Solid angle , never seen derivative

[zn52]

hello,
Please attached snapshot of an answer to an Ex. I was stunned at the formula for the derivative of the solid angle which is :

d(solid angle) = dv^x * dv^y

I would appreciate if somebody can provide some hints on how one can find it ?

Thank you,

 

[mathman]

Since I don't know what v^x and v^y are, it is hard to answer. However the solid angle can be envisioned as an area on a unit sphere, so these could be length differentials in the x and y directions on a unit sphere.

 

[zn52]

well as you can see in the same line, the v^x and v^y are velocities since they are equal to P over E in the 4-momentum . the p^x and p^y are momentum in the x and y directions of the incoming photos I guess ?

 

[Bill_K]

It's not a derivative of solid angle, it's a differential, it's the area of a small rectangle in velocity space, exactly analogous to the other line, d(cross-sectional area) = dx dy

 

[zn52]

I see . But how can we derive this expression ? why is it defined in the velocity space ? sometimes I wonder from where similar expressions come from ...

 

[mathman]

A solid angle is dimensionless, but the expression d(solid angle) = dv^x * dv^y has velocity squared on the right hand side. Something has to be included to make the expression dimensionally consistent.

 

[zn52]

yes it might be that the speed of light squared is in the denominator and it is considered as 1 ...