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Differential Geometry | Topology
Prove that The transition maps of a smooth surface are smooth.
Differential Geometry | Topology
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.
Differential Geometry | Topology
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)
Differential Geometry | Topology
Q. Compute the first fundamental form and second fundamental form of the elliptical paraboloid σ(u,v)=(u, v,〖 u〗^2+v^2)
Differential Geometry | Topology
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
Differential Geometry | Topology
Q. How to cover a patch(patch of sphere)?
Differential Geometry | Topology
Define surface.
Differential Geometry | Topology
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)
Differential Geometry | Topology
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.
Differential Geometry | Topology
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.
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