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 $I\!R^3$.

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)

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}

## 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}