Show that the vectors tangent to the embedding coordinate curves are given by⃗er =cos φ
(α + cos (θ ))sin φ
(α + cos (θ ))sin (θ )⃗ eθ = r− sin (θ ) cos φ
− sin (θ ) sin φ
cos (θ ) and⃗ eφ = r− sin φ
(α + cos (θ ))cos φ
(α + cos (θ ))0 .Show all steps of the calculation

For the given position vectors r(t) compute the unit tangent vector T(t) for the given value of t A) let r(t)=<cos2t,sin2t> Then T(pi/4) <__,__>B) let r(t)=<t^2,t^3> Then T(1)=<_,_>c) let r(t)= e^2t i+ e^-t j+ t K Then T(2)= _i+_j+_k

Consider the surface F embedded in R3 such that each point on F is parameterized according to⃗p = αr⃗ p1 + r⃗ p2,where 0 < r is a fixed real number, 1 < α is a constant, and⃗p1 =cos φ
sin φ
0 and⃗ p2 =cos (θ ) cos φ
cos (θ ) sin φ
sin (θ )where θ ∈ [0, 2π) and φ ∈ [0, 2π).Answer the following questions.[1.1] Use Mathematica to plot F embedded in R3, showing the coordinate grid on F whereconstant θ and φ are marked in different colours. Include a large, high quality graphic andidentify the coloured lines with each of the embedding coordinates.

Consider the helix r(t) =<cos (5t),sin(5t),-1t>. Compute, at t=pi/6 :A. The unit tangent vector T=<___,___,___> , , B. The unit normal vector N=<___,___,___> , , C. The unit binormal vector B=<___,___,___>

Compute the curvature and torsion as the orthodrom having these coordinates:Departure: 20 degrees longitude and 14 degrees latitudeArrival: 30 degrees longitude and 22 degrees latitudeusing differential geometry. Solve this correctly and detailed

Consider the helix r(t) =<cos (5t),sin(5t),-1t>. Compute, at t=pi/6 :A. The unit tangent vector T=<-5/squareroot 101, -5 squareroot3/squareroot 101, -2/squareroot 101> B. The unit normal vector N=<squareroot 3/2, -1/2, 0>C. The unit binormal vector B=<___,___,___>