x_{u} = ( -r cos v sin u, -r sin v sin u, r cos u)

x_{uu} = ( -r cos v cos u, -r sin v cos u, -r sin u)

x_{v} = ( -(a + r cos u) sin v, (a + r cos u) cos v, 0)

x_{vv} = ( -(a + r cos u) cos v, - (a + r cos u) sin v, 0)

x_{uv} = ( r sin u sin v, -r sin u cos v, 0)

E = [itex]x_u \cdot x_u [/itex]

F = [itex]x_u \cdot x_v [/itex]

G = [itex]x_v \cdot x_v [/itex]

l = [itex]x_{uu} \cdot n [/itex]

m = [itex]x_{uv} \cdot n [/itex]

n = [itex]x_{vv} \cdot n [/itex]

I'm not worried about the first fundamental forms. However, when I find the normal to use in computing the second fundamental forms, the equations are extremely cumbersome. Am I missing a simplification here or did I make an error in my derivation?