1.let S={(x,y,z) in R^3 | x =y}, consider σ : U -> R^3 with U={(u,v) in R^2 | u>v} and σ(u,v) =(u+v,u+v,uv). Is σ a regular surface patch for P?

2. let S be the surface z=x^2-y^2. show that the following are regular surface patches for S and find the parts of S covered by the patches
a) σ(u,v) = (u+v,u-v,4uv), (u,v) in R^2
σ(u,v) = ucosh(v), usinh(v), u^2), (u,v) in R^2, u≠0

3.let c=(0,L) -> R^3 be a smooth curve parametrised by arclength and having nonzero curvature. Fix a>0. Consider the parametrised surface
σ(s,u) = c(s) + a((cosu)N(s) + (sinu)B(s)) where N(s) and B(s) are the normal and binormal vectors along c.

i)find the conditions under which σ is a regular parametrised surface.
ii) find a normal unit vector along the surface

need help with these questions. willing to pay fg whoever can help.