This 1971 book by Avner Friedman presents the calculus which every mathematics student should learn in the first 2 years of University. The theorems are very precisely stated and proved in detail, but there is also strong emphasis on computations for explicit functions. It is surprisingly advanced, although at first sight it has the look-and-feel of an old-fashioned run-of-the-mill introduction to early undergraduate calculus. There's no Lebesgue integral in it, but I think that's a very good thing!At first I thought the word "advanced" in the title was an exaggeration, as if it was made up by the marketing department. But whenever I consulted my dozen (or two dozen) books on calculus to try to find a particular theorem or definition, this book was surprisingly often the only book which had what I was looking for.On the subject of the basic Riemann integral (pages 108-115), Friedman makes a notational distinction between the Darboux and Riemann integrals, and shows their equivalence. Most authors gloss over this and confuse the two integrals, but showing their equivalence gives a deeper understanding of analysis. Using intuition to gloss over significant logical distinctions leads down the path to serious errors later. Friedman gives special attention to necessary and sufficient conditions for Riemann integral well-definition, namely that the points of discontinuity have measure zero. Despite the strong emphasis on proving integration rules before using them, there is a wide range of practical computational exercises.On the subject of multiple integrals (pages 255-265), the very interesting concept of the "Jordan content" is introduced as a condition for use in integration-related theorems. (Only a handful of authors even define this.) The Jordan content is a finite-cover version of the Lebesgue measure, which uses countable covers. As an example, page 262 has a theorem that a bounded function on a closed rectangle in R^2 is integrable if its set of discontinuities has content zero.On the subject of integration for curves (pages 298-314), Friedman defines reparametrisations of curves in much the same way that I do, but I don't think I've seen this way of doing it elsewhere.On the subject of the Lebesgue integral, which does not appear in this book, at first I thought this indicated that it was an elementary book. But over the last couple of years, I have completely turned against the Lebesgue integral because of its heavy reliance on the axiom of choice. This book by Friedman shows how to do integration without the Lebesgue integral. I know why so many people prefer 20th century Lebesgue integration to the traditional 19th century Cauchy-Riemann-Darboux-Stieltjes family of integrals, but the greater power of the Lebesgue integral is purchased at a heavy price, and in practical computations it is the traditional integrals which are always used. This book is thankfully free of the axiom of choice, which I call the "tooth fairy axiom" because it only gives you a concrete solution if you construct it yourself.