# 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 understand that the blank in the metric tensor means it acts like a one-form which takes in a vector and outputs a real number. For this case it makes sense to me because it is a

$$\begin{bmatrix} 0 \\ 2 \end{bmatrix}$$ tensor. How do we understand the same for a mixed tensor like the one above ?

Reference: Chapter 3, Problem 30: A First Course in General Relativity, Second Edition, B. Schutz, pg 82.

Let us assume $$U^\alpha$$ is a vector, i.e. described by its contravariant components. The subscript $$_{, \beta}$$ means the partial derivative, that is $$_{, \beta} = \partial_\beta$$. Therefore $$U^\alpha _{, \beta} = \partial_\beta U^\alpha$$.
However in general that is not a tensor, as the partial derivative is not a tensor in arbitrary coordinates. To have a tensorial expression, you should replace the partial derivative with the covariant derivative, that is $$\partial_\mu$$ replaced by $$\nabla_\mu = _{; \mu}$$.
So the expression $$U^\alpha _{; \beta} = \nabla_\beta U^\alpha$$ is a tensor, contravariant in $$\alpha$$ and covariant in $$\beta$$.