Question

Let a∈R be a constant. The parametrization r(u,v)=⟨asin(u)cos(v),asin(u)sin(v),acos(u)⟩ for (u,v)∈[0,π/2]×[0,2π]

describes
Select one:
a. a cylinder with radius a


b. an ellipse with radius a


c. a circle with radius a


d. a cylinder with height a


e. a sphere with radius a


f. a hemisphere with radius a

Answer 1

The parametrization r(u,v)=⟨asin(u)cos(v),asin(u)sin(v),acos(u)⟩ is a spherical parametrization:

$x=a sin (u)cos (v), y=asin(u)sin(v), z=acos(u).$

Here u is the angle between the positive z-axis and the line segment from the origin to P, and v is the angle between the positive x-axis and the line segment from the origin to the Q (projection of P to the xy-plane). In spherical coordinates we have such restrictions: $0\le u \le \pi, 0\le v \le 2\pi.$

As $(u,v)∈[0,π/2]×[0,2π]$, then we consider only an upper hemisphere with radius a.

The answer: f. a hemisphere with radius a.