Consider the following definition of components of a covariant vector:
Definition: A set of quantities $(A_1,\ldots, A_n)$ is said to constitute the components of a contravariant vector at a point $P$ with coordinates $(x^1, \ldots, x^n)$ if, under the coordinate transformation given by $\bar{x}_i = \bar{x}_i(x_1,\ldots,x_n)$, these quantities transform according to the relations $$ \bar{A}_j = \sum_{h=1}^n \frac{\partial{}\bar{x_j}}{\partial x^h} A^h $$ in which the coefficients $\partial{}\bar{x_j} / \partial x^h$ are to be evaluated at $P$.
My question: How are the numbers $\bar{A}_1,\ldots,\bar{A}_n$ defined? The definition given about specifies what the numbers have to satisfy, but where do we get these numbers from?
