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- some modern notation and orientation towards linear algebra. Not just the old component-based approach, which seems more like an exercise in algebraic manipulation, without gaining any understanding

- general tensors, not just Cartesian, ideally with some differential geometry applications

- not geared primarily towards mathematicians, as my knowledge of analysis is not quite good enough. Rather oriented towards applied mathematicians, physicists, engineers, i.e. not an entire book just listing theorems and proofs (they have to be included to some extent, of course, but not making up 100% of the content)

- perhaps some differential forms, but this is really not essential

Does anyone know of a suitable book? Thanks.