28 2. The First Variation (b) C1[a, b], the space of real-valued functions that are continuous and that have continuous derivatives on the closed interval [a, b] (c) C2[a, b], the space of real-valued functions that are continuous and that have continuous first and second derivatives on the closed interval [a, b] (d) D[a, b], the space of real-valued functions that are piecewise con- tinuous on the closed interval [a, b] and (e) D1[a, b], the space of real-valued functions that are continuous and that have piecewise continuous derivatives on the closed in- terval [a, b]. A piecewise continuous function can have a finite number of jump discontinuities in the interval [a, b]. The right-hand and left-hand limits of the function exist at the jump discontinuities. A function that is piecewise continuously differentiable is continuous but may have a finite number of corners. We wish to find the extremum of a functional. Extremum is a word that was first introduced by Paul du Bois-Reymond (1879b). Du Bois-Reymond got tired of always having to say “maximum or minimum” and so he introduced a single term, extremum, to talk about both maxima and minima. The term stuck. We will take our lead from (ordinary) calculus. We will look for a condition analogous to setting the first derivative equal to zero in calculus. The resulting Euler–Lagrange equation is quite important, so much so that we will derive this equation in three ways. We will begin with Euler’s heuristic derivation (Euler, 1744) and then move on to Lagrange’s 1755 derivation (the traditional approach). We will then consider Paul du Bois-Reymond’s modification of Lagrange’s derivation (du Bois-Reymond, 1879a). 2.2. Euler’s approach Leonhard Euler was the first person to systematize the study of vari- ational problems. His 1744 opus, A Method for Finding Curved Lines Enjoying Properties of Maximum or Minimum, or Solution of Isoperi- metric Problems in the Broadest Accepted Sense, is a compendium of 100 special problems. The book also contains a general method for

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