Students construct two different rectangular coordinate systems and corresponding vector bases, then compare computations done with each.

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. \[\vert\vec{r}-\vec{r}^\prime\vert=\sqrt{(x-x^\prime)^2+(y-y^\prime)^2-(z-z^\prime)^2}\]

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.

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..

Cylindrical Coordinates:

Find the general form for \(d\vec{r}\) in cylindrical coordinates by determining \(d\vec{r}\) along the specific paths below.

• Path 1 from \((s,\phi,z)\) to \((s+ds,\phi,z)\): \[d\vec{r}=\hspace{35em}\]
• Path 2 from \((s,\phi,z)\) to \((s,\phi,z+dz)\): \[d\vec{r}=\hspace{35em}\]
• Path 3 from \((s,\phi,z)\) to \((s,\phi+d\phi,z)\): \[d\vec{r}=\hspace{35em}\]

If all three coordinates are allowed to change simultaneously, by an infinitesimal amount, we could write this \(d\vec{r}\) for any path as:

\[d\vec{r}=\hspace{35em}\]

This is the general line element in cylindrical coordinates.

Spherical Coordinates:

Find the general form for \(d\vec{r}\) in spherical coordinates by determining \(d\vec{r}\) along the specific paths below.

• Path 1 from \((r,\theta,\phi)\) to \((r+dr,\theta,\phi)\): \[d\vec{r}=\hspace{35em}\]
• Path 2 from \((r,\theta,\phi)\) to \((r,\theta+d\theta,\phi)\): \[d\vec{r}=\hspace{35em}\]
• Path 3 from \((r,\theta,\phi)\) to \((r,\theta,\phi+d\phi)\): (Be careful, this is a tricky one!) \[d\vec{r}=\hspace{35em}\]

If all three coordinates are allowed to change simultaneously, by an infinitesimal amount, we could write this \(d\vec{r}\) for any path as:

\[d\vec{r}=\hspace{35em}\]

This is the general line element in spherical coordinates.

Instructor's Guide

Main Ideas

This activity allows students to derive formulas for \(d\vec{r}\) in cylindrical, and spherical coordinates, using purely geometric reasoning. These formulas form the basis of our unified view of all of vector calculus, so this activity is essential. For more information on this unified view, see our publications, especially: Using differentials to bridge the vector calculus gap

Students' Task

Using a picture as a guide, students write down an algebraic expression for the vector differential in different coordinate systems (cylindrical, spherical).

Introduction

Begin by drawing a curve (like a particle trajectory, but avoid "time" in the language) and an origin on the board. Show the position vector \(\vec{r}\) that points from the origin to a point on the curve and the position vector \(\vec{r}+d\vec{r}\) to a nearby point. Show the vector \(d\vec{r}\) and explain that it is tangent to the curve.

It may help to do activity Vector Differential--Rectangular as an introduction.

Student Conversations

For the case of cylindrical coordinates, students who are pattern-matching will write \(d\vec{r} = dr\, \hat{r} + d\phi\, \hat{\phi} + dz\, \hat{z}\). Point out that \(\phi\) is dimensionless and that path two is an arc with arclength \(r\, d\phi\).

Some students will remember the formula for arclength, but many will not. The following sequence of prompts can be helpful.

• What is the circumference of a circle?
• What is the arclength for a half circle?
• What is the arclength for the angle \(\pi\over 2\)?
• What is the arclength for the angle \(\phi\)?
• What is the arclength for the angle \(d\phi\)?

For the spherical case, students who are pattern matching will now write \(d\vec{r} = dr\, \hat{r} + d\phi\, \hat{\phi} + d\theta\, \hat{\theta}\). It helps to draw a picture in cross-section so that they can see that the circle whose arclength gives the coefficient of \(\hat{\theta}\) has radius \(r\sin\theta\). It can also help to carry around a basketball to write on to talk about the three dimensional geometry of this problem.

Wrap-up

The only wrap-up needed is to make sure that all students have (and understand the geometry of!) the correct formulas for \(d\vec{r}\).

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. \(\boldsymbol{\vec H} = y\,\boldsymbol{\hat x} + x\,\boldsymbol{\hat y}\)

None

(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. \(\boldsymbol{\vec N}=\sin(2\pi s)\,\boldsymbol{\hat\phi}\)

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

Warmup

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.

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.

Students compute a vector line integral, then investigate whether this integral is path independent.

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).

Students explore path integrals using a vector field map and thinking about integration as chop-multiply-add.

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".

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 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.

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.

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.

Students are asked to review:

• Addition of matrices
• Multiplication of a matrix by a scalar
• Matrix multiplication
• Finding the determinant of a matrix

in preparation for an in-class quiz.

In this small group activity, students draw components of a vector in Cartesian and polar bases. Students then write the components of the vector in these bases as both dot products with unit vectors and as bra/kets with basis bras.

• Divergence of a vector field (at a point) is the flux per unit volume through an infinitesimal box.
• How to predict the sign and relative magnitude of the divergence from graphs of a vector field.
• (Optional) How to calculate the divergence of a vector field with computer algebra.

• A component of the curl of a vector field (at a point) is the circulation per unit area around an infinitesimal loop.
• How to predict the sign and relative magnitude of the curl from graphs of a vector field.
• (Optional) How to calculate the curl of a vector field using computer algebra.

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.

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).

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

Students compute vector line integrals and explore their properties.

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).

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

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.