How do I define a region in R3 with spherical/polar coords?

In summary, the student attempted to solve the homework equation using spherical and cylindrical coordinates but was unable to find the boundaries for x or y. They were able to find the boundaries for z using either coordinate system. However, they were not able to find the azimuthal angle phi. They were able to find the radius r for the sphere using cylindrical coordinates, but were not able to find the same for spherical coordinates.
  • #1
Phantoful
30
3

Homework Statement


TzpRq83.png


Homework Equations


x^2 + y^2 + z^2 = r^2
Conversion equations between the three coordinate systems

The Attempt at a Solution


I tried to solve this problem using spherical/cylindrical coordinates from the beginning, but that wouldn't work so I started with cartesian. However, I couldn't find the boundaries for x or y, but I believe z is 0≤z≤2. Using either spherical and cylindrical, I found that 0≤θ≤2π. This is all I could extract on my own, and I don't know if it's even possible to convert without knowing φ (This is the angle from the z-axis). Am I approaching this wrong, or missing some information I should have extracted? (Or are my statements about z and theta wrong?)
 

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  • #2
Hi, thanks for your inquiry. To do the spherical coordinates, rewrite the equation for the cone in terms of the polar angle theta (the angle from the z axis). You can then write an inequality for theta because all points in the region are inside the cone. Then write the equation for the sphere, in terms of the radius. You can then write an inequality for r because all points in the region are inside the sphere.

For cylindrical coordinates, you can write the equation for the cone as a relationship between rho (the distance from the z axis) and z. To be inside the cone, require an inequality in terms of rho and z. Then to do the sphere, write the radius r as a function of z and rho. You can then write an inequality where rho is less than a certain amount.

The azimuthal angle phi should not appear in any of your equations.

Good luck! Get back to me if you have more difficulty.
 
  • #3
Gene Naden said:
Hi, thanks for your inquiry. To do the spherical coordinates, rewrite the equation for the cone in terms of the polar angle theta (the angle from the z axis). You can then write an inequality for theta because all points in the region are inside the cone. Then write the equation for the sphere, in terms of the radius. You can then write an inequality for r because all points in the region are inside the sphere.

For cylindrical coordinates, you can write the equation for the cone as a relationship between rho (the distance from the z axis) and z. To be inside the cone, require an inequality in terms of rho and z. Then to do the sphere, write the radius r as a function of z and rho. You can then write an inequality where rho is less than a certain amount.

The azimuthal angle phi should not appear in any of your equations.

Good luck! Get back to me if you have more difficulty.

I'm a little confused with which symbols and angles you are talking about, my class uses a different set of symbols, and I think I actually also wrote them incorrectly in my original question (sorry!):

Spherical (ρ,θ,Φ):
spherical_coordinates.png


Cylindrical, (r, θ, z):
cylindrical_coordinates.png


In this notation, for Spherical I got 0 ≤ ρ ≤ 2 (top of the region W, correct?), and 0 ≤ θ ≤ 2π (goes all the way around x-y). For cylindrical I got 0 ≤ z ≤ 2 as well, and θ the same.

I'm not sure how I should be rewriting the equation for the cone in terms of Φ, the angle from the z-axis (do I use the coordinate conversion equations?), as well as rewriting the equation in terms of radius.

For the cylindrical part I didn't understand if you meant rho (ρ) as in actual distance from the z-axis or by angle. And again, how would you start rewriting the equations?
 

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  • #4
Yes, well, you are using a different convention for the symbols. Here is my hint using the symbols the way you have defined them.
Your theta should not appear in the equations.
Yes, you use the coordinate conversion equations. I believe they are as below but you should compare to your notes.

SPHERICAL:
rho < 2, you have that right.
z=rho*cos(phi).
x^2+y^2 = rho^2 * (sin(phi)^2).
You should find a function f such that f(phi) < some constant. Phi is between 0 and pi/2 for this problem.
Because f increases with phi, this should give you an equation phi < another constant

CYLINDRICAL:
0 < z < 2 is not correct.
x^2+y^2=r^2

Try to work with that and please get back to me :)
 

Related to How do I define a region in R3 with spherical/polar coords?

1. How do I convert Cartesian coordinates to spherical/polar coordinates?

To convert Cartesian coordinates (x, y, z) to spherical coordinates (r, θ, φ) in R3, use the following formulas:
r = √(x² + y² + z²)
θ = tan⁻¹(y/x)
φ = cos⁻¹(z/√(x² + y² + z²))

2. What are the ranges of the variables r, θ, and φ in spherical coordinates?

In spherical coordinates, r represents the distance from the origin, and can take any positive value. θ represents the angle between the positive x-axis and the projection of the point onto the xy-plane, and can range from 0 to 2π. φ represents the angle between the positive z-axis and the point, and can range from 0 to π.

3. How do I plot a region defined by spherical coordinates in R3?

To plot a region defined by spherical coordinates, you can use the "rgl" package in R. First, define the ranges for r, θ, and φ using the appropriate functions (e.g. seq() or seq.int()). Then, use the function "sphplot()" to create a 3D plot of the region. You can also use the "rgl.surface()" function to add a surface to the plot.

4. Can I define a region in R3 using only spherical/polar coordinates?

Yes, it is possible to define a region in R3 using only spherical or polar coordinates. However, it may be more convenient to use a combination of spherical/polar coordinates and Cartesian coordinates, depending on the shape of the region and the specific calculations you need to perform.

5. How do I calculate the volume of a region defined by spherical/polar coordinates?

To calculate the volume of a region defined by spherical/polar coordinates, you can use the triple integral formula in spherical or polar coordinates, depending on the shape of the region. Alternatively, you can also convert the region to Cartesian coordinates and use the standard triple integral formula in Cartesian coordinates.

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