Colloquium publications.

LECTURE I
FUNCTIONALS, DERIVATIVES AND VARIATIONAL EQUATIONS*
~ 1. FUNDAMENTAL NOTIONS; FUNCTIONALS OF PLANE CURVES
1. Continuity of a Functional. If we have two sets of values
x and y, we say that y is a function of x if to every value of x in
its set corresponds a value of y. Similarly, if we have a set of
functions cp(x) given between limits a and b, and a set of values y,
we say that y is a functional of p(x) if to every sp(x) of the set
corresponds one of the values y. This relation we may write in
some such form as the following:
b
(1) y = y[(zx)]
a
It is an obvious generalization if instead of a functional of
o(x), we speak of a functional of a curve (in a plane, or in space),
or of a surface. Of special interest are the closed curves. The
area inside a closed plane curve, the Green's function of which
the point arguments are fixed, and the solutions of boundary
value problems in physics are ready examples of functionals of
curves or surfaces. The maximum of a function is a functional
of that function.
* This lecture is based upon the following references:
Volterra, Sopra le funzioni che dipendono da altre funzioni; Sopra le funzioni
dipendenti da linee, Rendiconti della Reale Accademia dei Lincei, vol. 3
(1887), five notes;
Lemons sur les fonctions de lignes, Paris (1913), chapters I, II, III.
Hadamard, Sur l'equilibre des plaques elastiques encastrees, M6moires presentees a l'Academie des Sciences, vol. 33.
P. Levy, Sur les equations integro-differentielles definissant des fonctions de
lignes, Theses presentees a la Faculte des Sciences de Paris, no. d'ordre
1436;
Sur les equations aux derivees fonctionelles et leur application a la
Physique mathematique, Rendiconti del Circolo Matematico di Palermo,
vol. 33 (1912);
Sur l'integration des equations aux derivees fonctionelles partielles,
Rendiconti del Circolo Matematico di Palermo, vol. 37 (1914).
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