202
3 Sufficient Conditions of Extrema of Functionals
x
y
0
Fig. 3.1 Inherent curve field
x
y
0
Fig. 3.2 Non-inherent curve field
the other point of intersection in the domain D, then the family of curves y = y(x, c)
is called in D to form a central curve field.
Example 3.1.1 For the sine family of curves y = c sin x, according to the different
selection of regional D, which is divided into three conditions. Which forms an
inherent curve field in the domain D 1 =
(x, y)
0 < δ ≤ x ≤ a < π
−∞ < y < +∞
; Which
forms an central curve field taking the origin (0, 0) as the center in the domain
D 2 =
(x, y)
0 ≤ x ≤ a < π
−∞ < y < +∞
; Which does not form any curve field in the
domain D 3 =
(x, y)
0 ≤ x ≤ a 1 , a 1 > π
−∞ < y < +∞
, as shown in Fig. 3.3.
3 Sufficient Conditions of Extrema of Functionals
x
y
0
Fig. 3.1 Inherent curve field
x
y
0
Fig. 3.2 Non-inherent curve field
the other point of intersection in the domain D, then the family of curves y = y(x, c)
is called in D to form a central curve field.
Example 3.1.1 For the sine family of curves y = c sin x, according to the different
selection of regional D, which is divided into three conditions. Which forms an
inherent curve field in the domain D 1 =
(x, y)
0 < δ ≤ x ≤ a < π
−∞ < y < +∞
; Which
forms an central curve field taking the origin (0, 0) as the center in the domain
D 2 =
(x, y)
0 ≤ x ≤ a < π
−∞ < y < +∞
; Which does not form any curve field in the
domain D 3 =
(x, y)
0 ≤ x ≤ a 1 , a 1 > π
−∞ < y < +∞
, as shown in Fig. 3.3.