First off, sorry this is past the midnight deadline. Monday and Tuesday are absurdly busy for me on my schedule. I've forgone proper symbols to get this out quicker; my apologies for the messiness.
Section 1.6: Cool stuff - derivatives are straightforward, the easy proofs with Laplacian, curl, and divergence are impressive. Gets a little unclear towards the end. (1.89) is not manifestly covariant since it is not yet expressed in terms of tensors and we do not know how it changes under rotation. Is covariant here used in the sense that it transforms nicely under rotations or in the sense of covariant tensors/pseudovectors. But is B_ij a tensor or a pseudotensor? It is not obtained by tensor product or by Levi-Civita contraction, so we have not yet established a rule for determination.
Section 2.1-2.2: One question that immediately stands out is equation (2.1). We have a volumetric force F_i multiplied by mass (rho*dV) equal (in units) to the net force. That is, the units seem not to work out. How is this resolved? Another point lies in equation (2.6) - why are we computing surface integrals when we are told to integrate over volume? The rest pretty much made sense, though I'll be interested to see how this is presented in class. My mathematical instincts are a little frustrated by the "wishy-washiness" of the argument.
Section 2.3: Okay so F_i, though not mentioned before, is force per unit mass. That makes more sense now. This stuff is very cool and intuitive. Everything else pretty much makes sense.
Good stuff.
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