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Q. Compute the normal curvature of the circle γ(t)=(cost, sint, 1) on the elliptic paraboloid

σ(u,v)=(u,v,u^2+v^2)

σ(u,v)=(u,v,u^2+v^2)

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

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)?

Define surface.

Q. The unit sphere S^2 defined by

σ(θ,φ) =(cosθcosφ,cosθsinφ,sinθ)

σ͂(θ,φ) =(-cosθcosφ,-sinθ,-cosθsinφ)

σ(θ,φ) =(cosθcosφ,cosθsinφ,sinθ)

σ͂(θ,φ) =(-cosθcosφ,-sinθ,-cosθsinφ)

Q. Show that the circular cylinder S={(x,y,z)∈R^3 |x^2+y^2=1} can be covered by a single surface patch and so a surface.

Q. The hyperboloid of one sheet is S={(x,y,z)∈R^3 |x^2+y^2-z^2=1} show that for every θ,the straight line (x-z)cosθ=(1-y)sinθ,

(x+z)sinθ=(1+y)cosθ

Is contained in S and that every point of hyperboloid lies on one of these deduce that S can be covered by a single surface patch, and hence is a surface.

(x+z)sinθ=(1+y)cosθ

Is contained in S and that every point of hyperboloid lies on one of these deduce that S can be covered by a single surface patch, and hence is a surface.

(x+z)sinθ=(1+y)cosθ

Is contained in S and that every point of hyperboloid lies on one of these deduce that S can be covered by a single surface patch, and hence is a surface.