Computing inner products of spherical harmonics

The surface element has r^2\sin\theta\, d\phi\, d\theta.In summary, the speaker in the video is discussing the orthogonality of spherical harmonics and at around 37:10, he explains the inner product using an integral. However, the listener is confused about the use of \sin \theta in the integrand and the difference between a surface and volume integral. The speaker misspeaks and it is clarified that the integral is actually a surface integral, with the surface element being r^2\sin\theta\, d\phi\, d\theta.
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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 [itex]\sin \theta[/itex] 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 [itex]\phi, \theta[/itex]. So, this is a surface integral and the surface element (as I would've done it naively) is [itex]r^2\sin\theta\, d\phi\, d\theta[/itex]. I am aware that this isn't a function of [itex]r[/itex] 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 [itex]\sin\theta\, d\phi\, d\theta[/itex], 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 [itex]r[/itex].
 
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  • #2
He misspoke. It is not a volume integral, it is a surface integral.
 
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1. What are spherical harmonics?

Spherical harmonics are a set of mathematical functions used to represent the shape of a spherical object or field. They are commonly used in fields such as physics, mathematics, and computer science to describe the behavior of waves and fields in spherical systems.

2. How are inner products of spherical harmonics computed?

The inner product of two spherical harmonics is computed by integrating the product of the two functions over the entire sphere. This integral can be simplified using various mathematical techniques, such as orthogonality and symmetry properties of spherical harmonics.

3. What is the significance of computing inner products of spherical harmonics?

Computing inner products of spherical harmonics is important in a variety of applications, such as in quantum mechanics and signal processing. It allows us to determine the relationship between different spherical harmonics and can help us understand the behavior of waves and fields in spherical systems.

4. Are there any important properties to consider when computing inner products of spherical harmonics?

Yes, there are several important properties to consider when computing inner products of spherical harmonics. These include orthogonality, symmetry, and normalization. These properties can greatly simplify the computation and help us understand the relationship between different spherical harmonics.

5. How are inner products of spherical harmonics used in computer science?

In computer science, inner products of spherical harmonics are commonly used in various applications, such as computer graphics and sound processing. They can be used to represent and manipulate complex spherical shapes and fields, making them useful in virtual reality, video game development, and other computer-generated imagery.

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