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Activities

Problem

5 min.

##### Vectors

Let \begin{align} \boldsymbol{\vec a} &= \boldsymbol{\hat x}-3\boldsymbol{\hat y}-\boldsymbol{\hat z}\\ \boldsymbol{\vec b} &= \boldsymbol{\hat x}+\boldsymbol{\hat y}+2\boldsymbol{\hat z}\\ {\boldsymbol{\vec c}} &= -2\boldsymbol{\hat x}-\boldsymbol{\hat y}+\boldsymbol{\hat z}\\ \boldsymbol{\vec d} &= -\boldsymbol{\hat x}-\boldsymbol{\hat y}+\boldsymbol{\hat z} \end{align}

Use the dot product to determine which pairs (if any) of these vectors

1. Are perpendicular?
2. Are parallel?
3. Have an angle less than $\pi/2$ between them?
4. Have an angle of more than $\pi/2$ between them?

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

Small Group Activity

30 min.

##### Magnetic Vector Potential Due to a Spinning Charged Ring

Students work in small groups to use the superposition principle $\vec{A}(\vec{r}) =\frac{\mu_0}{4\pi}\int\frac{\vec{J}(\vec{r}^{\,\prime})}{\vert \vec{r}-\vec{r}^{\,\prime}\vert}\, d\tau^{\prime}$ to find an integral expression for the magnetic vector potential, $\vec{A}(\vec{r})$, due to a spinning ring of charge.

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

• Found in: Static Fields, AIMS Maxwell, Problem-Solving course(s) Found in: Power Series Sequence (E&M), E&M Ring Cycle Sequence sequence(s)

Problem

5 min.

##### Undo Formulas for Center of Mass (Geometry)

(Sketch limiting cases) Purpose: For two central force systems that share the same reduced mass system, discover how the motions of the original systems are the same and different.

The figure below shows the position vector $\vec r$ and the orbit of a “fictitious” reduced mass $\mu$.

1. Suppose $m_1=m_2$, Sketch the position vectors and orbits for $m_1$ and $m_2$ corresponding to $\vec{r}$. Describe a common physics example of central force motion for which $m_1=m_2$.
2. Repeat, for $m_2>m_1$.

• Found in: Central Forces course(s)

Small Group Activity

30 min.

##### Vector Differential--Curvilinear

In this small group activity, students are given a picture as a guide. They then write down an algebraic expression for the vector differential in different coordinate systems (cartesian, cylindrical, spherical).

Use Vector Differential--Rectangular as an introduction. This activity can be done simultaneously with Pineapples and Pumpkins where students or the instructor cut volume elements out of pineapples and/or pumpkins to show the geometry.

• Found in: Static Fields, AIMS Maxwell, Vector Calculus II, Surfaces/Bridge Workshop, Problem-Solving course(s) Found in: Integration Sequence sequence(s)

Small White Board Question

10 min.

##### Vector Differential--Rectangular

In this introductory lecture/SWBQ, students are given a picture as a guide. They then write down an algebraic expression for the vector differential in rectangular coordinates for coordinate equals constant paths.

This activity can be done as a mini-lecture/SWBQ as an introduction to Vector Differential--Curvilinear where students find the vector differential in cylindrical and spherical coordinates..

• Found in: Static Fields, AIMS Maxwell, Vector Calculus II, Surfaces/Bridge Workshop, Problem-Solving course(s) Found in: Integration Sequence sequence(s)

Problem

5 min.

##### Vector Sketch (Rectangular Coordinates)
Sketch each of the vector fields below.
1. $\boldsymbol{\vec F} =-y\,\boldsymbol{\hat x} + x\,\boldsymbol{\hat y}$
2. $\boldsymbol{\vec G} = x\,\boldsymbol{\hat x} + y\,\boldsymbol{\hat y}$
3. $\boldsymbol{\vec H} = y\,\boldsymbol{\hat x} + x\,\boldsymbol{\hat y}$
• Found in: AIMS Maxwell, Static Fields, Surfaces/Bridge Workshop, Problem-Solving course(s)

Problem

5 min.

