The History of Mathematics: A Brief Course

(^478) 16. THE CALCULUS
discussed in Chapter 17, along with the development of real analysis in general. For
the rest of the present section, we confine our discussion to power-series techniques.
The heat equation
du _ d^2 u

m=adx^2

was the first partial differential equation to which the power-series method was

applied. Fourier used this method to produce the solution

r=0
when a = 1, without realizing that this solution "usually" diverges.
It was not until the nineteenth century that mathematicians began to worry
about the convergence of their series solutions. Then Cauchy and Weierstrass pro-
duced proofs that the series do converge for ordinary differential equations, provided
that the coefficients have convergent series representations. For partial differential
equations, it turned out that the form of the equation had some influence. Weier-
strass' student Sof'ya Kovalevskaya showed that in general the power series solution
for the heat equation diverges if the intial temperature distribution is prescribed,
even when that temperature is an analytic function of position. She showed, how-
ever, that the series converges if the temperature and temperature gradient at one
point are prescribed as analytic functions of time. More generally, she showed that
the power-series solution of any equation in "normal form" (solvable for a pure
derivative of order equal to the order of the equation) would converge.

3.3. Calculus of variations. The notion of function lies at the heart of calculus.

The usual picture of a function is of one point being mapped to another point.

However, the independent variable in a function can be a curve or surface as well

as a point. For example, given a curve 7 that is the graph of a function y — f(x)

between ÷ = a and ÷ = b, we can define its length as

Ë(7)= f yj\ + {f'{x))^2 dx.
J a
One of the important problems in the history of geometry has been to pick out the
curve 7 that minimizes Ë(7) and satisfies certain extra conditions, such as joining
two fixed points Ñ and Q on a surface or enclosing a fixed area A. The calculus
technique of "setting the derivative equal to zero" needs to be generalized for such
problems, and the techniques for doing so constitute the calculus of variations.
The history of this outgrowth of the calculus has been studied very thoroughly in
a number of classic works, such as Woodhouse (1810),^5 Todhunter (1861), and
Goldstine (1980), as well as many articles, such as Kreyszig (1993).

(^5) The treatise of Woodhouse is a textbook as much as a history, and its last chapter is a set of
29 examples posed as exercises for the reader with solutions provided. The book also marks an
important transition in British mathematics. Woodhouse says in the preface that, "In a former
Work, I adopted the foreign notation...". The foreign notation was the Leibniz notation for
differentials, in preference to the dot above the letter that Newton used to denote his fluxions.
He says that he found this notation even more necessary in calculus of variations, since he would
otherwise have had to adopt some new symbol for Lagrange's variation. But he then goes on to
marvel that Lagrange had taken the reverse step of introducing Newton's fluxion notation into
the calculus of variations.