# Compute the first and second fundamental form

1. Sep 22, 2014

### Shackleford

xu = ( -r cos v sin u, -r sin v sin u, r cos u)

xuu = ( -r cos v cos u, -r sin v cos u, -r sin u)

xv = ( -(a + r cos u) sin v, (a + r cos u) cos v, 0)

xvv = ( -(a + r cos u) cos v, - (a + r cos u) sin v, 0)

xuv = ( r sin u sin v, -r sin u cos v, 0)

E = $x_u \cdot x_u$

F = $x_u \cdot x_v$

G = $x_v \cdot x_v$

l = $x_{uu} \cdot n$

m = $x_{uv} \cdot n$

n = $x_{vv} \cdot n$

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?

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