##### Vector Sketch (Curvilinear Coordinates)
(4pts) Sketch each of the vector fields below.
1. $\boldsymbol{\vec K}=s\,\boldsymbol{\hat s}$
2. $\boldsymbol{\vec L}=\frac1s\boldsymbol{\hat\phi}$
3. $\boldsymbol{\vec M}=\sin\phi\,\boldsymbol{\hat s}$
4. $\boldsymbol{\vec N}=\sin(2\pi s)\,\boldsymbol{\hat\phi}$
• Found in: Static Fields, Surfaces/Bridge Workshop, Problem-Solving course(s)

Small Group Activity

30 min.

##### Murder Mystery Method
For each of the following vector fields, find a potential function if one exists, or argue that none exists.
• $\boldsymbol{\vec{F}} = (3x^2 + \tan y)\,\boldsymbol{\hat{x}} + (3y^2 + x\sec^2 y) \,\boldsymbol{\hat{y}}$
• $\boldsymbol{\vec{G}} = y\,\boldsymbol{\hat{x}} - x\,\boldsymbol{\hat{y}}$
• $\boldsymbol{\vec{H}} = (2xy + y^2 \sin z) \,\boldsymbol{\hat{x}} + (x^2 + z + 2xy\sin z) \,\boldsymbol{\hat{y}} + (y + z + xy^2 \cos z) \,\boldsymbol{\hat{z}}$
• $\boldsymbol{\vec{K}} = yz \,\boldsymbol{\hat{x}} + xz \,\boldsymbol{\hat{y}}$

#### Main ideas

• Finding potential functions.

Students love this activity. Some groups will finish in 10 minutes or less; few will require as much as 30 minutes. *

#### Prerequisites

• Fundamental Theorem for line integrals
• The Murder Mystery Method

none

#### Props

• whiteboards and pens

#### Wrapup

• Revisit integrating conservative vector fields along various paths, including reversing the orientation and integrating around closed paths.

### Details

#### In the Classroom

• We recommend having the students work in groups of 2 on this activity, and not having them turn anything in.
• Most students will treat the last example as 2-dimensional, giving the answer $xyz$. Ask these students to check their work by taking the gradient; most will include a $\boldsymbol{\hat{z}}$ term. Let them think this through. The correct answer of course depends on whether one assumes that $z$ is constant; we have deliberately left this ambiguous.
• It is good and proper that students want to add together multivariable terms. Keep returning to the gradient, something they know well. It is better to discover the guidelines themselves.

#### Subsidiary ideas

• 3-d vector fields do not necessarily have a $\boldsymbol{\hat{z}}$-component!

#### Homework

A challenging question to ponder is why a surface fails to exist for nonconservative fields. Using an example such as $y\,\boldsymbol{\hat{x}}+\boldsymbol{\hat{y}}$, prompt students to plot the field and examine its magnitude at various locations. Suggest piecing together level sets. There is serious geometry lurking that entails smoothness. Wrestling with this is healthy.

#### Essay questions

Write 3-5 sentences describing the connection between derivatives and integrals in the single-variable case. In other words, what is the one-dimensional version of MMM? Emphasize that much of vector calculus is generalizing concepts from single-variable theory.

#### Enrichment

• The derivative check for conservative vector fields can be described using the same type of diagrams as used in the Murder Mystery Method; this is just moving down the diagram (via differentiation) from the row containing the components of the vector field, rather than moving up (via integration). We believe this should not be mentioned until after this lab.

When done in 3-d, this makes a nice introduction to curl --- which however is not needed until one is ready to do Stokes' Theorem. We would therefore recommend delaying this entire discussion, including the 2-d case, until then.

