) (around both the origin and (if time allows) \(\frac{\pi}{6}\)). The meaning of these coefficients and the convergence behavior for each approximation is discussed in the whole-class wrap-up and in the follow-up activity: Visualization of Power Series Approximations.

This small group activity is designed to provide practice with the chain rule and to develop familiarity with polar coordinates. Students work in small groups to relate partial derivatives in rectangular and polar coordinates. The whole class wrap-up discussion emphasizes the importance of specifying what quantities are being held constant.

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.

Consider the region \(D\) in the \(xy\)-plane shown below, which is bounded by \[u=9 \qquad u=36 \qquad v=1 \qquad v=4\] where \[u=xy \qquad v={y\over x}\] If you want to determine \(x\) and \(y\) as functions of \(u\) and \(v\), consider \(uv\) and \(u/v\).

• List as many methods as you can think of for finding the area of the given region.

  It is enough to refer to the methods by name or describe them briefly.

• For at least 3 of these methods, give explicitly the formulas you would use to find the area.

  You must put limits on your integrals, but you do not need to evaluate them.

• Using any 2 of these methods, find the area.

  One of these should be a method we have learned recently.

• Now consider the following integral over the same region \(D\):

  \(\int\!\!\int_D {y\over x} \>\>dA\)

• Which of the above methods can you use to do this integral?
• Do the integral.

Main ideas

• There are many ways to solve this problem!
• Using Jacobians (and inverse Jacobians)

Prerequisites

• Surface integrals
• Jacobians
• Green's/Stokes' Theorem

Warmup

Perhaps a discussion of single and double integral techniques for solving this problem.

Props

• whiteboards and pens

Wrapup

This is a good conclusion to the course, as it reviews many integration techniques. We emphasize that (2-dimensional) change-of-variable problems are a special case of surface integrals.

Here are some of the methods one could use to do these integrals:

• change of variables (at least 2 ways)
• Area Corollary to Green's Theorem (at least 2 ways)
• ordinary single integral (at least 2 ways)
• ordinary double integral (at least 2 ways)
• surface integral

Details

In the Classroom

• Some students will want to simply use Jacobian formulas; encourage such students to try to solve this problem both by computing \(\frac{\partial(x,y)}{\partial(u,v)}\) and by computing \(\frac{\partial(u,v)}{\partial(x,y)}\).
• Other students will want to work directly with \(d\boldsymbol{\vec{r}}_1\) and \(d\boldsymbol{\vec{r}}_2\). This works fine if one first solves for \(x\) and \(y\) in terms of \(u\) and \(v\).

• Students who compute \(d\boldsymbol{\vec{r}}_1\) and \(d\boldsymbol{\vec{r}}_2\) directly can easily get confused, since they may try to eliminate \(x\) or \(y\), rather than \(u\) or \(v\).

  Along the curve \(v=\hbox{constant}\), one has \(dy=v\,dx\), so that \(d\boldsymbol{\vec{r}}_1 = dx\,\boldsymbol{\hat{x}} + dy\,\boldsymbol{\hat{y}} = (\boldsymbol{\hat{x}} + v\,\boldsymbol{\hat{y}})\,dx\), which some students will want to write in terms of \(x\) alone. But one needs to express this in terms of \(du\)! This can be done using \(du = x\,dy + y\,dx = x (v\,dx) + y\,dx = 2y\,dx\), so that \(d\boldsymbol{\vec{r}}_1 = (\boldsymbol{\hat{x}} + v\,\boldsymbol{\hat{y}}) \,\frac{du}{2y}\). A similar argument leads to \(d\boldsymbol{\vec{r}}_2 = (-\frac{1}{v}\,\boldsymbol{\hat{x}}+\boldsymbol{\hat{y}})\,\frac{x\,dv}{2}\) for \(u=\hbox{constant}\), so that \(d\boldsymbol{\vec{S}} = d\boldsymbol{\vec{r}}_1\times d\boldsymbol{\vec{r}}_2 = \boldsymbol{\hat{z}} \,\frac{x}{2y}\,du\,dv = \boldsymbol{\hat{z}} \,{du\,dv\over2v}\). This calculation can be done without solving for \(x\) and \(y\), provided one recognizes \(v\) in the penultimate expression.

  Emphasize that one must choose parameters, both on the region, and on each curve, and that \(u\) and \(v\) are chosen to make the limits easy.

• Take time before the activity to gauge students' recollection of single variable techniques and the Jacobian. After the activity, be sure to set up more than one approach. People will be fine after the first couple of steps but shouldn't leave class feeling stuck.

Subsidiary ideas

• Review of Green's Theorem
• Review of single integral techniques
• Review of double integral techniques

Enrichment

• Discuss the 3-dimensional case, perhaps relating it to volume integrals.

Students consider the change in internal energy during three different processes involving a container of water vapor on a stove. Using the 1st Law of Thermodynamics, students reason about how the internal energy would change and then compare this prediction with data from NIST presented as a contour plot.

