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In summary: So in summary, it seems that most people prefer to use the symbol ##g_{\mu \nu}## instead of ##\bar g_{\mu \nu}## or ##g_{\bar \mu \bar \nu}## when referring to the metric coefficients in a coordinate transformation.

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I fully agree. The symbol itself is the field that does not care about coordinates and what actually is basis dependent are the components. Unfortunately, many people don't agree ...Pencilvester said:Personally, if I’m referring to the same tensor field simply in different coordinates, it usually makes more sense to me to put the bars over the indices.

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Pencilvester said:

Thank you. For this to be valid, we should consider the same region of the manifold before and after the parametrization, correct? Because otherwise the domain of the functions would be different, what in turn makes the functions different.Orodruin said:I fully agree. The symbol itself is the field that does not care about coordinates and what actually is basis dependent are the components. Unfortunately, many people don't agree ...

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It sounds like you’re thinking about overlapping charts on a manifold, in which case, yes, you should be looking at where the domains of the coordinate functions overlap. However, for many problems, the issue is not getting from one patch of spacetime to another, it’s what happens simply when we transform the coordinates we’re using (think of almost all problems in SR, where most reasonable choices of coordinates for inertial observers cover the entire manifold).kent davidge said:Thank you. For this to be valid, we should consider the same region of the manifold before and after the parametrization, correct? Because otherwise the domain of the functions would be different, what in turn makes the functions different.

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Thanks!Pencilvester said:It sounds like you’re thinking about overlapping charts on a manifold, in which case, yes, you should be looking at where the domains of the coordinate functions overlap. However, for many problems, the issue is not getting from one patch of spacetime to another, it’s what happens simply when we transform the coordinates we’re using (think of almost all problems in SR, where most reasonable choices of coordinates for inertial observers cover the entire manifold).

Tensors are mathematical objects that describe linear relations between vectors, scalars, and other tensors. They are used in physics and engineering to represent physical quantities and their transformations.

The bar symbol over a tensor function or index indicates that the quantity is a covariant tensor, meaning that it transforms in the same way as the coordinate system. It is also known as a contravariant index.

Scalars are quantities that have only a magnitude and no direction, while vectors have both magnitude and direction. Tensors, on the other hand, not only have magnitude and direction but also describe how they change under different coordinate systems.

Tensors are used in physics to describe the laws of nature, such as the laws of motion, electromagnetism, and general relativity. They provide a mathematical framework for understanding and predicting physical phenomena.

In engineering, tensors are used to model and analyze complex systems, such as stress and strain in materials, fluid dynamics, and structural mechanics. They also play a crucial role in computer vision and machine learning for image and signal processing tasks.

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