Preface
Understanding what happens in the real life surrounding us, in phenomena involving physics, chemistry, biology and so on, is not an easy task. What we can do as humans is to come up with some machinery, and perform measurements, recording quantities such as length, volume, temperature, pressure and so on, and then see how these quantities, called “variables”, and denoted [math]x,y,z,\ldots[/math] depend on each other, and change in time.
Calculus is the study of the correspondences [math]x\to y[/math] between such variables. Such correspondences are called “functions”, and are denoted [math]y=f(x)[/math], with [math]f[/math] standing for the abstract machinery, or mathematical formula, producing [math]y[/math] out of [math]x[/math].
The basics of calculus were developed by Newton, Leibnitz and others, a long time ago. The idea is very simple. The simplest functions [math]f:\mathbb R\to\mathbb R[/math] are the linear ones, [math]f(x)=a+bx[/math] with [math]a,b\in\mathbb R[/math], but of course not any function is linear. Miraculously, however, most functions [math]f:\mathbb R\to\mathbb R[/math] are “locally linear”, in the sense that around any given point [math]c\in\mathbb R[/math], we have a formula of type [math]f(c+x)\simeq a+bx[/math], for [math]x[/math] small. Why? Obviously, [math]a\in\mathbb R[/math] can only be the value of our function at that point, [math]a=f(c)[/math]. As for the number [math]b\in\mathbb R[/math], this can be taken to be the rate of change of [math]f[/math] around that point, called derivative of the function at that point, and denoted [math]b=f'(c)[/math].
So, this was the main idea of calculus, “functions are locally linear”. This idea applies as well to more complicated functions, such as the “multivariable” ones [math]f:\mathbb R^N\to\mathbb R^M[/math], relating vector variables [math]x\in\mathbb R^N[/math] to vector variables [math]y\in\mathbb R^M[/math], with the linear approximation formula [math]f(c+x)\simeq a+bx[/math] needing this time as parameters a vector [math]a=f(c)\in\mathbb R^M[/math], and a linear map, or beast called rectangular matrix, [math]b=f'(c)\in M_{M\times N}(\mathbb R)[/math].
Further ideas of calculus, which are more advanced, include the facts that: (1) the remainder [math]\varepsilon(x)[/math] given by [math]f(c+x)=a+bx+\varepsilon(x)[/math] can studied by using again derivatives, (2) in several variables, the geometric understanding of the derivatives [math]f'(c)\in M_{M\times N}(\mathbb R)[/math] is best done by using complex numbers, (3) in fact, the use of complex numbers is useful even for one-variable functions [math]f:\mathbb R\to\mathbb R[/math], and (4) in one variable at least, there is a magic relation between derivatives and weighted averages, called integrals and denoted [math]\int_a^bf(x)dx[/math], the idea being that “the derivative of the integral is the function itself”.
Calculus can be learned from many places, with this being mostly a matter of taste. Personally as a student I read the books of Rudin [1], [2], and this was a very good investment, never had any trouble with calculus since, be that for research, or teaching. And these are still the books that I recommend to my students, although in the present modern age there are so many alternative resources, for having the basics learned.
The present book is an introduction to calculus, based on lecture notes from various classes that I taught at Cergy, and previously at Toulouse. The material inside claims of course no originality, basically going back to Newton, Leibnitz and others. But in what regards the presentation, there are a few ideas behind it, none of these claiming of course originality either, but their combination being something original, I hope:
(1) One complex variable comes before several real variables. This is perhaps not that standard, but as a quantum physicist, I just love complex numbers.
(2) Applications to probability everywhere, scattered throughout the book. Again, coming from experience with mathematics, physics, and science in general.
(3) Combinatorics, binomials and factorials all over the place, with joy. With this being a quite popular approach, who in mathematics does not love binomials.
(4) Applications to physics too, including even the hydrogen atom, at the end. In short, read this book, and you'll understand how hydrogen [math]_1{\rm H}[/math] works.
In the hope that you will like this book. High-school or undergraduate students, wishing to learn calculus in a quick way, graduate students in math and science, wishing to fine-tune their calculus knowledge, or just math professionals like me, wishing to have a compact analysis book, so that they can grab the appropriate chapter, before going to class, no matter what the class is about. Hope you will all find this useful.
As already mentioned, the present book is based on lecture notes from classes at Toulouse and Cergy, and I would like to thank my students. Many thanks go as well to my cats, for useful pieces of advice, often complementary to the pieces of advice of my colleagues, and for some help with the underlying PDE and physics.
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Cergy, September 2024
Teo Banica\ \baselineskip=15.95pt \tableofcontents \baselineskip=14pt
General references
Banica, Teo (2024). "Calculus and applications". arXiv:2401.00911 [math.CO].