Coordinate Systems

Source lecture(s): PC604 Lec1-2

Intuition

Symmetry dictates which coordinates simplify a problem.

Formal Definition

A coordinate system assigns a unique triplet \((q_1,q_2,q_3)\) to each point. The Jacobian converts volume elements.

Mathematical Formulation

Cartesian: \(dV=dx\,dy\,dz\) Cylindrical: \(dV=r\,dr\,d\phi\,dz\) Spherical: \(dV=r^2\sin\theta\,dr\,d\theta\,d\phi\)

Derivation

In cylindrical coordinates \(\vec{r}=r\hat{e}_r+z\hat{e}_z\). The scale factors are \(h_r=1\), \(h_\phi=r\), \(h_z=1\), so \(dV=h_r h_\phi h_z\,dr\,d\phi\,dz=r\,dr\,d\phi\,dz\).

Worked Example

Sphere volume: \(V=\int_0^R r^2\,dr\int_0^\pi\sin\theta\,d\theta\int_0^{2\pi}d\phi=\frac{4}{3}\pi R^3\).

Common Mistakes

Forgetting the Jacobian when changing variables.
Swapping \(\theta\) and \(\phi\) in spherical coordinates.

Quiz

Q1. What is the Jacobian for cylindrical coordinates?

Answer

\(r\).

Q2. Which coordinate has \(\hat{e}_\phi\) depend on position?