High School

Express the following parametric equations and their constraints clearly:

1. [tex]x = 3 \cos(u) \sin(v)[/tex]
2. [tex]y = 3 \sin(u) \sin(v)[/tex]
3. [tex]z = 3 \cos(v)[/tex]

Constraints:
- [tex]0 \leq u \leq 1.57[/tex]
- [tex]0 \leq v \leq 1.57[/tex]

Answer :

The given parametric equations define a portion of a sphere of radius 3 centered at the origin.

To see this, notice that the x, y, and z coordinates are given in terms of spherical coordinates with radius r=3. Specifically,

x = 3cos(u)sin(v) = rcos(u)sin(v)

y = 3sin(u)sin(v) = rsin(u)sin(v)

z = 3cos(v) = rcos(v)

where r = 3 is the fixed radius of the sphere.

The limits of the parameters u and v determine the portion of the sphere that is being described. The range 0 ≤ u ≤ 1.57 (or π/2 in radians) corresponds to a quarter of the sphere, specifically the part of the sphere in the first and second quadrants where x is positive. The range 0 ≤ v ≤ 1.57 corresponds to the upper half of the sphere, where z is positive.

In summary, the given equations describe a quarter of a sphere of radius 3 centered at the origin, in the first and second quadrants where x is positive, and in the upper hemisphere where z is positive.

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