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| (1768-1830) |
The history of the interaction of mathematics and physics has not just been one of unidirectional influence. Certainly the development of mathematics has been of great importance to physics, by providing it new and improved tools for modeling natural phenomena. Yet, physics has also offered inspiration and spur for development of new mathematical tools. The tale of Fourier’s Théorie analytique de la chaleur is of the latter sort.
Fourier’s starting point was the revolutionary use of mathematics in understanding nature, instigated by the works of Descartes and completed, in a sense, in the works of Isaac Newton. What they did was to extend the use of mathematics from mere tool for studying of figures into a tool for studying the motions of bodies. The success of Descartes and Newton inspired others to investigate whether mathematics could be useful in studying other natural phenomena.
One obvious candidate was the propagation of heat through a substance. Whatever heat was thought to be - often it was considered a distinct caloric substance that permeated all objects - it certainly appeared to “flow” through these objects, touching at first only one spot of the object in question and gradually spreading through the object and finding a point of equilibrium. The physical model was simple enough, all that was needed was to express the movement of the heat mathematically.
Fourier noted, firstly, that the movement of heat in an object was dependent on three things specific to the object, its constitution and its relation to its environment: the capacity of the object to assimilate heat (heat capacity), the capacity of its parts or molecules to transmit heat to one another (thermal conductivity) and the capacity of the environment to transmit heat to the object in question. In practice, we can limit our attention to the first two, because they form the basis of his theory of heat flow and the question of one object transmitting heat to another merely complexifies the basic theory. For simplicity’s sake, Fourier regarded these two quantities as simple constants, although the heating of an object might in reality affect them.
While physical objects are, of course, three-dimensional, we can first concentrate on the simple case of one-dimensional transfer of heat, e.g. within a barlike object. Clearly, the more distant a point in the object is from the source of the heat, the colder the point is and the opposite end from the source remains coldest. Now, Fourier supposed that the temperature of a point, at a given time, is in a sense proportional to the distance from the heat source. To put it more precisely, if at some time the temperature at the heat source is a and temperature at the opposite end of the bar is b and e is a given unit of temperature, then at a given point of the bar, with a distance z from the heat source, the temperature at that point can be calculated by subtracting z(a-b)/e from a.
Now, the temperatures a and b do not remain same, since heat is continuously flowing from one end of the bar to another and a keeps decreasing while b increases. To make the situation easier to handle, Fourier supposes that temperatures a and b are artificially kept constant, e.g. through an external heat source warming a. This means, he continues, that all the temperatures between the extremes of the bar also remain constant, that is, the temperature v always decreases while we move away from the heat source, at a rate (dv/dz) opposite to (a-b)/e. At the same time, heat is constantly flowing through the bar, and this flow, Fourier argues, is at least partially expressed by the formula (a-b)/e, that is, the greater the difference between the ends of the bar, the more heat flows from the warmer to the colder end.
If we forget the assumption of a and b being constant, we might say that (a-b)/e or −(dv/dz) partially represents the flow of heat characteristic of a bar of certain substance at a certain point of time. This expression cannot be the whole truth of the notion of flow of heat, because different substances have different capacities for conducting heat through them. Then again, he concludes, this expression together with the constant K describing the thermal conductivity of the substance describes completely the flow F of heat through a bar: F = K(a-b)/e or in terms of an infinitesimal change of temperature dv through an infinitesimally long length of bar dz, F = −K(dv/dz).
Next step in Fourier's argument is generalization of this formula to three-dimensional flow of heat. He begins by considering a prism, with one corner having the highest temperature A and heat flowing from this corner to all the other directions. Just like with the case of the bar, the further one goes from the source of hear, the less is the temperature, although now we have to account for three dimensions, when counting temperature at a given point at a given time - that is, the formula for counting the temperature of point (x, y, z) of prism looks something like A − ax − by − cz. Again, just like with the bar, by keeping the temperature constant at the limiting surfaces of the prism the temperature remains constant at all the points within the prism. By restricting then the investigation to heat flows in one dimension, heat flows in lines crossing planes perpendicular to x-, y- and z-axes will then be respectively −K(dv/dx), −K(dv/dy) and −K(dv/dz).
