## Activity: Curvilinear Coordinates Introduction

AIMS Maxwell AIMS 21 Central Forces Spring 2021 Static Fields Winter 2021
First, students are shown diagrams of cylindrical and spherical coordinates. Common notation systems are discussed, especially that physicists and mathematicians use opposite conventions for the angles $\theta$ and $\phi$. Then students are asked to check their understanding by sketching several coordinate equals constant surfaces on their small whiteboards.
• This activity is used in the following sequences
What students learn
• The names and notations for variables in cylindrical $(s, \phi, z)$ and spherical $(r, \theta, \phi)$ coordinates;
• The differences between physicists' $(r, \theta, \phi)$ and mathematicians' $(r, \phi, \theta)$ notations for spherical coordinates;
• That specifying the value of a single coordinate in 3-d results in a 2-d surface;
• The range of values taken on by each of the coordinates in cylindrical and spherical coordinates.
• Media
• accessibility_new Curvilinear Basis Vectors

accessibility_new Kinesthetic

10 min.

##### Curvilinear Basis Vectors
AIMS Maxwell AIMS 21 Central Forces Spring 2021 Static Fields Winter 2021

Curvilinear Coordinate Sequence

Students use their arms to depict (sequentially) the different cylindrical and spherical basis vectors at the location of their shoulder (seen in relation to a specified origin of coordinates: either a set of axes hung from the ceiling of the room or perhaps a piece of furniture or a particular corner of the room).
• assignment Magnetic Field and Current

assignment Homework

##### Magnetic Field and Current
AIMS Maxwell AIMS 21 Consider the magnetic field $\vec{B}(s,\phi,z)= \begin{cases} 0&0\le s<a\\ \alpha \frac{1}{s}(s^4-a^4)\, \hat{\phi}&a<s<b\\ 0&s>b \end{cases}$
1. Use step and/or delta functions to write this magnetic field as a single expression valid everywhere in space.
2. Find a formula for the current density that creates this magnetic field.
3. Interpret your formula for the current density, i.e. explain briefly in words where the current is.
• group Scalar Surface and Volume Elements

group Small Group Activity

30 min.

##### Scalar Surface and Volume Elements
AIMS Maxwell AIMS 21 Static Fields Winter 2021

Integration Sequence

Students use known algebraic expressions for length elements $d\ell$ to determine all simple scalar area $dA$ and volume elements $d\tau$ in cylindrical and spherical coordinates.

This activity is identical to Vector Surface and Volume Elements except uses a scalar approach to find surface, and volume elements.

• group Total Charge

group Small Group Activity

30 min.

##### Total Charge
AIMS Maxwell AIMS 21 Static Fields Winter 2021

Integration Sequence

In this small group activity, students integrate over non-uniform charge densities in cylindrical and spherical coordinates to calculate total charge.
• format_list_numbered Curvilinear Coordinate Sequence

format_list_numbered Sequence

##### Curvilinear Coordinate Sequence
The curvilinear coordinate sequence introduces cylindrical and spherical coordinates (including inconsistencies between physicists' and mathematicians' notational conventions) and the basis vectors adapted to these coordinate systems.
• assignment Sphere in Cylindrical Coordinates

assignment Homework

##### Sphere in Cylindrical Coordinates
AIMS Maxwell AIMS 21 Find the surface area of a sphere using cylindrical coordinates.
• group Vector Surface and Volume Elements

group Small Group Activity

30 min.

##### Vector Surface and Volume Elements
AIMS Maxwell AIMS 21

Integration Sequence

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

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

• assignment Cone Surface

assignment Homework

##### Cone Surface
AIMS Maxwell AIMS 21 Static Fields Winter 2021

Using integration, find the surface area of a cone with height $H$ and radius $R$. Do this problem in both cylindrical and spherical coordinates.

• assignment Distance Formula in Curvilinear Coordinates

assignment Homework

##### Distance Formula in Curvilinear Coordinates

Ring Cycle Sequence

AIMS Maxwell AIMS 21 Static Fields Winter 2021

The distance $\left\vert\vec r -\vec r\,{}'\right\vert$ between the point $\vec r$ and the point $\vec r'$ is a coordinate-independent, physical and geometric quantity. But, in practice, you will need to know how to express this quantity in different coordinate systems.

