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I'm trying to understand what exactly it means by some tensor field to be '

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.

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

**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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