## chapter 1 of ‘Mathematics of Classical and Quantum Physics’

Posted by peeterjoot on March 15, 2009

This is probably the largest Dover book on my shelf, weighing in at about 650 pages. I bought it for the Green’s function content. I’d gotten through bits of it, but reading it out of sequence turns out to be hard. Some of what I’m seeing now in QM books I know is covered nicely here, and on whim I grabbed it off my bookshelf in the morning this weekend, and started giving it a read from the beginning.

Now, I’d intended to skip the first chapter on Vectors in classical physics, but I am glad that I didn’t. This chapter is actually packed with a lot of good stuff.

– Rotations. Calculation of coordinates in original and rotated coordinate systems are covered. The difference between the rotation matrix for active alteration of a vector and the change of basis (expressing a vector in a rotated coordinate system) is nicely explained. I’d become confused about this a few times. Each time usually resulted in me re-deriving the rotation identities, to make sure that I got them right for the use I intended.

– A nice example of a Newton’s law (projectile trajectories) done in two coordinate systems. Working through that in detail was an interesting exercise.

– I even found the section on dot product interesting. As an exercise I did the proof that the dot product in cosine form is linear, which was fun, and illuminating.

– Levi-Civita tensor use in some cross product identities. A summation identity that looked familiar was covered, but I hadn’t ever used in a cross product context. I did the proof initially with a perl script, then realized that it was really just the dot product of bivectors expressed in coordinates.

– grad, div, curl, and Laplacian in curvalinear coordinates. This is something that I’d tried to blunder through myself, and is also covered in Geometric Algebra for Physicists, but like so much in that book, not covered in a form that one could learn from if it wasn’t already known. This is an excellent treatment of both the general curvilinear case as well as the case of an orthonormal basis. One aspect of this treatment that I found interesting was that both divergence and curl are defined in terms of integrals. Hestenes did something like this for the gradient, in one of the papers on his website, and I’d like to dig that up to compare.

– The Laplacian as a measure of the difference of the average value of the field in a neighborhood and at a point is interesting. One of the problems I noticed is to give a physical interpretation of the Helmholtz equation, and I’ll try to apply that idea to the problem. I also didn’t fully understand the description of the wave equation in terms of this average and will have to think that through a bit better.

– There’s some nice examples of tensors here that would have been good to see before trying to struggle through GAFP, and Hestenes’s NFCM. Many of the examples are the same as those covered in a GA context in those books, but learning of them without having to simultaneously learn the GA stuff would have been easier.

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