Colloquium publications.

FUNCTIONALS AND THEIR APPLICATIONS.

17

b
z[x(t)], and we obtain the equation
a
b rb
dz = z[(t)] = [x(t), z t]6x(t)dt,
or
b
z'[x(t) I M] = f [x(t), z M],
a
in which z'[x(t) I M] denotes the functional derivative.of z.
In the total differential equation there are certain conditions of
integrability on the fi which must be satisfied in order to make
the fi possible partial derivatives of some function z(x...* x).
Similar conditions must therefore be expected for these new
equations. Their nature can be best described in terms of an
additional concept.
14. Adjoint Linear Functionals. Consider a closed curve C
and let E1[ulM] and E2[v M] be two linear functionals* of the
functional arguments u and v, defined on C; functionals which
depend also on the point argument M, on C. The functionals
E1 and E2 will be said to be adjoint (with respect to M or s) if for
every pair of functions u(M), v(M) in the field considered the
relation
(26) v(M)El[u I M]ds = u(M)E2[V lM]ds
is satisfied. The field of functions u, v will be that of all continuous functions, or continuous with their first k derivatives,
as the conditions of the problem demand. It follows from (26)
that the linear functionals must be homogeneous (i. e., E[ M]
0) if they are to have adjoints.
From (26) it follows immediately that no linear functional
adnrits more than one adjoint. If, in particular, El[u M] is
merely a differential expression in u with regard to the variable s,
on the curve, the equation (26) yields the well-known relations
between the coefficients of the given and adjoint expressions.t
* E[u] is a linear functional if E[clul + C2U2] = c1E[ul] + c2E[u2], where
c1 and C2 are arbitrary constants.
t If C is not a closed curve there will be involved relations among the end
values of u and its derivatives.