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.

The command used to plot the surface f in the default domain (−2π,2π)(−2𝜋,2𝜋).a.fsurff(f)b.esurf(f)c.surf(f)d.ezsurf(f)

he surface determined by the parametric equations x = z(cos u + u sin u), y = z(sin u − u cos u),0 ≤ u, z ≤ 1

Match the following surfaces to their correct definition as used in geometric modelingRuled SurfaceAnswer 1PlaneAnswer 2Answer 3

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

Let f (x, y, z) = sin(xy + z), and P be the point (0, −2, π3 ).(a) (6 pts) Compute ∇f (x, y, z).(b) (2 pts) At P , find the direction along which f obtains maximum directional derivative.(c) (4 pts) Calculate the directional derivative ∂f∂u (P ), where u is a unit vector making an angleπ6 with the gradient ∇f (P ).(d) (3 pts) The level surface f (x, y, z) = √32 defines z implicitly as a function of x and y near P .Compute ∂z∂x at P .