## Student handout: Curvilinear Coordinates Introduction

Static Fields 2022 (9 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.
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.

## 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}

• accessibility_new Curvilinear Basis Vectors

accessibility_new Kinesthetic

10 min.

##### Curvilinear Basis Vectors
Static Fields 2022 (8 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 2022 (8 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.

• assignment Magnetic Field and Current

assignment Homework

##### Magnetic Field and Current
Static Fields 2022 (3 years) 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
Static Fields 2022 (6 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 2022 (5 years)

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.
• group Vector Surface and Volume Elements

group Small Group Activity

30 min.

##### Vector Surface and Volume Elements
Static Fields 2022 (3 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 Sphere in Cylindrical Coordinates

assignment Homework

##### Sphere in Cylindrical Coordinates
Static Fields 2022 (3 years) Find the surface area of a sphere using cylindrical coordinates.
• assignment Cone Surface

assignment Homework

##### Cone Surface
Static Fields 2022 (5 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 Memorize $d\vec{r}$

assignment Homework

##### Memorize $d\vec{r}$
Static Fields 2022 (2 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}

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