Tangent spaces and the orientation of parametrized surfaces.

LetSbe the surface given by the cartesian equation

y : 6352+:2 ' (a) Let P be a point on ,5'. As we saw in class, a normal vector to 5' at the point P is given by Vf(P), where $2+z2 _ f($,y,z)=e y is the equation dening 3. Use this to nd the cartesian equation for the tangent plane to S at the point (1, e, 0). (b) A parametrization for S is the smooth map q; ; R2 _) R3 given by 2 2 @(uw) = (u, e\" +\" ,v). Like any graph parametrization, you can check that (I) is regular in the sense that (Eu (u, v) and '1)\" (u, v) are linearly independent for all (u, v) E R2.As we saw in class, we have an associated smooth nowherevanishing normal eld N: S > R3 given by N((u1v)) =