### Introduction

We have seen how changing the variable of integration of a single integral or changing the

coordinate system for multiple integrals can make integrals easier to evaluate. In this Section we introduce the Jacobian. The Jacobian gives a general method for transforming the coordinates of any multiple integral.

#### Prerequisites

- have a thorough understanding of the various techniques of integration
- be familiar with the concept of a function of several variables
- be able to evaluate the determinant of a matrix

#### Learning Outcomes

- decide which coordinate transformation simplifies an integral
- determine the Jacobian for a coordinate transformation
- evaluate multiple integrals using a transformation of coordinates

#### Contents

1 Changing variables in multiple integrals2 The Jacobian

3 The Jacobian in 3 dimensions

4 Engineering Example 3

4.1 Volume of liquid in an ellipsoidal tank