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Appendix: All Solutions to the Problems
The symmetric tensor of T i j is
S i j =
1
2
(T i j + T
c
i j ) =
⎡
⎣
2 2 1
2 3 2
1 2 5
⎤
⎦
The antisymmetric tensor of T i j is
A i j =
1
2
(T i j − T
c
i j ) =
⎡
⎣
0 2 −3
−2 0 0
3 0 0
⎤
⎦
Therefore T i j can be expressed as
T i j = S i j + A i j =
⎡
⎣
2 2 1
2 3 2
1 2 5
⎤
⎦ +
⎡
⎣
0 2 −3
−2 0 0
3 0 0
⎤
⎦
Chapter 2: Solutions to the Problems in Variational
Problems with Fixed Boundaries
2.1 Let the functions F = F(x, y, y
) ∈ C
1 , y = y(x) ∈ C
2 , find
(1) The differential d F; (2) The variation δ F.
Solution (1) d F = (F x + F y y
+ F y y
)dx; (2) δ F = F y δy + F y δy
.
2.2 Find the first variation of the following functions, where, a, b, c and d are all
constants.
(1) F = ax + by + cy
2
+ dy
2 ; (2) F = y
1 + y 2 ; (3) F =
a + by + cy 2 .
Solution
(1) δ F = bδy + 2cyδy + 2dy
δy
(2) δ F =
1 + y 2 δy + y
y
δy
√
1+y 2
(3) δ F = δ
a + by + cy 2 =
1
2
bδy
+2cy
δy
√
a+by +cy 2
2.3 Find the first variation of the following functionals
(1) J [y] =
x 1
x 0
(ay + by
+ cy
2
)dx; (2) J [y] =
x 1
x 0
y
2
1 + y 2 dx;
(3)
J [y]
=
x 1
x 0
(y
2
− y
2
− 2y cosh x)dx;
(4)
J [y]
=
x 1
x 0
(ax
2 y
+ bx y
2
+ c)dx;
(5) J [y] =
x 1
x 0
(x y + y
2
− 2y
2 y
)dx.
where, a, b, c are all constants.