Math Help - Torus coordinate patch

1. Torus coordinate patch

Hi,

I'm working on a differential geometry question at the moment which seems to involve some horrific differentiation and I'm starting to wonder if I'm approaching it wrong. Just looking for advice?

Cover the torus with the coordinate patch

PHI(u,v)=((R+rcos)cosv,(R+rcosu)sinv,rsinu)
Show that the shape operater S on the torus is given by

S(PHI_u)=PHI_u/r

S(PHI_v)=(cosu/(R+rcosu))PHI_v

Right after much incorrect calculating I got nBar=(cosucosv,-cosusinv,-sinu)

I need to find the covariant derivative of the vector field ie (vBAR(X_1),vBar(X_2),vBAR(X_3) where vBAR(X_i) are directional derivatives. How do I do this? Where do I find directional derivatives? I thnk I need to choose a curve with a(o)=p and a'(0)=vBAR but how does one choose this?

2. seems to involve some horrific differentiation

Yeap, that's differential geometry alright.

If your calculations are OK, you should have

$S(\Phi_u)=-N_u, \ S(\Phi_v)=-N_v$.