Fourier series: Introduction
Fourier's law of heat conduction
Jean-Baptiste Joseph Fourier (1768-1830) was a versatile man, but his most memorable work lies in the field of mathematics and physics. In physics he especially researched heat conduction. He discovered the greenhouse effect, studied thermometers and central heating systems. The law of heat conduction is named after him. In one dimension, this law can be stated as follows: \[\frac{Q}{A}=-k\,\frac{\dd T}{\dd x}\] where \(T\) and \(Q\) are the temperature and the heat flow at a certain position \(x\), respectively , \(k\) is the coefficient of thermal conductivity, and \(A\) is the surface area of the cross-section at position \(x\) through which heat flows. We have deliberately written down the equation in this form because the left-hand side of the equation is then equal to the heat flux density \(q\), also abbreviated as heat flux, (in fluid dynamics, flux is a scientific name for the flow of a quantity per cross-sectional area through which flow occurs. The right-hand side of the equation is a scalar multiple of the negative temperature gradient. Heat flows from a high to a low temperature, that is, at a negative temperature gradient we have a positive heat flow density in the positive \(x\) direction because of the minus sign in the formula.
With this law one can also derive a partial differential equation, the so-called heat equation. In Fourier's time, there was no general method to solve this equation. But in his main work Théorie analytique de la chaleur Fourier elaborated on the bright idea to describe solutions through infinite sums of sines and cosines. This approach revolutionized the mathematics of the nineteenth and twentieth centuries and its applications. Fourier analysis has become one of the basic techniques in the theory of signals and systems, with applications such as medical imaging, and anyone who owns a smart phone makes indirectly use of results for which Fourier laid the foundation.
In this chapter we take a brief look at the theory of Fourier series and Fourier integrals.