# Solid angle , never seen derivative

• zn52
In summary, the conversation discusses the formula for the derivative of solid angle, which is equal to the product of the differentials of velocity in the x and y directions. The participants also discuss the concept of solid angle and how it relates to velocity, as well as the inclusion of the speed of light squared to make the expression dimensionally consistent.
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) = dvx * dvy

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

Thank you,

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Since I don't know what vx and vy 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.

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

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

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 ...

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

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

## 1. What is a solid angle?

A solid angle is a unit of measurement used in three-dimensional geometry to quantify the size of an object or the amount of space it occupies. It is often compared to a two-dimensional angle, but in three dimensions, it measures the amount of space that a cone encloses.

## 2. How is solid angle different from regular angle?

A regular angle measures the amount of rotation between two intersecting lines, while a solid angle measures the amount of space that is enclosed by a cone with its apex at the intersection of those lines. So, while a regular angle is a two-dimensional measurement, a solid angle is a three-dimensional measurement.

## 3. What is the unit of measurement for solid angle?

The standard unit of measurement for solid angle is the steradian (sr). It is equivalent to the solid angle subtended by a surface that is equal in area to a square with sides of length equal to the radius of a sphere.

## 4. How is solid angle used in science?

Solid angle is used in various fields of science, including physics, astronomy, and engineering. It is used to calculate the amount of radiation emitted or received by an object, to measure the intensity of a light source, and to determine the direction of a light beam or radio wave.

## 5. What is the relationship between solid angle and the derivative?

The derivative of a function with respect to a variable represents the rate of change of that function with respect to that variable. In the case of solid angle, its derivative is not commonly used, as solid angle is a three-dimensional measurement and does not have a variable that it is dependent on. However, in some cases, the derivative of solid angle can be used to calculate the flux of a vector field through a surface.

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