Differentiation

After a short presentation of what a derivative is and the rules of differentiation, differential calculus is illustrated by the modern version of Newton’s deduction of planetary motion from the physical postulates. In principle, high school mathematics suffices for this, but it turns out that a complete understanding of the things that happen in the mathematical model requires the mean value theorem and the theorem of differentiability of inverse functions. Short but complete proofs are given and now the text requires some background in college mathematics. Then follows a section about systems of ordinary differential equations. There are not many examples but this section is not difficult, and the material is very important for everyone who wants to understand and use mathematics. Those who went through the contraction theorem in Banach spaces of Chapter 4 get an existence and uniqueness proof which is certain to impress everyone who sees it for the first time. After this, the scene changes to differential calculus in several variables. An introductory section states the simplest properties of partial derivatives, Taylor’s formula, smooth bijections (here the contraction theorem is used again), and implicitly defined functions. Partial differential equations are treated in passing. Finally, there is a section on differential forms, followed by a section on differential calculus on manifolds.