1. Find the distance $\left\vert\vec r -\vec r\,{}'\right\vert$ between the point $\vec r$ and the point $\vec r'$ in rectangular coordinates.
2. Show that this same distance written in cylindrical coordinates is: \begin{equation} \left|\vec r -\vec r\,{}'\right| =\sqrt{s^2+s\,{}'^2-2ss\,{}'\cos(\phi\,{}'-\phi) +(z\,{}'-z)^2} \end{equation}
3. Show that this same distance written in spherical coordinates is: \begin{equation} \left\vert\vec r\,{}' -\vec r\right\vert =\sqrt{r\,{}'^2+r^2-2rr\,{}' \left[\sin\theta\sin\theta\,{}'\cos(\phi\,{}'-\phi) +\cos\theta\,{}'\cos\theta\right]} \end{equation}
4. Now assume that $\vec r\,{}'$ and $\vec r$ are in the $x$-$y$ plane. Simplify the previous two formulas.

• assignment The Gradient for a Point Charge

assignment Homework

##### The Gradient for a Point Charge
AIMS Maxwell AIMS 21 Static Fields Winter 2021

The electrostatic potential due to a point charge at the origin is given by: \begin{equation} V=\frac{1}{4\pi\epsilon_0} \frac{q}{r} \end{equation}

1. Find the electric field due to a point charge at the origin as a gradient in rectangular coordinates.
2. Find the electric field due to a point charge at the origin as a gradient in spherical coordinates.
3. Find the electric field due to a point charge at the origin as a gradient in cylindrical coordinates.

## Instructor's Guide

### Introduction

First, show students diagrams of cylindrical and spherical coordinates. Discuss common notation systems, especially that mathematicians and physicists use opposite notations for the angles $\theta$ and $\phi$. Don't forget to discuss the ranges of each of the coordinates.

It can be very helpful to have a set of coordinate axes, perhaps suspended from the ceiling somewhere in the room, to refer to as needed. Coordinate axes, both freestanding and hanging. Attach a string to the $z$-axis at the origin so that it can move freely and demonstrate, by moving the string appropriately, how the various angles in cylindrical and spherical coordinates change.

### Student Conversations

Ask students to check their understanding by sketching several coordinate equals constant surfaces on their small whiteboards. Appropriate figures are attached. Beware, this part of the activity can take longer than you expect; you can try to speed things up by limiting the coordinate equals constant sketches to $s$=constant and $\phi$=constant in cylindrical and $\theta$=constant in spherical.

• When sketching $s=$ constant in cylindrical coordinates, some students sketch a sphere.
• When sketching $\phi=$ constant in cylindrical (or spherical) coordinates, many students are puzzled about why the answer is a half-plane instead of a whole plane. This is a good opportunity to emphasize the ranges for the various variables.
• Cylindrical Coordinates: \begin{align} 0\le &s<\infty\\ 0\le &\phi <2\pi\\ -\infty < &z <\infty \end{align}
• Spherical Coordinates: \begin{align} 0\le &r<\infty\\ 0\le &\theta<\pi\\ 0\le &\phi <2\pi\\ \end{align}
• When sketching $\theta=$ constant in spherical coordinates, most students do not understand that the answer is a cone.

## Cylindrical Coordinates

For the cylindrical coordinate system shown below, draw three surfaces: one for constant $s$, one for constant $\phi$, and one for constant $z$. \begin{align} x&=s\cos\phi\\ y&=s\sin\phi\\ z&=z \end{align} \begin{align} 0\le &s<\infty\\ 0\le &\phi <2\pi\\ -\infty < &z <\infty \end{align}

## Spherical Coordinates

For the spherical coordinate system shown below, draw three surfaces: one for constant $r$, one for constant $\theta$, and one for constant $\phi$. \begin{align} x&=r\, \sin\theta\, \cos\phi\\ y&=r\, \sin\theta\, \sin\phi\\ z&=r\, \cos\theta \end{align} \begin{align} 0\le &r<\infty\\ 0\le &\theta<\pi\\ 0\le &\phi <2\pi\\ \end{align}

Author Information
Corinne Manogue, Tevian Dray, Ed Price
Keywords
Cylindrical coordinates spherical coordinates curvilinear coordinates
Learning Outcomes