Help with Vector Analysis in spherical velocity-space coordinates

[phrozenfearz]

I'm looking at integrating a solid angle multiplied by the vector representing that solid angle over an entire sphere (4pi).

I've split the terms into sin(\theta)cos(\phi)i, sin(\theta)sin(\phi)j, cos(\theta)k, but I am still integrating phi over [0,2pi] and theta over [0,pi]. This doesn't seem right and yields a value of zero for each term. I'm unsure of how to change my bounds of integration here.

Or is...
\int{\overline{\Omega}d\overline{\Omega}}
where \Omega=\{\theta,\phi\}
...similar to a dot-product operation and therefore would equal zero?

Thanks!

edit: I just noticed this forum isn't for homework questions... I seem unable to move it myself, so if any admins see this please move it to the homework forum.

 

[tiny-tim]

Welcome to PF!

Hi phrozenfearz! Welcome to PF!

(have a theta: θ and a phi: φ and a pi: π and an integral: ∫ )

Your limits are fine, but what are you integrating?

 

[phrozenfearz]

I have a solid angle being integrated over the surface area of a unit sphere. The only way I can interpret the answer I'm getting is that the integral of a vector w.r.t. a vector is equivalent to doing a dot-product. Therefore, the dot-product of omega with omega is zero.

Thanks for the symbols!

 

[tiny-tim]

I have a solid angle being integrated over the surface area of a unit sphere.

Yes, but what is your integral …

how are you writing it (in θs and φs)?

 

[phrozenfearz]

Ah, sorry.. let me figure out how to export it from MathType...

\[<br /> \int {\Omega d\Omega } = \int {(\Omega _x e_x + \Omega _y e_y + \Omega _z e_z )d\Omega } = \int_0^{2\pi } {d\varphi } \int_0^\pi {\sin \theta d\theta (\Omega _x e_x + \Omega _y e_y + \Omega _z e_z )} <br /> \]<br /> \[<br /> \begin{array}{l}<br /> where \\ <br /> \Omega _x = \sin \theta \cos \varphi \\ <br /> \Omega _y = \sin \theta \sin \varphi \\ <br /> \Omega _z = \cos \theta \\ <br /> \end{array}<br /> \]<br />

Separating each component of Omega and integrating separately gives me zero for each term. In x and y, it's the ∫dφ over [0,2pi] reducing the term to zero, and ∫sinθcosθdθ over [0,pi] reducing the z term to zero. Thus, the integral of that vector/solid-angle is reducing to zero?

 

[HallsofIvy]

Of course, the integral over the entire sphere will be 0! Essentially what you are doing is calculating the flow of some vector quantity through the sphere, constant in time. Whatever goes into the sphere on one side, comes out on the other. There is no net "inflow" or "outflow" so the integral over the entire sphere is 0.

 

[tiny-tim]

ah i see what you're doing …

no, if you're trying to integrate Ω over the whole sphere to get 4π, your integrand is just 1 …

∫₀^2π∫₀^π sinθ dθdφ

 

[phrozenfearz]

ah i see what you're doing …

no, if you're trying to integrate Ω over the whole sphere to get 4π, your integrand is just 1 …

∫₀^2π∫₀^π sinθ dθdφ

Yeah, if the integrand were one, it would be simple. But the problem (as stated earlier, I posted this in the wrong forum) asks for the integrand to be Ω. Hence, my confusion. Is it just a dot-product, or am I doing something dreadfully wrong?

 

[tiny-tim]

… the problem … asks for the integrand to be Ω.

Are you sure?

If the integrand is Ω times the outward unit vector, then the integral is obviously 0, since the vectors at opposite points on the sphere cancel each other!

What is the exact question (if it says "integrate the solid angle Ω over the whole sphere", it probably means ∫ dΩ, not something like ∫ Ω dΩ)?

 

[phrozenfearz]

Can you help me to visualize this? It doesn't seem obvious to me, but integrals have never been a strong point once they move beyond x-y plots.

The exact question reads:

Integrate ∫ Ω dΩ. Hint: Ω = (iΩx + jΩy + kΩz)

 

[tiny-tim]

Can you help me to visualize this? It doesn't seem obvious to me, but integrals have never been a strong point once they move beyond x-y plots.

The exact question reads:

Integrate ∫ Ω dΩ. Hint: Ω = (iΩx + jΩy + kΩz)

hmm … that doesn't make sense to me …

at each point P of the sphere, what is the integrand as a function of P?

(It can't be Ω, since you can't have Ω "of a point" …

Ω is a solid angle, and points don't have them! )

If it's thinking of, say, the electric field from a charge inside a sphere, so that the flux through any small surface area is kΩ , then the whole flux is 4πΩ, but that's from ∫ k (e_r.dΩ)

 

[phrozenfearz]

It's meant to represent the angle in which a neutron is travelling. Omega is made up of theta and phi and if represented as a vector has a length of one. I'm not sure about that last bit you mentioned, but I don't think flux is relevant since we're only talking about one neutron.

 

[tiny-tim]

It's meant to represent the angle in which a neutron is travelling.

So this has nothing to do with the whole sphere?

Omega is made up of theta and phi and if represented as a vector has a length of one.

If it has a length of one, then it has nothing to do with solid angle, it's just the unit vector in the normal direction, e_r.

 

[darkside00]

use greens theorem