Volume spherical coordinates

In summary, the formula for calculating the volume in spherical coordinates is V = ρ³sin(φ)dρdφdθ, where ρ represents the radial distance, φ represents the polar angle, and θ represents the azimuthal angle. To convert from Cartesian coordinates to spherical coordinates, you can use the equations ρ = √(x² + y² + z²), φ = arctan(y/x), and θ = arccos(z/ρ). The range for the values of ρ is from 0 to infinity, φ is from 0 to π, and θ is from 0 to 2π. To calculate the volume of a spherical shell in spherical coordinates, you can use
  • #1
fishingspree2
139
0
Say I have a solid given using polar coordinates
I want to compute its volume.

We know that when switching from cartesian to polar, dV becomes [tex]\rho^{2}\sin\phi d\rho d\theta d\phi[/tex]

But I am not converting from cartesian to polar, I am already in polar coordinates.

do I still have to carry the [tex]\rho^{2}\sin\phi[/tex] that comes from the dV conversion formula?
 
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  • #2
Yes, you do. That formula for dV is the volume element in spherical coordinates, whether you're converting into spherical coordinates or staying in spherical coordinates.
 

What is the formula for calculating the volume in spherical coordinates?

The formula for calculating the volume in spherical coordinates is: V = ρ³sin(φ)dρdφdθ, where ρ represents the radial distance, φ represents the polar angle, and θ represents the azimuthal angle.

How do I convert from Cartesian coordinates to spherical coordinates?

To convert from Cartesian coordinates (x,y,z) to spherical coordinates (ρ,φ,θ), you can use the following equations: ρ = √(x² + y² + z²), φ = arctan(y/x), and θ = arccos(z/ρ).

What is the range for the values of ρ, φ, and θ in spherical coordinates?

The range for the values of ρ is from 0 to infinity, φ is from 0 to π, and θ is from 0 to 2π.

How do I calculate the volume of a spherical shell in spherical coordinates?

To calculate the volume of a spherical shell in spherical coordinates, you can use the formula: V = 2π∫φ1φ2∫θ1θ2∫ρ1ρ2 ρ²sin(φ)dρdφdθ, where φ1 and φ2 are the limits of the polar angle, θ1 and θ2 are the limits of the azimuthal angle, and ρ1 and ρ2 are the inner and outer limits of the radial distance.

What is the relationship between spherical and cylindrical coordinates?

Spherical and cylindrical coordinates are related by the following equations: x = ρcos(θ), y = ρsin(θ), and z = z. This means that the radial distance ρ and the azimuthal angle θ in spherical coordinates correspond to the cylindrical coordinates (ρ, θ, z).

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