Consider f(u,v)=((a+bcosu)cosv,(a+bcosu)sinv,bsinu) What is the domain and target of f ? Show that f defines a regular parametrized surface. Find the tangent space and normal space at any f(u,v). Note that in this specific example, we have a tangent frame globally defined over the whole surface. This usually does not exist in general: even a nowhere-vanishing global vector field does not exist over the sphere - it has Euler characteristic two, meaning that any vector field vanishes at two points (counting multiplicity). Consider f(u,v)=((a+bcosu)cosv,(a+bcosu)sinv,bsinu) What is the domain and target of f ? Show that f defines a regular parametrized surface. Find the tangent space and normal space at any f(u,v). Note that in this specific example, we have a tangent frame globally defined over the whole surface. This usually does not exist in general: even a nowhere-vanishing global vector field does not exist over the sphere - it has Euler characteristic two, meaning that any vector field vanishes at two points (counting multiplicity)