...= abc (sin v)^2 / A
f = -
...= (abc/A) sin u cos u sin v cos v - (abc/A) sin u cos u sin v cos v
...= 0
g = -
....= [abc (cos u)^2 (sin v)^2 + abc (sin u)^2 (sin v)^2 + abc (cos v)^2] / A
....= abc / A
Second fundamental form II(u,v) =
(e f)
(f g)
4. Computation of Gaussian curvature
(1) Determinant of first fundamental form
det I(u,v)
= EG - F^2
= c^2 (b^2 (cos u)^2 + a^2 (sin u)^2) (sin v)^4 + a^2 b^2 (cos u)^4 (cos v)^2 (sin v)^2
...+ a^2 b^2 (sin u)^4 (cos v)^2 (sin v)^2 + 2a^2 b^2 (cos u)^2 (sin u)^2 (cos v)^2 (sin v)^2
= c^2 (b^2 (cos u)^2 + a^2 (sin u)^2) (sin v)^4 + a^2 b^2 ((cos u)^2 + (sin u)^2)^2 (cos v)^2 (sin v)^2
= [c^2 (b^2 (cos u)^2 + a^2 (sin u)^2) (sin v)^2 + a^2 b^2 (cos v)^2] (sin v)^2
= A^2 (sin v)^2
(2) Determinant of second fundamental form
det II(u,v) = eg - f^2 = a^2 b^2 c^2 (sin v)^2 / A^2.
The Gaussian curvature K = det II(u,v) / det I(u,v) = a^2 b^2 c^2 / A^4, or,
K = a^2 b^2 c^2/[a^2 b^2 (cos v)^2 + c^2(a^2 (cos u)^2 + b^2 (sin u)^2) (sin v)^2]^2.