## Activity: Curvilinear Coordinates Introduction

Static Fields 2023 (11 years)
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

## 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}

• 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.
• accessibility_new Curvilinear Basis Vectors

accessibility_new Kinesthetic

10 min.

##### Curvilinear Basis Vectors
Static Fields 2023 (10 years)

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).
• group Vector Differential--Curvilinear

group Small Group Activity

30 min.

##### Vector Differential--Curvilinear
Vector Calculus II 23 (11 years)

Integration Sequence

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.

• group Scalar Surface and Volume Elements

group Small Group Activity

30 min.

##### Scalar Surface and Volume Elements
Static Fields 2023 (7 years)

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
Static Fields 2023 (6 years)

Integration Sequence

In this small group activity, students integrate over non-uniform charge densities in cylindrical and spherical coordinates to calculate total charge.
• group Vector Surface and Volume Elements

group Small Group Activity

30 min.

##### Vector Surface and Volume Elements
Static Fields 2023 (4 years)

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 vector approach to find directed surface and volume elements.

• assignment Cone Surface

assignment Homework

##### Cone Surface
Static Fields 2023 (6 years)

• Find $dA$ on the surface of an (open) cone in both cylindrical and spherical coordinates. Hint: Be smart about how you coordinatize the cone.
• Using integration, find the surface area of an (open) cone with height $H$ and radius $R$. Do this problem in both cylindrical and spherical coordinates.

• assignment Sphere in Cylindrical Coordinates

assignment Homework

##### Sphere in Cylindrical Coordinates
Static Fields 2023 (4 years) Find the surface area of a sphere using cylindrical coordinates. Note: The fact that you can describe spheres nicely in cylindrical coordinates underlies the equal area cylindrical map project that allows you to draw maps of the earth where everything has the correct area, even if the shapes seem distorted. If you want to plot something like population density, you need an area preserving map projection.
• assignment Memorize $d\vec{r}$

assignment Homework

##### Memorize $d\vec{r}$
Static Fields 2023 (3 years)

Write $\vec{dr}$ in rectangular, cylindrical, and spherical coordinates.

1. Rectangular: \begin{equation} \vec{dr}= \end{equation}
2. Cylindrical: \begin{equation} \vec{dr}= \end{equation}
3. Spherical: \begin{equation} \vec{dr}= \end{equation}

• assignment Distance Formula in Curvilinear Coordinates

assignment Homework

##### Distance Formula in Curvilinear Coordinates

Ring Cycle Sequence

Static Fields 2023 (6 years)

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.

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