Problems with the definition of a vector field in do Carmo's book.

Question

The vector field is defined in terms of the functions $a_i$, which have domain on a subset $U$ of $\mathbb{R}^n$, but in the book, it evaluates these functions at a point $p$ in the manifold. Can someone explain? Thank you.

Answer 1

Yeah...that should probably be $$ X(\mathbf{x}(u)) = \sum a_i(u) \frac{\partial}{\partial x_i}(\mathbf{x}(u)) $$ where $u \in U$ is a point in the parameter space. If we let $p$ temporarily denote $\mathbf{x}(u)$, then you get the formula DoCarmo wrote.