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Appendix: All Solutions to the Problems
Substituting Eq. (6) into Eq. (4), the best probability distribution density can be
obtained
p(x) =
1
2a
(8)
It can be seen that the best probability distribution density is a constant. The
information entropy of the best probability distribution density is
J [ p(x)] = −
a
−a
1
2a
ln
k
2a
dx = − ln
k
2a
(9)
5.13 Using spherical coordinates (r, ϕ, θ), find the geodesic line from point
A(R, 0, 0) to point B
0,
√
2
2
R,
√
2
2
R
on the sphere x
2
+ y
2
+ z
2
= R
2 .
Solution On the spherical coordinates, r = R is a constant, the derivative of it with
respect to the coordinate is zero. The differential of arc is
(ds)
2
= (Rdϕ)
2
+ (R sin ϕdθ)
2
The objective functional is
J [ϕ(θ)] =
θ 1
θ 0
R
ϕ 2 + sin
2
ϕdθ
Making the auxiliary functional
J
∗
=
θ 1
θ 0
[R
ϕ 2 + sin
2
ϕ + λ(r − R)]dθ
The Euler equation of the functional is
sin ϕ cos ϕ
ϕ 2 + sin
2
ϕ
−
d
dθ
ϕ
ϕ 2 + sin
2
ϕ
= 0
Expanding the above equation and simplifying, we obtain
sin ϕ cos ϕ sin
2
ϕ − ϕ
sin
2
ϕ + 2ϕ
2 sin ϕ cos ϕ
ϕ 2 + sin
2
ϕ(ϕ 2 + sin
2
ϕ)
= 0
Since ϕ and the denominator of the above equation are both not zero, there is
ϕ
sin ϕ − 2ϕ
2 cos ϕ − cos ϕ sin
2
ϕ = 0
Putting u = cot ϕ =
cos ϕ
sin ϕ
, then