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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.

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

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.