I'm learning about the transformation rules for vectors. In the image above, the author addresses that the components of the metric transform covariantly. However, I'm confused as to why the metric components comprise base vectors in this explanation. Aren't the components the differentials $dx^a$ and $dx^b$?
The full link is here.
Those aren’t the components. In local coordinates, the metric takes the form $$g=\sum\limits_{i,j} g_{ij}dx^i\otimes dx^j.$$ Note that the components are $g_{ij}=g\left(\frac{\partial}{\partial x^i},\frac{\partial}{\partial x^j}\right)$.
Remember that, in general, if we have a representation of a vector in a given basis, the components are given by the dual basis acting on that vector.