Determinant in Transformation from spherical to cartesian space

[Nikitin]

Homework Statement

Evaluate the appropriate determinant to show that the Jacobian of the transformation from Cartesian (this is a typo, they mean spherical) pψθ-space to Cartesian xyz-space is ρ²sin(ψ).

Homework Equations

The Attempt at a Solution

Uhm, I am lost. I'm supposed to prove that when a function F(p,ψ,θ) is transformed into a function H(x,y,z), then the jacobian is ρ²sin(ψ).

But, to do that I am supposed to solve a determinant which involves the partial derivatives of p(x,y,z), ψ(x,y,z) and θ(x,y,z) with respect to x,y,z, namely J(x,y,z)?? That would take an hour, so I assume I am not understanding the problem properly?? As far as I know, evaluating the determinant J(p,ψ,θ) will make ρ²sin(ψ) pop out - but this is for the opposite transformation, from cartesian to spherical space.

I'm confused. Help pls?

 

[voko]

First of all, it is not "spaces" that are transformed. It is coordinates.

Write x = x(p, ψ, θ), y = y(p, ψ, θ), z = z(p, ψ, θ) and find the Jacobian of this.

 

[Nikitin]

But that's the jacobian for the opposite transformation (cartesian into spherical, I want to find the jacobian of the reverse transformation)! How does that apply here?

 

[voko]

The transformation in #2 is from the spherical coordinates into Cartesian.

 

[Nikitin]

Well, it's also the jacobian for the cartesian -> spherical transformation..

How can that be? Can somebody explain this jacobian stuff to me?

 

[voko]

I think you are confused by the word "transform". This may be because I and whoever gave you the problem use it to mean something different. Could you restate your problem in more details without using this term?

 

[Nikitin]

"Evaluate the appropriate determinant to show that the Jacobian of the transformation from Cartesian pψθ-space to Cartesian xyz-space is ρ²sin(ψ)."

This is the exact wording from the book. I am not confused about transformation.

What I am confused about is how ρ²sin(ψ) can be the jacobian determinant for both the xyz --> pψθ transformation, and the pψθ ---> xyz transformation.

 

[voko]

The Jacobian for p(x,y,z), ψ(x,y,z) and θ(x,y,z) is not ρ²sin(ψ), simply because it ought to be a function of (x, y, z). But even if expressed via (p, ψ, θ) it will be 1/(ρ²sin(ψ)).

 

[Nikitin]

Let's say we have a sphere x^2+y^2+z^2=1.

Its volume can be expressed as -1∫1 -1∫1 -1∫1 dxdydz. If it is transformed into spherical coordinates, then ∫∫∫x^2+y^2+z^2dxdydz --> 0∫2pi 0∫pi 0∫1 J(p,ψ,θ) dpdψdθ,

where the Jacobian J(p,ψ,θ) = p²*sin(ψ).

Where is the flaw in my logic? Here a xyz---> pψθ transformation happened, right?

 

[voko]

Let's say we have a sphere x^2+y^2+z^2=1.

Its volume can be expressed as -1∫1 -1∫1 -1∫1 x^2+y^2+z^2 dxdydz

I do not understand what this means.

If it is transformed into spherical coordinates, then ∫∫∫x^2+y^2+z^2dxdydz

This is not the volume of the sphere. The volume of the sphere is ∫∫∫dxdydz, where the integration domain is the interior of the sphere.

In spherical coordinates, that becomes ∫∫∫J(ρ,ψ,θ)dρdψdθ, where the integration domain is the "brick" 0 ≤ ρ ≤ 1, 0 ≤ ψ ≤ ∏, 0 ≤ θ ≤ 2∏.

Here a xyz---> pψθ transformation happened, right?

You can say you "transformed" the integration problem from Cartesian to spherical coordinates. However, this uses the "coordinate transformation" from spherical coordinates to Cartesian. And the Jacobian is that of the "coordinate transformation" from spherical coordinates to Cartesian. The word transformation may mean very different things depending on what it is attached to.

 

[Nikitin]

Oh, sorry. Youre completely right on the volume thing - my mind is exhausted after 10 hours of cramming. And yeh, i think ill return to this problem tomorrow