• Work out the Murder Mystery Method using polar basis vectors, by reversing the process of taking the gradient in this basis.
• Revisit the example in the Ampère's Law lab, using the Fundamental Theorem to explain the results. This can be done without reference to a basis, but it is worth computing $\boldsymbol{\vec\nabla}\phi$ in a polar basis.
• Found in: Vector Calculus II course(s)

Small Group Activity

30 min.

##### Magnetic Field Due to a Spinning Ring of Charge

Students work in small groups to use the Biot-Savart law $\vec{B}(\vec{r}) =\frac{\mu_0}{4\pi}\int\frac{\vec{J}(\vec{r}^{\,\prime})\times \left(\vec{r}-\vec{r}^{\,\prime}\right)}{\vert \vec{r}-\vec{r}^{\,\prime}\vert^3} \, d\tau^{\prime}$ to find an integral expression for the magnetic field, $\vec{B}(\vec{r})$, due to a spinning ring of charge.

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

• Found in: Static Fields, AIMS Maxwell, Problem-Solving course(s) Found in: Power Series Sequence (E&M), E&M Ring Cycle Sequence sequence(s)

Kinesthetic

30 min.

##### The Distance Formula (Star Trek)
A short improvisational role-playing skit based on the Star Trek series in which students explore the definition and notation for position vectors, the importance of choosing an origin, and the geometric nature of the distance formula. $\vert\vec{r}-\vec{r}^\prime\vert=\sqrt{(x-x^\prime)^2+(y-y^\prime)^2-(z-z^\prime)^2}$
• Found in: Static Fields, AIMS Maxwell course(s) Found in: E&M Ring Cycle Sequence sequence(s)

Small Group Activity

60 min.

##### The Wire
Students compute a vector line integral, then investigate whether this integral is path independent.
• Found in: Vector Calculus II, Surfaces/Bridge Workshop, Problem-Solving course(s) Found in: Workshop Presentations 2023 sequence(s)

Small Group Activity

30 min.

##### Which Way is North?
Students construct two different rectangular coordinate systems and corresponding vector bases, then compare computations done with each.
• Found in: Vector Calculus II, Surfaces/Bridge Workshop course(s)

Small Group Activity

30 min.

##### Outer Product of a Vector on Itself
Students compute the outer product of a vector on itself to product a projection operator. Students discover that projection operators are idempotent (square to themselves) and that a complete set of outer products of an orthonormal basis is the identity (a completeness relation).
• Found in: Quantum Fundamentals course(s) Found in: Completeness Relations sequence(s)

Small Group Activity

30 min.

##### Vector Integrals (Contour Map)
Students explore path integrals using a vector field map and thinking about integration as chop-multiply-add.

Kinesthetic

10 min.

Students are shown a topographic map of an oval hill and imagine that the classroom is on the hill. They are asked to point in the direction of the gradient vector appropriate to the point on the hill where they are "standing".
• Found in: Static Fields, AIMS Maxwell, Surfaces/Bridge Workshop, Problem-Solving course(s) Found in: Geometry of Vector Fields Sequence, Gradient Sequence sequence(s)

Small Group Activity

5 min.

##### Acting Out Flux
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.
• Found in: AIMS Maxwell, Static Fields, Surfaces/Bridge Workshop, Problem-Solving course(s) Found in: Gauss/Ampere Sequence (Integral Form), Flux Sequence sequence(s)

Small Group Activity

30 min.

##### Vector Surface and Volume Elements

Students use known algebraic expressions for vector line elements $d\boldsymbol{\vec{r}}$ to determine all simple vector area $d\boldsymbol{\vec{A}}$ and volume elements $d\tau$ in cylindrical and spherical coordinates.

This activity is identical to Scalar Surface and Volume Elements except uses a vector approach to find directed surface and volume elements.

• Found in: AIMS Maxwell, Static Fields, Surfaces/Bridge Workshop, Problem-Solving course(s) Found in: Integration Sequence sequence(s)

Small Group Activity

30 min.

