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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.

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    $\begingroup$ Not sure about a book, but deriving the del operator (gradient, curl, divergence) in different coordinate systems will give you a good grasp over new coordinate systems. $\endgroup$
    – Sidd
    Commented Dec 21, 2023 at 5:42
  • $\begingroup$ How is great rigor going to give you intuition and let you “see” them? For me, those things are diametrically opposite. My opinion is that you haven’t seen, or drawn for yourself, enough diagrams. In any case, the “rigor” is just simple definitions like $\hat r\equiv(x\hat x+y\hat y+z\hat z)/(x^2+y^2+z^2)^{1/2}$ in terms of Cartesian components and Cartesian unit vectors. $\endgroup$
    – Ghoster
    Commented Dec 21, 2023 at 6:31
  • $\begingroup$ @Ghoster: Right, just any source that really discusses the above in detail would help. Any inputs, please? $\endgroup$
    – S_M
    Commented Dec 21, 2023 at 6:35
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    $\begingroup$ I’ll let others recommend sources, since I don’t find rigor illuminating. What I don’t understand is why you aren’t yet able to visualize these things. For example, the $\hat r$ unit vector points away from the origin. On the surface of the Earth, with the origin at the center of the Earth, this is the direction called “up”. You can close your eyes and visualize that, can’t you? $\endgroup$
    – Ghoster
    Commented Dec 21, 2023 at 7:28
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    $\begingroup$ Similarly, $\hat\theta$ is “south” and $\hat\phi$ is “east”. They form an orthonormal triad at each point on the surface (or above or below it). This kind of thing is visualization and intuition, not rigor. This is the kind of understanding you need to do physics, IMHO, not a bunch of formulas that you can look up. $\endgroup$
    – Ghoster
    Commented Dec 21, 2023 at 7:31

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I recommend : Methods of Theoretical Physics: V1, V2. written by Morse&Feshbach. Although it is old one, As many problem was analytically solved in that time, It explains mathematical physics in greater depth than Arfken.

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