I was recently intereseted on Continuum mechanics and it's relationship with integration, manifolds and General Relativity. I'm interested in the connection of Continuum mechanics with General Relativity, specially with the relationship of the stress tensor σij\sigma_{ij} with the energy-momentum-stress tensor TμνT_{\mu \nu} . I want to deeply understand the stress tensor, in order to understand it's generalization to GR. Also, I'm interested in ways of deriving continuity equations or conservations laws using the fact that the volumen of integration is chaning with time in order to derive the integral theorems (like Reynolds transport theorem). In differential geometry this theorem can be quickly derived using parametrizations: ddτ∫Σ(t)ω=∫Σ0ddτ(θt∗ ω),\frac{d }{d \tau} \int_{\Sigma(t)} \omega = \int_{\Sigma_0} \frac{d }{d \tau} \left ( \theta_t^* \, \omega \right), so if the book introduces to the reader to differential geometry, it would be nice, but maybe after some chapters, as an alternative way of understanding the topic. I really like the style of the Landau Lifshitz - Theory of elasticity . Specially it's index notation style of equations (I've tried to read continuum mechanics books with vector style notation, but I find it very obfuscated). So, to sum up, I'm searching a Landau Lifshitz style kind of book (but more modern), that explains the meaning of the stress tensor, also makes emphasis on the integral theorems and maybe starts with the "classical way" of explaining in continuum mechanics (like, for example, defining velocity as v(x,y,z,t)v(x, y, z, t) ) and then introducts the reader to a "manifold/differential geometry" way of explaining the topic (like defining movements on the manifold, pullpacks or pushforwards, etc...). "Too long, didn't read" sum up: Emphasis on the stress tensor Emphasis on the integral theorems Landau Lifshitz - Theory of elasticity style (specially it's notation) Later connection with manifolds and differential geometry Applications to general relativity (optional)