Compute the Gauss curvature of the ellipsoid

x^2/a^2 + y^2/a^2 + z^2/c^2 = 1

I know how to find K = ln-m^2/W but I don't know where to start with this problem in terms of parametrization.

Please help, thanks.

1. The parametrization

r(u,v) = (a cos u sin v, b sin u sin v, c cos v), 0 ≤ u ≤ 2π, 0 ≤ v ≤ π.

2. Computation of first fundamental form

(1) First partial derivatives of r(u,v)

r_u = (a sin u sin v, b cos u sin v, 0)
r_v = (-a cos u cos v, b sin u cos v, -c sin v)

(2) Coefficients of first fundamental form

E =
....= a^2 (sin u)^2 (sin v)^2 + b^2 (cos u)^2 (sin v)^2
....= [a^2 (sin u)^2 + b^2 (cos u)^2] (sin v)^2

F = ,
...= -a^2 sin u cos u sin v cos v + b^2 cos u sin u sin v cos v
...= (b^2 - a^2) sin u cos u sin v cos v

G =
....= a^2 (cos u)^2 (cos v)^2 + b^2 (sin u)^2 (cos v)^2 + c^2 (sin v)^2
....= [a^2 (cos u)^2 + b^2 (sin u)^2] (cos v)^2 + c^2 (sin v)^2

First fundamental form I(u,v) =

(E F)
(F G)

3. Computation of the second fundamental form

(1) Outward unit normal N(u,v)

r_u x r_v
= (-bc cos u (sin v)^2, ac sin u (sin v)^2, ab (sin u)^2 sin v cos v + ab (cos u)^2 sin v cos v)
= (-bc cos u (sin v)^2, ac sin u (sin v)^2, ab sin v cos v)

||r_u x r_v||
= √(b^2 c^2 (cos u)^2 (sin v)^4 + a^2 c^2 (sin u)^2 (sin v)^4 + a^2 b^2 (sin v)^2 (cos v)^2)
= sin v √(a^2 b^2 (cos v)^2 + c^2(b^2 (cos u)^2 + a^2 (sin u)^2) (sin v)^2)
= A sin v,

where A = √(a^2 b^2 (cos v)^2 + c^2(b^2 (cos u)^2 + a^2 (sin u)^2) (sin v)^2).

N = r_u x r_v/||r_u x r_v|| = (-(bc/A) cos u sin v, (ac/A) sin u sin v, (ab/A) cos v)

(2) Second partial derivatives of r(u,v)

r_uu = (a cos u sin v, -b sin u sin v, 0)
r_uv = (a sin u cos v, b cos u cos v, 0)
r_vv = (a cos u sin v, -b sin u sin v, -c cos v)

(3) Coefficients of second fundamental form

e = -