Book Recommendation: One that has a lot of problems and theory associated with polar coordinates and spherical polar coordinates

Question

I would like to "master" polar coordinates and spherical polar coordinates. In the sense, I would like to become as well versed with them as I am with cartesian coordinates.

I have gone through many physics books, Boas, Marsfield, Griffiths, but they really don't get into this stuff deep enough. For instance, I just don't have any intuition for unit vectors (r, theta), the fact that they can keep changing as a particle moves along a certain trajectory troubles me greatly. I simply cannot "see" these vectors.

In truth, I may be able to solve problems, but I lack understanding and insight.

Hence, this request. My ultimate goal: (1) To be able to truly appreciate the full power of these coordinate systems -- when, why, and how to use them, especially in the context of Classical Mechanics, Electrodynamics, and Quantum Mechanics.

In other words, I would appreciate book(s) that treat(s) the above coordinate systems with great rigour, so that I may be able to invoke them with impunity in the areas of physics. I do not want any book(s) that treat the above with "lazy" rigour as I see most physics books do.

So, please give me titles of books that truly discuss the above in detail, and not just burn through them without giving them their due diligence, as most applied mathematics/mathematical physics/engineering books I know of.

Answer 1

I have yet to find a book discussing spherical coordinates in the rigorous manner you are asking for. Regardless I can share other books/resources that discuss this coordinate system and apply it to quantum mechanics. Seeing some more images and calculations might show the spherical coordinate system's usefulness and help your intuition.

I would advise you to solve the Schrödinger equation for a hydrogen atom with cartesian and spherical coordinates separately. I believe this would convince you why spherical coordinates are easier to use for spherical symmetric systems.

Here is a general calculus book, definitely not in debt or rigorous. Either way it might be a different point of view.

1. Adams, R. A., & Essex, C. (2016). Calculus: A Complete Course (9th ed.). Prentice Hall. (p.605-607)

Here are some more chemistry-related books that show spherical harmonics and or hydrogen-like orbitals:

1. Efthimiou, C., & Frye, C. (2014). Spherical harmonics in P dimensions. World Scientific Publishing Company Incorporated. (p.1-12)
2. Atkins, P. W., De Paula, J., & Keeler, J. (2018). Atkins’ physical chemistry (11th ed.). Oxford University Press. (p.285-287)
3. Oxtoby, D. W., Gillis, H. P., & Butler, L. J. (2015). Principles of modern chemistry (8th ed.). Brooks Cole. (p.178-194)

These websites might also be helpful:

1. LibreTexts, Spherical Coordinates
2. Medium, Mathematics for Quantum Mechanics: Coordinate Systems
3. Mathematics for Quantum Physics, Coordinate systems

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As a final note, the notation for the azimuthal and polar angles is not standard. Mathematicians typically use $\theta$ and $\phi$ respectively while Physicists do it the other way around.

Answer 2

I recall my first experience with a "crash course" in vector calculus that included introduction to cylindrical coordinates and spherical coordinates. The professor presented several heuristic developments in that course, especially when discussing topics such as the divergence theorem and Stokes' theorem.

So to address your quest, almost any text book on Vector Calculus will suffice. For example, "Introduction to Vector Analysis," by Davis and Snider has an entire chapter dedicated to Generalized Orthogonal Coordinates, which contains a separate section on cylindrical and spherical coordinates.

In addition, many books on Advanced Calculus will also contains chapters on vector analysis. The book "Advanced Calculus," by John Olmsted, for example, has a chapter on vector analysis with separate sections on curvilnear coordinates and orthogonal coordinates. Those sections cover cylindrical and spherical coordinates.

These were only two examples out of a voluminous set of references that one can find with a literature review. Best of luck in your pursuits.