Created by Dong-Woo (Dom) Ko
Email: dk1713@ic.ac.uk
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VC: Volume Elements in different Coordinate Systems

Learning Objectives

• To aid in visualisation of volume elements in different coordinate systems of $I\!R^3$.

Table of Contents

1. Introduction
2. Cylindrical
3. Spherical

1. Introduction

We will explore volume elements in Cylindrical and Spherical Coordinates Systems. (In the form of Riley Diagrams)
This visualisation is designed to help you with triple integration.

A volume element is the differential element dV, and when integrated over specified range, the result will give you the volume of the solid.

For Cartesian,

Volume Element:

$$ dV = dx * dy * dz. $$

Volume integral:

\begin{align} V &= \iiint_V{dV}\\ &= \iiint_V \,dx\,dy\,dz \end{align}

Note:

• These volume elements are infinitesimally small. Hence consider them as a small cube.
• Different coordinates systems may be used depending on the shape of the solid.

2. Cylindrical

Volume Element, dV:

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

Riley Diagram:

3. Spherical

Volume Element, dV:

\begin{align} dV &= dr * rd\theta * r\sin\theta d\phi\\ &= r^2 \; \sin\theta \; dr \; d\theta \; d\phi. \end{align}

Riley Diagram: