How to graph spherical coordinates

[bfusco]

Homework Statement

given I=∫∫∫ρ^3 sin^2(∅) dρ d∅ dθ
the bounds of the integrals:
left most integral: from 0 to pi
middle integral: from 0 to pi/2
right most integral: from 1 to 3

i have no idea how to graph this, i was hoping someone would be able to recommend some techniques.

 

[DryRun]

Do you know what ρ, ∅ and θ mean? What do they represent in terms of spherical coordinates?

Check the attachment. By the way, your description is confusing. Instead of "left most", "middle", etc, use specific symbols to represent those values, for clarity.

For example, ρ varies from 1 to 3 or better put as: $1\leqρ\leq 3$

In your description: "left most integral: from 0 to pi" would indicate that ρ varies from 0 to pi, which is incorrect, as ρ is not an angle. ρ is the magnitude/length of the chord extending from the origin to the surface of the sphere.

 

[bfusco]

Do you know what ρ, ∅ and θ mean? What do they represent in terms of spherical coordinates?

I believe they mean:
θ means the rotation from the x axis
∅ means rotation from the z axis
ρ is like the length

does spherical coordinates literally mean that we are putting the function into a sphere?

Check the attachment. By the way, your description is confusing. Instead of "left most", "middle", etc, use specific symbols to represent those values, for clarity.

For example, ρ varies from 1 to 3 or better put as: $1\leqρ\leq 3$

In your description: " right most integral: from 1 to 3" would indicate that θ varies from 1 to 3, which is likely incorrect, as it's an angle and it's more conventional to state its range of values in terms of pi.

kk well then i will rewrite what the bounds are: (<_ is going to mean less than or equal to)
0<_θ<_pi, 0<_∅<_pi/2, 1<_ρ<_3

 

[DryRun]

It should be quite obvious now. Just draw the sphere. Try to imagine it first. Starting from the positive x-axis, draw an angle of pi (anti-clockwise), which would represent the required angle θ. Plotting ∅ should be simple, starting from the positive z-axis until the plane z = 0. For ρ, you need to plot two surfaces; one at radius = 1 and the other surface at radius 3 units from the origin. The required spherical volume is a quarter sphere with radius = 3, but hollowed from the centre until radius = 1. See the attachment.

 

[bfusco]

i am still not understanding, if i am only focusing on the bounds what is the function given for? also, if i am eventually just drawing two spheres, one with a radius of 1 and the other 3, what is the point of doing anything with theta, and phi?

 

[DryRun]

i am still not understanding, if i am only focusing on the bounds what is the function given for?

Your question is: "how to graph spherical coordinates" which is exactly what I've explained.

Are you trying to evaluate the value of I? If yes, then you'll need to make use of ρ^3 sin^2(∅) by proceeding with the triple integration, and using the given bounds as the limits.

also, if i am eventually just drawing two spheres, one with a radius of 1 and the other 3, what is the point of doing anything with theta, and phi?

Without θ, you would not know if the required section of the sphere does a complete revolution about the z-axis, or not.
Without ρ, you would not know the radius of the sphere.

 

[bfusco]

Your question is: "how to graph spherical coordinates" which is exactly what I've explained.

Are you trying to evaluate the value of I? If yes, then you'll need to make use of ρ^3 sin^2(∅) by proceeding with the triple integration, and using the given bounds as the limits.

well eventually i was going to express the iterated integral in terms of both cartesian coordinates and cylindrical coordinates, and i thought that by drawing it out that would help me. i also was expecting to get a shape as you would get if you did a function in cartesian coordinates, not really just a line. just seeing a line and 2 angles i have no idea how that helps me towards my eventual goal of expressing this triple integral in cartesian and cylindrical coordinates.

Without θ, you would not know if the required section of the sphere does a complete revolution about the z-axis, or not.
Without ρ, you would not know the radius of the sphere.

 

[DryRun]

well eventually i was going to express the iterated integral in terms of both cartesian coordinates and cylindrical coordinates, and i thought that by drawing it out that would help me.

Indeed, it is always a good idea to draw the required 3D region. It'll clarify and enhance your understanding.

i also was expecting to get a shape as you would get if you did a function in cartesian coordinates, not really just a line.

If you express the given spherical coordinates into Cartesian/cylindrical coordinates, you should still get the exact same shape/volume. In this case, spherical coordinates has been used as it is simply more appropriate (since it deals with the volume of a sphere). Using other (less appropriate) coordinates systems would make the problem more complex (the limits wouldn't be so clearly defined) and the resulting triple integration tedious.

just seeing a line and 2 angles i have no idea how that helps me towards my eventual goal of expressing this triple integral in cartesian and cylindrical coordinates.

That would be another question. You are asking how to convert spherical coordinates to Cartesian/cylindrical coordinates. This is the point where you'll need to use the 3D plot (from earlier) to set the new limits for the different coordinates systems.

 

[LCKurtz]

This is a new thread which should have been left as a continuation of

https://www.physicsforums.com/showthread.php?t=659296