Colloquium publications.

FUNCTIONALS AND THEIR APPLICATIONS.

3

10(x) I < (, in the interval xo -h < x < xo + h, setting 0(x) = 0
rb
otherwise, and let us form the ratio Ay/a, where a = J 0(x)dx.
Jat
If this ratio approaches a limit as e and h approach zero in an
arbitrary manner, the limit is defined as the functional derivative
b
of y[kp(x)] at the point xo:
a
(2) y'[9(x) xIo] lim.
a e=0, h=O
A similar definition applies for the derivative of a functional
of a plane curve, as the accompanying diagram shows; the
derivative may be denoted by the symbol y'[C\ 11], or even y'(M),
( ~~~-D
FIG. 1
if there is no ambiguity. The point M denotes in this connection
a point on the curve C. The quantity ao is considered as positive
or negative according as it lies on the same side of the curve,
or not, as the positive normal (which we take on a closed curve
as directed towards the interior).*
3. Additive and Non-Additive Functionals of Plane Curves.
Let C1 and C2 be two closed plane curves exterior to each other
except for a common portion C', with directions such that C'
is traversed in opposite ways on the two curves; and let C3 be
the curve composed of C1 and C2 with the omission of C', the
* It is not desirable, for what follows, to consider any curves whose running
point co-ordinates are not functions of finite variation of a parameter t; in
fact we need consider only " standard " curves (see Art. 49). For simplicity of geometrical treatment, we thus make a distinction with respect to
generality when the argument of a functional is a curve, instead of a function
(see, for instance, Lecture III). The case where the curves have vertices
involves no special consideration.