## Chapter 5MULTIPLE INTEGRALS

‘One by one, or all at once.’

W. S. GILBERT, *The Yeomen of the Guard*

### 5·01. Distance; neighbourhoods; curves; regions.

In this chapter we shall be concerned with functions of two or more variables, say *x*, *y*, *z*. These will often be rectangular coordinates of a point in the ordinary sense, but not always. We shall generally take them to be two or three in number, but extensions to more will be obvious. Some simple ideas from Euclidean geometry can be extended usefully to the cases where the variables are not rectangular coordinates. One of the chief of these is *distance*. The distance *r* between two points given by rectangular coordinates (*x*, *y*, *z*) (*x*' *y*', *z*') is the non-negative value that satisfies

If the variables ...