Sketching solids given spherical coordinate inequalities

[yango_17]

Homework Statement

Sketch the solid whose spherical coordinates (ρ, φ, θ):
0≤ρ≤1, 0≤φ≤(pi/2)

Homework Equations

The Attempt at a Solution

I was thinking that since ρ represented the distance from the point of the origin and φ represented the angle between the positive z-axis and the ray through the origin and any point, that the surface represented by such an inequality would be something like the top hemisphere of a 3d sphere. Any feedback would be appreciated.

 

[Mark44]

Homework Statement

Sketch the solid whose spherical coordinates (ρ, φ, θ):
0≤ρ≤1, 0≤φ≤(pi/2)

Homework Equations

The Attempt at a Solution

I was thinking that since ρ represented the distance from the point of the origin and φ represented the angle between the positive z-axis and the ray through the origin and any point, that the surface represented by such an inequality would be something like the top hemisphere of a 3d sphere. Any feedback would be appreciated.

So far, so good. For a better description, what is the radius of the hemisphere, and where is its center (i.e., the center of the sphere that the hemisphere is half of)?

For extra credit, what does it mean that θ doesn't appear in the inequalities?

 

[yango_17]

The radius of the hemisphere would be 1, and the center would be on the z-axis. I'm not entirely sure what the absence of θ signifies.

 

[Mark44]

The radius of the hemisphere would be 1, and the center would be on the z-axis. I'm not entirely sure what the absence of θ signifies.

The center is at a particular point on the z-axis. Care to guess which one?

The fact that θ is absent means that there are no constraints on θ -- it is a free variable. For a given value of θ, you would get a semicircle.