CUP editors and typesetters have shown almost infinite patience and allowed me many proof readings, but after a while, staring at the same words, the experience resembles the occasions when you repeat the same word over and over again—the word becomes meaningless, nonsensical. So errors remain, and I will list these here as I find them, or even better when readers let me know of errors they have found.
comments:
It is notable that div applied to a
rank-two tensor is ambiguous. If I used the common
convention in fluid mechanics that the indices on the Cauchy stress
tensor are reversed then the differentiation implied in the
divergence is taken with respect to the first, not the second,
index. The same applies to the formula in the footnote: the
indices j and n would need to be exchanged; albeit in
that expression no explicit differentiation is indicated, and the
form is valid in any orthogonal coordinate system.
The convention that the first index of the Cauchy stress tensor
refers to the direction of the traction vector and the second, the
normal to the element of area over which it acts, seems to be most
common in solid mechanics. Certainly that's how Professors Eshelby
and Bilby taught it. Then the equilibrium condition reads,
It was Professor Bilby who first showed me how to get the Navier-Stokes equations from the condition of elastic equilibrium of an isotropic solid. Indeed I found my old undergraduate notes and it appears that he actually did reverse the order of the indices when he made the passage into fluid mechanics. I don't imagine that he intended to confuse the students and maybe he had fused two sets of lecture notes. My choice has been to start with the solid mechanics convention and stick to it!