Students work in small groups to use completeness relations to change the basis of quantum states.

Students use a plastic surface representing the potential due to a charged sphere to explore the electrostatic potential, equipotential lines, and the relationship between potential and electric field.

Students use a completeness relations to write hydrogen atoms states in the energy and position bases.

Students are asked to explore the parameters that affect orbit shape using the supplied Maple worksheet or Mathematica notebook.

Students are asked to draw lines of constant \(u\) and \(v\) in a \(u,v\) coordinate system. Then, in the same coordinate system, students must draw lines of constant \(x\) and constant \(y\) when

\[x(u,v)=u \] and \[y(u,v)=\frac{1}{2}u+3v. \]

Students consider how changing the volume of a system changes the internal energy of the system. Students use plastic graph models to explore these functions.

This small group activity is designed to help students visualize the cross product. Students work in small groups to determine the area of a triangle in space. The whole class wrap-up discussion emphasizes the geometric interpretation of the cross product.

Students construct the volume element in cylindrical and spherical coordinates.

In https://paradigms.oregonstate.edu/act/2525 you learned about an experiment in which rubidium atoms are dropped from a trap into an optical two-slit experiment. During this experiment the atoms fall a total of 1.5 meters. What is the de Broglie wavelength of an atom after falling from rest 1.5 m? \begin{align} \lambda &= \frac{2\pi\hbar}{p} \end{align}

In this small group activity, students determine various aspects of local points on an elliptic hill which is a function of two variables. The gradient is emphasized as a local quantity which points in the direction of greatest change at a point in the scalar field.

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.

This small whiteboard question (SWBQ) serves as a quick review of the dot product. It is also an opportunity to help students see the advantages of knowing many different representations of and facts about a physical concept.

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

Students work in small groups to use Coulomb's Law \[\vec{E}(\vec{r}) =\frac{1}{4\pi\epsilon_0}\int\frac{\rho(\vec{r}^{\,\prime})\left(\vec{r}-\vec{r}^{\,\prime}\right)}{\vert \vec{r}-\vec{r}^{\,\prime}\vert^3} \, d\tau^{\prime}\] to find an integral expression for the electric field, \(\vec{E}(\vec{r})\), everywhere in space, due to a ring of charge.

In an optional extension, students find a series expansion for \(\vec{E}(\vec{r})\) 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.

• Students need to understand that the surface represents the electric potential in the center of a parallel plate capacitor. Try doing the activity Electric Potential of Two Charged Plates before this activity.
• Students should know that
  1. objects with like charge repel and opposite charge attract,
  2. object tend to move toward lower energy configurations
  3. The potential energy of a charged particle is related to its charge: \(U=qV\)
  4. The force on a charged particle is related to its charge: \(\vec{F}=q\vec{E}\)

Students examine a plastic "surface" graph of the electric potential due to two charged plates (near the center of the plates) and explore the properties of the electric potential.

• 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.

Students calculate probabilities for a particle on a ring using three different notations: Dirac bra-ket, matrix, and wave function. After calculating the angular momentum and energy measurement probabilities, students compare their calculation methods for notation.

This lecture is one step in motivating the form of the Planck distribution.

Students are prompted to consider the scalar superposition of the electric potential due to multiple point charges. First a single point charge is discussed, then four positive charges, then an electric quadrupole. Students draw the equipotential curves in the plane of the charges, while also considering the 3D nature of equipotentials.

In this hour-long activity, students establish classroom norms for being respectful when working in small groups. This is particularly helpful in the first course a cohort of students encounters.

Students practice identifying whether events on spacetime diagrams are simultaneous, colocated, or neither for different observers. Then students decide which of two events occurs first in two different reference frames.

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.

Students calculate the expectation value of energy and angular momentum as a function of time for an initial state for a particle on a ring. This state is a linear combination of energy/angular momentum eigenstates written in bra-ket notation.

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.

Eigenvalues and Eigenvectors

Each group will be assigned one of the following matrices.

\[ A_1\doteq \begin{pmatrix} 0&-1\\ 1&0\\ \end{pmatrix} \hspace{2em} A_2\doteq \begin{pmatrix} 0&1\\ 1&0\\ \end{pmatrix} \hspace{2em} A_3\doteq \begin{pmatrix} -1&0\\ 0&-1\\ \end{pmatrix} \]
\[ A_4\doteq \begin{pmatrix} a&0\\ 0&d\\ \end{pmatrix} \hspace{2em} A_5\doteq \begin{pmatrix} 3&-i\\ i&3\\ \end{pmatrix} \hspace{2em} A_6\doteq \begin{pmatrix} 0&0\\ 0&1\\ \end{pmatrix} \hspace{2em} A_7\doteq \begin{pmatrix} 1&2\\ 1&2\\ \end{pmatrix} \]
\[ A_8\doteq \begin{pmatrix} -1&0&0\\ 0&-1&0\\ 0&0&-1\\ \end{pmatrix} \hspace{2em} A_9\doteq \begin{pmatrix} -1&0&0\\ 0&-1&0\\ 0&0&1\\ \end{pmatrix} \]
\[ S_x\doteq \frac{\hbar}{2}\begin{pmatrix} 0&1\\ 1&0\\ \end{pmatrix} \hspace{2em} S_y\doteq \frac{\hbar}{2}\begin{pmatrix} 0&-i\\ i&0\\ \end{pmatrix} \hspace{2em} S_z\doteq \frac{\hbar}{2}\begin{pmatrix} 1&0\\ 0&-1\\ \end{pmatrix} \]

