I'm trying to understand what exactly it means by some tensor field to be '(adsbygoogle = window.adsbygoogle || []).push({}); well-defined'on a manifold. I'm looking at some informal definition of a manifold taken to be composed of open sets ##U_{i}##, and each patch has different coordinates.

The text I'm looking at then talks about how in order for a tensor field to be defined globally there are certain transition laws that must be obeyed in intersecting regions of ##U_{i}##

From this, my interpretation of well-defined is that you have in variance in certain patches, and so because each patch has its own coordinates, you have in variance in any coordinates.So, a tensor means you have invariance with respect to a change in coordinate system?Are these thoughts correct?

Is this literal agreement? I.e- the value of a scalar? the components of a matrix (representing a tensor as a matrix)?

Thanks in advance.

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# Tensor well-defined on a manifold, basic concepts.

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