I am reading "Calculus on Manifolds" by Michael Spivak.
The author defined $\langle F,G\rangle$ as $\langle F,G\rangle(p)=\langle F(p), G(p)\rangle$ for vector fields $F$ and $G$.
But I think $\langle F,G\rangle(p)=\langle F(p), G(p)\rangle_p$ is correct.
Am I right?
The author defined $\nabla$ as $\nabla=\sum_{i=1}^n D_i\cdot e_i$ but I think $\nabla=\sum_{i=1}^n D_i\cdot (e_i)_p$ is correct.
For example $\mathrm{div} F$ is defined as $\mathrm{div} F=\langle\nabla, F\rangle$.
$F$ is a vector field such that $F(p)\in\mathbb{R}_p^n$ for each $p\in\mathbb{R}^n$, so I think $\nabla$ must be a vector field.
Am I right?