1. Review

Fill in the table of coordinate transforms between spherical, rectangular, and cylindrical systems. Do this by extracting the relevant equations from the book and putting onto a single (nice) reference page either in the template below or your own reference system.

This will be handy for quick reference later.
To→
↓From		 cartesian cylindrical spherical





cartesian





cylindrical





spherical

2. Notes

TrigonometryTriangle2
Figure 1. Remember your trigonometry? [1]
\[\tan\theta = \dfrac{a}{b}\]
\[\tan^{-1} \left(\frac{a}{b}\right) = \theta\]

What is b when θ=+90°?

Inverse tangent has several issues to be aware of:

  • The denominator can be zero.

  • It can’t distinguish between e.g. \(\frac{+1}{+1}\) and \(\frac{-1}{-1}\).

  • It can’t distinguish betteen e.g. \(\frac{+1}{-1}\) and \(\frac{-1}{+1}\).

Python, Matlab, and other languages provide a 2-argument version that allows the function to detect the difference between opposite quadrants like I and III:

atan2(y, x) or atan2d(y, x)

This yields the 4-quadrant result with a 2π or 360° range.

1. Modified version from https://en.wikiversity.org/wiki/File:TrigonometryTriangle.svg