# Joseph Fourier

Joseph Fourier (March 21, 1768 - May 16, 1830) was a French mathematician and physicist noted for his investigations of heat flow. His work on this topic led him to develop Fourier series which are today a central tool in numerous branches of mathematics, physics, and related fields.

## Biography

Fourier was born into a poor family and orphaned at the age of 10. For his early advocacy of the French Revolution, he was rewarded with a seat at the École Normale Supérieure, a leading university. Fourier enlisted in Napoleon's army and fought in the Egyptian campaign, after which he was left as the governor of part of Egypt. During this time Fourier published a number of mathematical papers, which earned him enough recognition to be appointed secretary of the French Academy of Sciences. More importantly, Fourier began to carry out physical experiments on heat transfer which would form the basis for his later theoretical work.

The most important paper of Fourier was Théorie analytique de la chaleur (The Analytical Theory of Heat), published in 1822. This contents of this work are described in more detail in the following section.

Through his research, Fourier became convinced that heat possessed miraculous healing properties, and he was known to keep his body wrapped in numerous blankets at all times. It is also rumored that he kept his house heated to temperatures so hot that no others could stand them. Fourier died after falling down the stairs at his home, possibly the result of tripping on his blankets.

## Work

Fourier's most important work was centered on the dynamics of heat transfer. A foundational problem is the following: suppose that an infinitely long rod (roughly one-dimensional) is unevenly heated, and then allowed to cool off as heat transfers throughout the rod. Let u(x,t) denote the temperature of the rod at position x along the rod and time t: thus for any given time t0, the function of one variable u(x,t0) describes the distribution of heat at time t0. By considering an infinitesimal version of Newton's law of cooling, Fourier arrived at the now-famous heat equation, which says that the flow should be determined by the partial differential equation

ut = uxx, where the subscripts denote partial derivatives.

Fourier's key insight was that the function u(x,t), and indeed any periodic function, could be decomposed as an infinite linear combination of sines and cosines, so that

$u(x,t) = \sum_{n = 1}^\infty a_n(t) \sin(nx) + \sum_{n=0}^\infty b_n(t) \cos(nx)$.

This is the so-called Fourier expansion, which plays a foundational role in modern math and physics. Once the initial condition is expressed in this form, Fourier concluded that it was easy to solve for the functions an(t) and bn(t) and thus solve the heat equation. Today it is understood that Fourier's work was essentially correct, though deficient in certain technical details which could not be fixed until the advent of modern techniques of analysis.