# What's a good primer from linear algebra to spherical harmonics?

I need a topic, a primer, that will be able to introduce me to spherical harmonics and how to translate and use them with the usual tools of linear algebra and calculus, namely matrices, polynomials and derivatives for example .

In other words, I would like to know enough to handle and compute harmonics.

What topics do you suggest I should touch to get up and running with spherical harmonics starting with a linear algebra and calculus background ?

## 2 Answers

Having studied calculus and linear algebra is a good start, but a key component of spherical harmonics is their relation to harmonic analysis and PDEs, (eg, if one wants to use harmonic analysis on the $n-$dimensional sphere, you do it in terms of spherical harmonics). I would recommend studying some basics on Fourier analysis, lie algebras and Laplace-Beltrami operators. For this, check out the following references:

(If you need some differential geometry references, please let me know). You should then have the background to move onto spherical harmonics. Check out the following references:

Check out the Wikipedia page for spherical harmonics (specifically the section "Laplace's spherical harmonics"): https://en.wikipedia.org/wiki/Spherical_harmonics

and the Wolfram page: http://mathworld.wolfram.com/SphericalHarmonic.html

I would then recommend the following books to study:

The following online resources are quite good for what you are looking for:

Hope that helps!

• Are you sure that Claus Muller is a good first reference? I am trying to understand basic relations about Linear independence of Spherical Harmonics and Legendre Polinomials and suddenly (first chapter defninition 1) we are confronted with spherical harmonics of order $n$ in q dimensions. – Conrado Costa Jul 22 '15 at 14:25
• I would look at the other books by Groemer first and the the notes by Mohlenkamp. Check out the following notes: math.utk.edu/~freire/m435f07/m435sphericalharmonics.pdf – user230715 Jul 22 '15 at 14:34

You should read about these topics:

• Vector spaces (I think you already know what this is)
• Metric spaces (A vector space + a norm)
• Hilbert spaces (A metric space with a finite or infinite number of dimensions)
• Fourier transform (The projection of a vector belonging to a Hilbert space in an orthogonal basis)
• Spherical coordinates (Really easy to understand, check Wikipedia)

Once you get that, you'll be able to understand the spherical harmonics:

An orthogonal basis for the Hilbert space of functions in spherical coordinates that only vary with $\theta$ and $\phi$ under the norm $L_2$.

P.S. The spherical harmonics are the sinus and cosinus of functions in 3D that don't change with $r$.

I'm not being very rigorous because I wanted to give you an eagle view on the subject.

• Thank you, were do you think I might find the fact that any spherical harmonics of degree $n$ are obtained as linear combinations of a single spherical harmonic of degree $n$? – Conrado Costa Jul 24 '15 at 2:56
• The spherical harmonics are orthogonal under two indices: m and l (order and degree). That implies that your statement is false. Can you provide the source? – Moises Arizpe Rojo Jul 24 '15 at 3:18
• – Conrado Costa Jul 24 '15 at 3:19