Created by Dong-Woo (Dom) Ko
Email: dk1713@ic.ac.uk
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VC: Plane Polar

Learning Objectives

• To aid in visualisation of basis vectors in plane polar.
• To aid in visualisation of area element in plane polar.

Table of Contents

1. Introduction
2. Basis Vectors
3. Area Element

1. Introduction

This is a brief summary of plane polar coordinate system. A reminder, before you move on to coordinate systems in $I\!R^3$.

This coordinate system comprises of (rho, phi):
   $\rho$ = radial coordinates ($\rho \geq 0$),
   $\phi$ = angular coordinates ($0 \leq \phi \leq 2\pi$),

In terms of cartesian coordinate (x, y):

\begin{align} \rho &= \sqrt{x^2 + y^2},\\ \phi &= \tan^{-1}\left(\frac{y}{x}\right). \end{align}

Cartesian Vector:

$$ {\bf{r}} \equiv \left(\begin{matrix} x \\ y \end{matrix}\right) = \left(\begin{matrix} \rho\cos\phi \\ \rho\sin\phi \end{matrix}\right) $$

2. Basis Vectors

$$ \widehat{\rho} \equiv \frac{\frac{d\bf{r}}{d\rho}}{\left|\frac{d\bf{r}}{d\rho}\right|} = \left(\begin{matrix} \cos\phi \\ \sin\phi \end{matrix}\right), \quad \widehat{\phi} \equiv \frac{\frac{d\bf{r}}{d\phi}}{\left|\frac{d\bf{r}}{d\phi}\right|} = \left(\begin{matrix} -\sin\phi \\ \cos\phi \end{matrix}\right). $$

Note: Unlike cartesian, orientation of basis vectors is not fixed.

3. Area Element

\begin{align} dA &= d\rho * \rho d\phi\\ &= \rho \; d\rho \; d\phi. \end{align}

Please note that area element is infinitesimally small and can be considered as a small square.

Diagram: