# Ann Varela: Joseph Fourier- Mathematician Par Excellence

An Interview with Ann Varela: Joseph Fourier- Mathematician Par Excellence

Michael F. Shaughnessy –

1) What do we know about his early life- and what propelled him into mathematics?

Jean-Baptiste-Joseph Fourier was born into a very large family in the year 1768 in the village of Auxerre, Burgundy, Kingdom of France. Joseph was the ninth of twelve children to a French tailor and his second wife. Tragically, by the time he was about nine or ten years old, both of his parents were deceased.

His earliest educational experience was at a local convent called Pallais upon recommendation of the local bishop in Auxerre. Although Fourier was a good student and showed talent in mathematics at a young age, he elected to prepare for the priesthood and began his studies with the Benedictines in 1787. Fourier had a change of plans in 1789 and decided to leave the religious life and pursue mathematics full time as a teacher at the Benedictine college, École Royale Militaire of Auxerre. Just a few years later, Fourier became involved in politics and affiliated himself with the local Revolutionary Committee. His hope was to launch a free government, liberated from kings and priests.

Fourier tried to extract himself from the Revolutionary Committee because he was displeased with the Terror resulting from the French Revolution. His efforts, however, were futile and his freedom, perhaps even his life, was in jeopardy because of his defense of certain factions while in Orléans. Fourier did indeed land himself in prison for his actions in Orléans. Fortunately, he did not go to the guillotine due to timely political changes.

2) What is the Fourier series and what does it have to do with heat?

A Fourier series is a way to express an intermittent (periodic) function as the sum of simple sine waves. In 1807, the purpose of the Fourier series was to solve the heat equation in a metal plate. As a result of his research, Fourier insightfully discovered that a trigonometric series can be used to represent a random function. In other words, Fourier’s series was comprised of the summation of other simple functions.

Fourier’s idea was to create a model of a complicated heat source as a linear mixture of simple sine and cosine waves, and to write the solution as a superposition of the corresponding simple solutions. A superposition for all linear systems states that the net response instigated by two or more inputs is the summation of the responses that would have been instigated by each input individually. For example, if input x_{1} produces response A and input x_{2} produces response B then input (x_{1}+x_{2}) produces response (A+B).

This superposition principle is observed in physics and engineering applications. Keep in mind that physical systems are roughly linear, so the superposition principle is only an estimated calculation of the actual physical behavior.

3) Although Fourier never called it the “greenhouse effect”, he seemed to have some theories about this. What can you tell us about this?

Fourier’s calculations concerning the temperature of the Earth based on solar radiation showed a discrepancy when compared with the expected temperature of the planet. Fourier proposed that the Earth’s raised temperature was warmer than expected due to interstellar radiation, after reading published articles relating to this subject. Fourier’s conclusion that there may be another explanation for the increased temperature of the Earth involved the Earth’s atmosphere acting as an insulator.

An experiment conducted by de Saussure, demonstrated how the atmosphere acts as in insulator. First, he coated a vessel with blackened cork to seal it. Next, several layers of glass were inserted, separated by pockets of air. Last, sunlight shone through the top of the vessel, penetrating each layer of glass. The temperature was recorded for each air pocket and it was confirmed how the temperature became higher in the innermost compartments of the vessel. One can see how the layers of glass are similar to modern greenhouses, thus the term “greenhouse effect“ is now utilized. Fourier later thought the elevation in temperature was due to convection (heat transfer due to molecular movement).

4) When Fourier spoke of “dimensional homogeneity in equations” what was he talking about and why is it important?

Dimensional homogeneity is one of the rudimentary rules of dimensional analysis. Dimensional homogeneity refers to physical quantities having identical dimensions, as only proportional quantities may be compared, equated, added, or subtracted.

It is possible to take ratios of dimensions that are not proportional and then either multiply or divide them. With this particular case, the factor-label method for converting units is frequently used. See Figure 1.

Figure 1. Factor-label Method.

Dimensional analysis has applications in mathematics, finance, economics, accounting, and fluid mechanics. One of the most common applications of dimensional analysis in operation is interest rates. Although interest rates tend to be expressed as percentages, they actually possess the dimensions of percent per year.

Fourier’s book on heat flow entitled, *The Analytical Theory of Heat*, explained an important physical contribution. That contribution was the theory of dimensional homogeneity in equations. In other words, an equation can be justifiably correct only if the dimensions match on both sides of the equality.

5) One of his equations is supposedly taught to every student of mathematical physics. What is it and why is it important?

Drum roll please…The answer is the partial differential equation for conductive diffusion of heat. The heat equation describes the circulation of heat in a given area over time.

Differential equations contain one or more derivatives (terms representing the rates of change of continuously varying quantities). In fields of quantitative study, it is possible to observe and measure fluctuations in a system. These fluctuations are rates of change. The solution to a differential equation is typically a function that can be used to predict the behavior of the original system, within specific restrictions.

There are two main categories of differential equations, ordinary and partial. Ordinary differential equations involve equations involving a single variable, while partial differential equations involve several independent variables.

Fourier’s heat equation describes a temperature function, which changes over time as heat dissipates throughout space. As time passes, the trend is for apexes to flatten and dips to fill in. If the function becomes linear, a steady-state is reached, meaning the function is unchanging.

The heat equation occurs when illustrating numerous events and is often used in financial mathematics in the modeling of options. Problems involving pressure diffusion, image analysis, and machine-learning also utilize the heat equation.

6) What is the Fourier transform?

The Fourier transform dissects a function of time, space or some other variable, into the individual frequencies (components) of which it is comprised. The Fourier Transform shows that any waveform can be rewritten as the sum of sinusoidal functions. One can see the component frequencies of the sines and cosines (indicated with apexes) sweeping across the frequency spectrum in Figure 2. Cell phone signals and TV signals are examples of waveforms that have a continuous range of frequencies.

Figure 2. Component Frequencies.

http://pgfplots.net/media/tikz/examples/PNG/fourier-transform.png

The mathematical process called Fourier transform analysis has been extensively used to help interpret X-ray crystallography diffraction data to obtain molecular structures of crystallized molecules, like proteins or DNA. Knowing the structures of molecules like protein or DNA, etc., allows their functions to be known and studied with meticulous detail. This structure and function knowledge permits the development of new medicines, biotechnology products, nanotechnology applications (like handling atoms and molecules), and diagnostic kits, for example.

7) What is the Fourier–Motzkin elimination?

The Fourier-Motzkin Elimination Method is a mathematical algorithm for eliminating variables from a system of linear inequalities. The goal is to eliminate all variables from the system, resulting in a system of constant inequalities, in order to determine if there is a solution for the original system. When one arrives at a constant inequality, one must determine if the resulting inequality is true or false. Inequalities ending up as true statements indicate that the original system, in fact, has solutions.

The Fourier-Motzkin Eliminating Method has applications in information theory and is used for solving linear programming problems.

8) I understand that he received many recognitions- and if one visits the Eiffel Tower in Paris, his name is somewhere indicated. What other recognitions did he receive?

Gustave Eiffel commemorated seventy-two prominent French scientists on plaques around the first stage of the Eiffel tower. Fourier is among the honorees. He became a member of the Academy of Sciences in 1817. In 1823 he was honored with his election into the Royal Society. In addition, Fourier joins fellow mathematicians, scientists, and physicists from this interview series, Archimedes, Babbage, Bernoulli, Boole, Cantor, and von Kármán, in having a lunar crater named after him. Joseph Fourier University is named after him. Finally, a bronze statue honoring Fourier was also raised in Auxerre, but it was melted down for weaponries during World War II.

9) What have I neglected to ask?

Did you know that Fourier was an Egyptologist? He accompanied Napoleon on an expedition to Egypt in 1798 and remained there for three years. His research and duties included Egyptian antiquities, engineering and diplomatic endeavors, and serving as secretary of the Institut d’Égypte in Cairo. The information gathered during this tenure in Egypt resulted in a publication entitled *Description de I’**É**gypte.*