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Tensor calculus (tensor analysis) is a systematic extension of vector calculus to multivector and tensor fields in a form that is independent of the choice of coordinates on the relevant manifold, but which accounts for respective sub-spaces, their symmetries, and their connections.

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Notation issue for mixed tensors

When I am asked to evaluate, $\mathbf{U^{\alpha}_{~,~~\beta}}$ for all $\alpha$ and $\beta$, what does it mean? I have not able to understand this notation. In case of $\mathbf{g(~~,~\bar{A})}$ I …
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