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