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Q no. 3) let
f(t)= { e^-1/t^2 if t not equal to 0
{0. If t=0
You may assume f is C ^infinity with f^(n) =0 for all n. Let a(t) be given by
a(t) = {(t,f(t),0) if t<0
{(0,0,0) if t=0
{(t,0,f(t) if t>0
a) prove that a is regular and c^infinity .
b) show that k=0 at t=0
a consist of curves in two different planes joined together at a point where k=0.
Q no. 4) let a(s) be a unit speed curve
A) prove that the tangent spherical image of alpha (a) is a constant curve iff. a is straight line.
B) prove that the binormal spherical image of a is a constant curve iff. a is a plane curve.
C) prove that the normal spherical image of a is never constant.

Qno. 6 ) let a(s) be a unit speed curve k>0 . Let s star be arc length on the normal spherical image. Prove k= | ds star /ds| iff. a (alpha) is a plane curve.
if A = 5t2 + tj-t3k and B = sin t1-costj evaluate d/dt (AXB)
Prove that The transition maps of a smooth surface are smooth.
Prove that If γ(t)=σ(u(t),v(t)) is a unit speed curve on a surface patch σ, its normal curvature is given by k_n=Lu ̇^2+2M(u ) ̇v ̇+Nv ̇^2
Where L〖du〗^2+2Mdudv+N〖dv〗^2 is the second fundamental form of σ.
Q. Compute the normal curvature of the circle γ(t)=(cost, sint, 1) on the elliptic paraboloid
Q. Compute the first fundamental form and second fundamental form of the elliptical paraboloid σ(u,v)=(u, v,〖 u〗^2+v^2)
Q. Calculate the first fundamental forms of the following surfaces:
1) Sphere: σ(θ,φ) =(cosθcosφ, cosθsinφ,sinθ)
2) A generalized cylinder: σ(u,v) =γ(u)+Va
Q. How to cover a patch(patch of sphere)?
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