What is the meaning of ##d\Omega## in solid angle integration?

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Discussion Overview

The discussion revolves around the meaning and application of the differential solid angle element, ##d\Omega##, in the context of integrating products of unit vectors. Participants are exploring the mathematical implications and properties of these integrals.

Discussion Character

  • Exploratory
  • Mathematical reasoning
  • Debate/contested

Main Points Raised

  • One participant asks how to perform the integrals ##\int d\Omega n_{i}n_{j}## and ##\int d\Omega n_{i}n_{j}n_{k}n_{l}##, where ##n## represents a unit vector.
  • Another participant expresses skepticism about the integrals, suggesting that without additional context, they seem nonsensical, particularly questioning the relationship ##n_in_j = \delta_{ij}##.
  • A further reply proposes that the result of the integrals should be proportional to a symmetrized product of Kronecker deltas, but questions how to demonstrate this.
  • Another participant requests clarification on the meaning of ##d\Omega## and whether a unit vector is considered a constant in the integration process.

Areas of Agreement / Disagreement

Participants do not appear to reach a consensus, as there are differing interpretations of the integrals and the role of the unit vector in the context of solid angle integration.

Contextual Notes

There are unresolved assumptions regarding the properties of the unit vectors and the definition of ##d\Omega##, which may affect the interpretation of the integrals.

PreposterousUniverse
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Anyone have any idea how to perform the following two integrals?

##\int d\Omega n_{i}n_{j}## and ##\int d\Omega n_{i}n_{j}n_{k}n_{l}##

where the n is a unit vector.
 
Last edited:
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Not if you don't provide some more context. The way it looks now (to me) it's nonsense (##n_in_j = \delta_{ij}## ? )
 
BvU said:
Not if you don't provide some more context. The way it looks now (to me) it's nonsense (##n_in_j = \delta_{ij}## ? )
Actually, I think it should be proportional to some symmetrized product of kronecker deltas. But how can one show that?
 
PreposterousUniverse said:
But how can one show that?
First you have to show what you mean. Isn't a unit vector a constant of the integration ?

Describe ##\;d\Omega##
 

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