For your matrix:

1. Find the eigenvalues.
2. Find the (unnormalized) eigenvectors.
3. Describe what this transformation does.
4. Normalize your eigenstates.

If you finish early, try another matrix with a different structure, i.e. real vs. complex entries, diagonal vs. non-diagonal, \(2\times 2\) vs. \(3\times 3\), with vs. without explicit dimensions.

Instructor's Guide

Main Ideas

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.

Students' Task

Introduction

Give a mini-lecture on how to calculate eigenvalues and eigenvectors. It is often easiest to do this with an example. We like to use the matrix \[A_7\doteq\begin{pmatrix}1&2\cr 9&4\cr\end{pmatrix}\] from the https://paradigms.oregonstate.edu/activities/2179 https://paradigms.oregonstate.edu/activities/2179 Finding Eigenvectors and Eigenvalues since the students have already seen this matrix and know what it's eigenvectors are. Then every group is given a handout, assigned a matrix, and then asked to: - Find the eigenvalues - Find the (unnormalized) eigenvectors - Normalize the eigenvectors - Describe what this transformation does

Student Conversations

• Typically, students can find the eigenvalues without too much problem. Eigenvectors are a different story. To find the eigenvectors, they will have two equations with two unknowns. They expect to be able to find a unique solution. But, since any scalar multiple of an eigenvector is also an eigenvector, their two equations will be redundant. Typically, they must choose any convenient value for one of the components (e.g. \(x=1\)) and solve for the other one. Later, they can use this scale freedom to normalize their vector.
• The examples in this activity were chosen to include many of the special cases that can trip students up. A common example is when the two equations for the eigenvector amount to something like \(x=x\) and \(y=-y\). For the first equation, they may need help to realize that \(x=\) “anything” is the solution. And for the second equation, sadly, many students need to be helped to the realization that the only solution is \(y=0\).

Wrap-up

The majority of the this activity is in the wrap-up conversation.

The [[whitepapers:narratives:eigenvectorslong|Eigenvalues and Eigenvectors Narrative]] provides a detailed narrative interpretation of this activity, focusing on the wrap-up conversation.

• Complex eigenvectors: connect to discussion of rotations in the Linear Transformations activity where there did not appear to be any vectors that stayed the same.
• Degeneracy: Define degeneracy as the case when there are repeated eigenvalues. Make sure the students see that, in the case of degeneracy, an entire subspace of vectors are all eigenvectors.
• Diagonal Matrices: Discuss that diagonal matrices are trivial. Their eigenvalues are just their diagonal elements and their eigenvectors are just the standard basis.
• Matrices with dimensions: Students should see from these examples that when you multiply a transformation by a real scalar, its eigenvalues are multiplied by that scalar and its eigenvectors are unchanges. If the scalar has dimensions (e.g. \(\hbar/2\)), then the eigenvalues have the same dimensions.

Commutation Relations for Spin Operators

A commutator of two observables is defined as:

\[[\hat{A},\hat{B}] = \hat{A}\hat{B} - \hat{B}\hat{A}\]

Determine the results of the following commutators:

1. \([\hat{S}_x,\hat{S}_y]\)
   
2. \([\hat{S}_y,\hat{S}_z]\)
   
3. \([\hat{S}_z,\hat{S}_x]\)
   
4. \([\hat{S}_y,\hat{S}_x]\)
   
5. \([\hat{S}_z,\hat{S}_y]\)
   
6. \([\hat{S}_x,\hat{S}_z]\)
   

Remember that the matrix representation of the spin operators written in the \(S_z\) basis is: \begin{align*} \hat{S}_x \doteq \frac{\hbar}{2}\begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix} \quad \hat{S}_y \doteq \frac{\hbar}{2}\begin{bmatrix} 0 & -i \\ i & 0 \end{bmatrix} \quad \hat{S}_z \doteq \frac{\hbar}{2}\begin{bmatrix} 1 & 0 \\ 0 & -1 \end{bmatrix} \end{align*}

Activity: Introduction

Divide students into groups to work out whether the spin operators commute.

Activity: Wrap-up

Groups should find that none of the quantum operators commute and therefore do not share the same basis for their respective eigenvectors. Because of this, it provides mathematical evidence for many properties that have so far been only observed. Since none of them commute, none of them have the same basis, nor can the spin operators be measured simultaneously.

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}