Question

Problem 4 (20pts) Let $\Phi: D \subset S \subset \mathbb{R}^3$ be a parameterized surface given by $\Phi(u, v) = (x(u, v), y(u, v), z(u, v)).$ a) Show that $\|\partial_u \Phi \times \partial_v \Phi\| = \sqrt{\|\partial_u \Phi\|^2 \|\partial_v \Phi\|^2 - (\partial_u \Phi \cdot \partial_v \Phi)^2}$ b) Use this formula to compute the area of the helicoid $\Phi(u, v) = (u \cos(v), u \sin(v), v)$, $(u, v) \in [0, 1] \times [0, 2\pi]$.

          Problem 4 (20pts)
Let $\Phi: D \subset S \subset \mathbb{R}^3$ be a parameterized surface given by
$\Phi(u, v) = (x(u, v), y(u, v), z(u, v)).$
a) Show that
$\|\partial_u \Phi \times \partial_v \Phi\| = \sqrt{\|\partial_u \Phi\|^2 \|\partial_v \Phi\|^2 - (\partial_u \Phi \cdot \partial_v \Phi)^2}$
b) Use this formula to compute the area of the helicoid $\Phi(u, v) = (u \cos(v), u \sin(v), v)$, $(u, v) \in [0, 1] \times [0, 2\pi]$.

Problem 4 (20pts)
Let Φ: D ⊂ S ⊂ℝ^3 be a parameterized surface given by
Φ(u, v) = (x(u, v), y(u, v), z(u, v)).
a) Show that
Φ×Φ = √(Φ^2 Φ^2 - (Φ·Φ)^2)
b) Use this formula to compute the area of the helicoid Φ(u, v) = (u cos(v), u sin(v), v), (u, v) ∈ [0, 1] × [0, 2π].

Added by Ryan D.

Calculus: Early Transcendentals

James Stewart 8th Edition

Instant Answer

Solved by Expert Supreeta N

12/05/2023

Step 1

O is the position vector of the point (x,y,z) and O4 is the fourth partial derivative of D with respect to u. Show more…

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Problem 4 (20pts) Let : D -> S C R3 be a parameterized surface given by D(u,v)=(x(u,v),y(u,v),z(u,v)) a) Show that |O4xO|=V1|O|2|O|2-(O4O)2 b) Use this formula to compute the area of the helicoid (u,v) = (ucos(v),usin(),v) (u,v) E [0,1] x [0,2T].