94
2 Variational Problems with Fixed Boundaries
Fig. 2.3 The two curves
only with the zero-th order
approach degree
O
x
y
A
B
Fig. 2.4 The two curves
with the zero-th order and
first order approach degree
O
x
y
A
B
approach degree, moreover Fig. 2.4 is the two curves with the zero-th order and first
order approach degree. The continuation of a functional can be accurately defined
with the concept of approach degree.
Let the function y(x) ∈ F = C
n
[a, b], J [y(x)] is a functional of the domain F.
If for an arbitrary given positive number ε, δ > 0 can always be found, as long as
d n [y(x), y 0 (x)] < δ, i.e. y(x) ∈ N n [δ, y 0 (x)] ⊂ F
there is that
|J [y(x)] − J [y 0 (x)]| < ε
holds, then the functional J [y(x)] is called the continuous functional with n th
order δ approach degree at the function y 0 (x).