The final ingredient to be added to a general theory of heat is time. We again remove the assumption that the limits of a solid would have constant temperatures. Then the temperature of points within solid change as time goes on and heat flows in some manner through the solid, that is, temperature becomes a function of spatial coordinates and time: v = f(x, y, z, t). Fourier’s next move is to restrict the attention to flow through an infinitesimally small circle o at an infinitesimal instant dt, where due to extreme smallness the condition of the precious paragraph then apply. If we suppose the circle to be situated perpendicularly to z-axis, the heat flow going through it should be, Fourier says, −K(dv/dz)odt.
The notion of infinitesimals is, undoubtedly, unclear gibberish according to more modern understanding of differential and integral calculus, but even greater gibberish is to follow. Supposing the ring o to have an infinitesimal thickness, dz, we can distinguish between the heat flow coming within o and heat flow leaving o. The former is, expectedly, −K(dv/dz)odzdt, while the latter is almost the same, differing from this already infinitesimal quantity by “an even smaller infinitesimal”. The difference of the two quantities - that is, the amount of heat left within the ring after the instantaneous flow of heat - is just this type of “second-grade” infinitesimal, namely, K(d2v/dz2)odzdt. Here the expression (d2v/dz2) and its relation to (dv/dz) - rate of change of temperature, when moving through z-axis at moment dt - can be understood through an analogy with the relation of acceleration and velocity. In effect, (d2v/dz2) describes how the value of (dv/dz) itself changes, when moving through z-axis at moment dt.
Now, Fourier notes that the shape of o is not important and that we might as well take instead an infinitely small rectangle dxdy, making the flow through that rectangle, at instant dt, −Kdxdy(dz/dv)dt. Consider then an infinitely small cube of size dxdydz. The heat flow forming within that cube, at instant dt, is the sum of heat flows left within the cube, when heat flows coming in and going out from and to all three directions have been accounted for, namely K(d2v/dx2 + d2v/dy2 + d2v/dz2)dxdydzdt.
Now that the quantity of heat accumulating within an infinitesimal point of a solid at an infinitesimal instant of time has been, in a sense, determined, we can answer the question how does the temperature of that point develop over a period of time. It is not just a matter of dropping dt out from the formula, since heat and temperature are not completely same thing. Instead, we finally need the notion of heat capacity C of substance, which Fourier defines as the relation how much heat is required for increasing temperature of an object of certain weight. In order to get the required quantity of weight, we also need to take into account density D, that is, the relation how much certain volume of this substance weighs. By putting all these ingredients together, we find out that the rate of change of temperature over time, dv/dt, equals (K/(CD))(d2v/dx2 + d2v/dy2 + d2v/dz2).
What Fourier’s complex argument has provided is a position, where we can continue with purely mathematical methods. It is still unclear what the function f(x, y, z, t) determining the temperature of a point within solid at a certain time should be. We do know that the equation Fourier has found could correspond to infinitely many functions, but that certain additional conditions might be enough for determining the function. What really interests us is the method Fourier uses in solving the function from given conditions. To put it shortly, Fourier starts with an assumption that the function in question can be expressed in terms of simpler functions. To be more precise, he assumes that the function can be expressed as a sum of a possibly infinite series of trigonometric functions. The assumption happens to make sense in the context of heat transmission, because this physical process is not too erratic. In other words, the changes in the transmission can be approximately described with sums of cosines and sines. All Fourier then needs is a systematic method for determining these constituent functions, which is a simple enough task.
The idea of using sums of trigonometric functions as a way to determine heat functions was not completely novel. Yet, Fourier was the first person to assume that this method could be used in so extensive manner. While trying to solve a physical problem - how to describe movement of heat - he launched a completely new area of mathematical studies, the so-called Fourier series.
