This is not a review of undergraduate mathematics or a distillation of the wisdom of many lecture courses into a few pages. Certainly, nobody should use it to understand new material. Mathematics is not learnt from crib-sheets and brief compendia but by careful study of definitions, theorems and – most importantly, perhaps – proofs, by elucidating the intuition behind ideas and grasping the interconnectedness between what might seem disparate concepts at first glance. There are no shortcuts and no cherry-tasting knowledge capsules to help you along your path …

A conscious attempt has been made throughout the volume not to take for granted any knowledge that an advanced mathematics undergraduate is unlikely to possess. If we need it, we explain it. However, every book has to start from somewhere.

Unless you have a basic knowledge of the first two years of university or college mathematics, this appendix will not help you and, indeed, this is the wrong book for you. However, it is not unusual for students to attend a lecture course, study material, absorb it, pass an exam with flying colours – and yet, a year or two later, a concept is perhaps not entirely forgotten but resides so deep in the recesses of memory that it cannot be used here and now. In these circumstances a virtuous reader consults another textbook or perhaps her lecture notes. A less virtuous reader usually means to do so – not just yet – but in the meantime plunges ahead with a decreased level of comprehension. […]