Computing inner products of spherical harmonics

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SUMMARY

The discussion centers on the computation of inner products of spherical harmonics, specifically addressing the confusion surrounding the use of the \sin \theta term in the integrand. The participant clarifies that the integral in question is indeed a surface integral over the sphere, requiring the surface element r² sin θ dφ dθ. The video explanation incorrectly categorizes this as a volume integral, leading to misunderstandings about the role of the radius r in the context of spherical harmonics. The consensus is that the correct interpretation is a surface integral, not a volume integral.

PREREQUISITES
  • Understanding of spherical harmonics
  • Familiarity with surface integrals in spherical coordinates
  • Knowledge of orthogonality in mathematical functions
  • Basic concepts of integration in three-dimensional space
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  • Study the properties of spherical harmonics and their applications
  • Learn about surface integrals in spherical coordinates
  • Explore the concept of orthogonality in functional analysis
  • Investigate the relationship between volume and surface integrals in multivariable calculus
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Mathematicians, physicists, and students studying advanced calculus or quantum mechanics who are interested in the properties and applications of spherical harmonics.

beefbrisket
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In this video, at around 37:10 he is explaining the orthogonality of spherical harmonics. I don't understand his explanation of the \sin \theta in the integrand when taking the inner product. As I interpret this integral, we are integrating these two spherical harmonics over the surface of a sphere, hence varying only \phi, \theta. So, this is a surface integral and the surface element (as I would've done it naively) is r^2\sin\theta\, d\phi\, d\theta. I am aware that this isn't a function of r and it isn't even really defined here so that brings some problems. However his explanation is that this is a volume integral and the volume element is \sin\theta\, d\phi\, d\theta, which doesn't really make sense to me either. Can someone help me out with what exactly is happening here? Mostly why the surface/volume element, and furthermore his entire explanation here, is missing any reference to r.
 
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He misspoke. It is not a volume integral, it is a surface integral.
 
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