##### Number of Paths
Student discuss how many paths can be found on a map of the vector fields $\vec{F}$ for which the integral $\int \vec{F}\cdot d\vec{r}$ is positive, negative, or zero. $\vec{F}$ is conservative. They do a similar activity for the vector field $\vec{G}$ which is not conservative.

Small Group Activity

30 min.

##### Visualization of Divergence
Students predict from graphs of simple 2-d vector fields whether the divergence is positive, negative, or zero in various regions of the domain using the geometric definition of the divergence of a vector field at a point as flux per unit volume (here: area) through an infinitesimal box surrounding that point. Optionally, students can use computer algebra to verify their predictions.
• Found in: Static Fields, AIMS Maxwell, Vector Calculus II, Surfaces/Bridge Workshop, Problem-Solving course(s) Found in: Geometry of Vector Fields Sequence, Flux Sequence sequence(s)

Small Group Activity

30 min.

##### Visualization of Curl
Students predict from graphs of simple 2-d vector fields whether the curl is positive, negative, or zero in various regions of the domain using the definition of the curl of a vector field at a point as the maximum circulation per unit area through an infinitesimal box surrounding that point. Optionally, students can use computer algebra to verify their predictions.
• Found in: Static Fields, AIMS Maxwell, Vector Calculus II, Surfaces/Bridge Workshop, Problem-Solving course(s) Found in: Geometry of Vector Fields Sequence 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

10 min.

##### Velocity and Acceleration in Polar Coordinates
Use geometry to find formulas for velocity and acceleration in polar coordinates.
• Found in: Central Forces course(s)

Small Group Activity

60 min.

##### Visualizing Plane Waves

Each small group of 3-4 students is given a white board or piece of paper with a square grid of points on it.

Each group is given a different two-dimensional vector $\vec{k}$ and is asked to calculate the value of $\vec{k} \cdot \vec {r}$ for each point on the grid and to draw the set of points with constant value of $\vec{k} \cdot \vec{r}$ using rainbow colors to indicate increasing value.

Small Group Activity

30 min.

##### The Hillside
Students work in groups to measure the steepest slope and direction at a given point on a plastic surface and to compare their result with the gradient vector, obtained by measuring its components (the slopes in the coordinate directions).
• Found in: Vector Calculus I course(s) Found in: Gradient Sequence, Workshop Presentations 2023 sequence(s)

Small Group Activity

30 min.

##### Flux through a Cone
Students calculate the flux from the vector field $\vec{F} = C\, z\, \hat{z}$ through a right cone of height $H$ and radius $R$ .
• Found in: Static Fields, AIMS Maxwell course(s) Found in: Integration Sequence, Flux Sequence sequence(s)

Small Group Activity

30 min.

##### Working with Representations on the Ring
This activity acts as a reintroduction to doing quantum calculations while also introducing the matrix representation on the ring, allowing students to discover how to index and form a column vector representing the given quantum state. In addition, this activity introduces degenerate measurements on the quantum ring and examines the state after measuring both degenerate and non-degenerate eigenvalues for the state.
• Found in: Central Forces course(s)

Small Group Activity

60 min.

##### The Valley
Students compute vector line integrals and explore their properties.
• Found in: Vector Calculus II, Surfaces/Bridge Workshop course(s)

Small Group Activity

30 min.

##### The Hillside (Updated)
Students work in groups to measure the steepest slope and direction on a plastic surface, and to compare their result with the gradient vector, obtained by measuring its components (the slopes in the coordinate directions).
• Found in: Surfaces/Bridge Workshop, Problem-Solving course(s) Found in: Workshop Presentations 2023 sequence(s)

Small Group Activity

10 min.

##### Angular Momentum in Polar Coordinates
Students learn how to express Angular Momentum as a vector quantity in polar coordinates, and then in Cylindrical and Spherical Coordinates
• Found in: Central Forces course(s)