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fresh_42

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Note that one can speak of tensors in a tensor space (and vectors are a form of tensor) or in a more complex setting one can work with tensor fields (which is what happens in say SR and especially GR). The "best" way to think of tensors in my opinion is as specific representations of the group of general linear transformations, but to do that requires you get more familiar with groups and their representations. You start with scalars (transforming trivially) and vectors (with the fundamental defining representation). Thence all the other enumerated representations defines the other ranks of tensors. But if you want to bypass the group theory you can think of tensors as composites of vectors (you can e.g. think of a matrix as a row vector of column vectors or vise versa and hence it is a "bivector" of sorts. A rank 3 vector as a vector of vectors of vectors etc.)

Now with tensor

So in this setting, in a physical application you can describe the dynamic evolution of a system by its motion in the space-time manifold and its transformation (rotation or or other general linear transformation and change in direction of motion) as its position moves. (Think, for example, of spinning asteroids flying through space as time passes.)

And I'll finally mention, what you were getting at with your question, when we are dealing with a flat manifold, we can treat it as a vector space and this allows us to lump position vectors and displacement vectors and force vectors and so on. We then can work in linear coordinates (vectors) which, when working in unified space-time means we are working within an inertial frame. But we may in those cases and are forced to in the case where the manifold is no longer necessarily flat,... <breathe> to work with arbitrary space-time coordinates. We then need to connect vectors and tensors representing e.g. forces, directions of motion, moments of inertia etc, at any one point to those at other points. We have vectors and tensors at each coordinate point and a geometric structure which sets the framework by which we compare these at different points. This provides a very general setting to describe what is happening (as well as describe artifacts of our choice of description which are not physical). Hence we can formulate rules and physical laws in a most general way without the necessity of assuming such things as "inertial frames" are even defined globally much less that we are actually working within one.

Sorry if this answer is overkill.

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It is a lot to absorb. It will take me time to digest it, but thanks for the explanation.

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Nugatory

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Yes, and that's why they're so useful.

A reference frame implies a particular convention for assigning coordinates to events, so it is not surprising that tensor equations which work with all coordinate systems would work in all frames.

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