What's New in the Second Edition
The main change is that we treat Lebesgue integration rather
than improper integrals in Section 4.11. Other changes include:
- More exercises, more examples, and more figures, including
approximately 270 additional exercises and 50 additional examples.
- We have added pictures of mathematicians and more historical
- There are now end-of-section exercises, as well as review
exercises for Chapters 1--6.
- More difficult material from Chapter 0 was moved to the Appendix.
- Chapter 0 now includes a new section on functions; some material
from Section 1.3 was moved to this section. The section on complex
numbers was expanded.
- The inverse and implicit function theorems have been rewritten.
- In Chapter 3 we simplified the definition of a manifold,
and we now begin with the general case and discuss curves and
surfaces as examples. Similarly, in Chapter 5, we eliminated
the separate sections on arc length and surface area; we now
have one section on volume of manifolds.
- In Chapter 6, we rewrote the discussion of orientation and
changed the definition of a piece-with-boundary of a manifold,
to make it both simpler and more inclusive.
- Some useful formulas are listed on the back cover.