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  • Print publication year: 2002
  • Online publication date: June 2012

9 - Differentials

Summary

This is a comparatively late stage to introduce the subject of differentials but there are good reasons for the delay. Readers will probably already have met the notation dx and dy in applied subjects and have been told that these represent ‘very small changes in x and y ’. There are even those who tell their students that dx and dy stand for ‘infinitesimally small changes in x and y ’. Such statements do not help very much in explaining why manipulations with differentials give correct answers. Indeed, in some contexts, such statements can be a positive hindrance. In this chapter we have tried to provide a more accurate account of the nature of differentials without attempting anything in the way of a systematic theoretical discussion. In studying this account, readers may find it necessary to put aside some of the preconceptions they perhaps have about differentials. Those who find this hard to do may take comfort in the fact that everything which is done in this book using differentials may also be done by means of techniques described in other chapters. For instance, three alternative methods of solving a problem are given in Example 1 before the method of differentials is used. However, this is not a sound reason for neglecting differentials. Their use in both applied and theoretical work is too widespread for this to be sensible.