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 ?
 A: 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:


*

*Fourier Analysis With Applications to BVPs (Schaum Outline Series) by M. Spiegel
http://www.icmc.usp.br/pessoas/menegatt/Brown-Churchill.pdf

*PDEs: An Introduction by W. Strauss 
https://zr9558.files.wordpress.com/2013/11/partial-differential-equations.pdf

*Introduction to Lie Groups and Lie Algebras by A. Kirillov Jr
https://www.math.stonybrook.edu/~kirillov/mat552/liegroups.pdf

*Laplace-Beltrami Operators Wikipedia Page:
https://en.wikipedia.org/wiki/Laplace%E2%80%93Beltrami_operator

*The Laplace-Beltrami Operator on Riemannian Manifolds by F. Schmidt
file:///home/g/gw/gws21/Downloads/lb2_slides.pdf

*The Laplace-Beltrami Operator on Riemannian Manifolds by F. Schmidt
file:///home/g/gw/gws21/Downloads/lb2_report.pdf
(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:


*

*Spherical Harmonics by C. Müller

*Geometric Applications of Fourier Series and Spherical Harmonics by H. Groemer 
http://ebooks.cambridge.org/ebook.jsf?bid=CBO9780511530005
The following online resources are quite good for what you are looking for:


*

*A User's Guide to Spherical Harmonics by M. Mohlenkamp:
http://www.ohio.edu/people/mohlenka/research/uguide.pdf

*Notes on Spherical Harmonics and Linear Representations of Lie Groups by J. Gallier
http://www.cis.upenn.edu/~cis610/sharmonics.pdf
Hope that helps! 
A: 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.
