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}


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