## Calculus in a Nutshell: the Definite Integral of a Monovariate Function

The definite integral is the key tool in calculus for defining and calculating quantities important to mathematics and science, such as areas, volumes, lengths of curved paths, probabilities, and the weights of various objects, just to mention a few.

The idea behind the integral is that we can effectively compute such quantities by breaking them into small pieces and then summing the contributions from each piece.

## The Lissajous-Bowditch Curves

Try to glue a small mirror to an end of a bent piece of wire fixed to a stable platform and let the laser beam of a laser pointer reflect on it. Entangled spires of an ephemeral dragon of light will perform a hypnotic dance on the wall of your room. This voluptuous dance is the results of two mutually perpendicular harmonic oscillations produced by the oscillations of the elastic wire.

The curved patterns are called Lissajous-Bowditch figures and named after the French physicist Jules Antoine Lissajous who did a detailed study of them (published in his Mémoire sur l’étude optique des mouvements vibratoires, 1857). The American mathematician Nathaniel Bowditch (1773 – 1838) conducted earlier and independent studies on the same curves and for this reason, the figures are also called  Lissajous-Bowditch curves. Lissajous invented different mechanical devices consisting of two mirrors attached to two oriented diapasons (or other oscillators) by double reflecting a collimated ray of light on a screen, produce these figures upon oscillations of the diapasons.  The diapason can be substituted with elastic wires, speakers, pendulum or electronic circuits. I the last case, the light is the electron beam of a cathodic tube (or its digital equivalent)  of an oscilloscope. This blog is about these curves and shows demonstrations and applications.

## The Magic Imaginary Numbers

Complex numbers may appear a difficult subject given the name. However, there is nothing of really complicated about complex numbers. However, they definitively add a pinch of \em magic \em in the mathematics manipulations that you can do with them!

## Berechnung der Konstante von Madelung

Die gesamte Coulomb-Potentialenergie eines Kristalls ist die Summe der einzelnen Terme der elektrostatischen Potentialenergie

$\displaystyle V_{AB} = \frac{e^2}{4\pi\epsilon_0} \frac{Z_AZ_B}{r_{AB}} \hfill (1)$

zum Laden von Ionen  ${q_A}$ e ${q_B}$ und  getrennt nach Entfernung ${r_{AB}}$.

Die Summe erstreckt sich auf alle im Festkörper vorhandenen Ionenpaare für alle kristallinen Strukturen.

Die Summe konvergiert sehr langsam, weil die ersten Nachbarn des Zentralatoms einen substanziellen Beitrag zur Summe mit einem negativen Term liefern, während die benachbarten Sekunden nur mit einem etwas weicheren positiven Term beitragen, und so weiter. Auf diese Weise wird der Gesamteffekt sicherstellen, dass eine totale Initation der Anziehung zwischen Kationen und Anionen vorherrscht mit einem (negativen) Beitrag, der für die Gesamtenergie günstig ist.

## Frosinone and the Ciociaria

Glie musèu  Tra glie campanìle i l'Annunziàta,'Ntòcce prima della uìa ‘Ngelòne,Stamatìna sò fatte 'na fermàtaPè rattaccà le strengh'a strucenòne. Me uànne gli'òcchie doppe 'na utràtaI uède, tutt'a giòrne, nè salòneAddò 'na raccòta sta urdenàtaDe prete, crete, pièzze de matòne, Cule de uàse, de pile, de pignàte,Màneche de recciòle i de bicchièraAncòra prima de Nuè 'mpastàte. Quante sècule,frà, che sò passàte'Nche gli'Ome a trebbulà 'ncim'a sta TeraI ch'è remàste ? Ddù cocce smenuzzàte.               Irèno Da Vialìra (Poeta ciociaro)

I am European of Italian origin. I very proud of my background but, unfortunately, my carrier put me in an orbit that does intersect my country only during the vacation time. In these close encounters, my landing site is Frosinone. When not Italian acquaintance wants to know about my Heimat, most of them are puzzled about the name of my hometown Frosinone and its location. Usually, I help them to overcome the understandable impasse by giving as reference Rome and telling them that my birthplace is somewhere 80 km in the South of the Caput Mundi. Last year (2018) my hometown soccer team (Frosinone Calcio, nicknamed Canarini, The Canaries, for their home colours)) moved in the first league (A) of the national soccer championship. So, let see if this success will help to raise its notoriety! In this article, I won’t write about the success of the Canarini football team but about my hometown and the rural place in central Italy where it is located.

## Calculus in a Nutshell: functions and derivatives

When I was about thirteen, the library was going to get ‘Calculus for the Practical Man.’ By this time I knew, from reading the encyclopedia, that calculus was an important and interesting subject, and I ought to learn it.

Richard P. Feynman, from What Do You Care What Other People Think?

## Introduction

Calculus is an important branch of mathematics that deals with the methods for calculating derivatives and integrals of functions and using this information to study the properties of functions. It was independently invented by I. Newton and W. Leibniz in the 18${^{th}}$ century and it was further developed by other great mathematicians in the centuries that follows (see Figure below).

It comprises two areas:

• Differential calculus ${\rightarrow}$ It concerns the study of the rate of variation of functions.
• Integral calculus ${\rightarrow}$ It concern the study of the area under functions.

Depending on the nature of the functions involved in the calculations, we can further distinguish between the single- and multi-variable calculus. In this chapter, the main concepts and methods of the single-variable calculus are summarised.