Try to glue a small mirror to the 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 red dragon will perform a hypnotic dance on the wall of your room. This voluptuous dance results from 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. In the last case, the light is the electron beam of a cathodic tube (or its digital equivalent)  of an oscilloscope. This article is about the mathematical theory behind these curves introduced with a demonstration program and an example of education application proposed in 1827 by C. Wheatstone . Finally, we will give a look to the equivalent of the L-B curves in 3D by exploring the spherical L-B curves.

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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.

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La serie di Taylor è un utilissimo strumento matematico. In questo blog, ne darò una breve descrizione dando qualche esempio di applicazione.

Chi è il signor Taylor?

Brook Taylor (1685 – 1731) era un matematico britannico del XVII secolo che ha dimostrato la formula che porta il suo nome, e l’argomento di questo blog, nel volume Methodus Incrementorum Directa et Inversa (1715). Maggiori informazioni si possono trovare nella corrispondente pagina della wikipedia.

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