Danilo Roccatano

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.

I am European of Italian origin. I very proud of my origin but, unfortunately, my carrier put me in a orbit that does intersect my country only during the vacation time. In these close encounters, my landing site is Frosinone. When not Italian acquaintance want 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 comprehensible 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, nickenamed 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 blog, I won’t write about the success of the Canaries Canarini (although I am proud of that even if I am not such a soccer game fan) but about my hometown and the bucolic place in the central Italy where it is located.

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 century and it was further developed by other great mathematicians in the centuries that follows (see Figure below).

It comprises two areas:

• Differential calculus  It concerns the study of the rate of variation of functions.
• Integral calculus 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.