Consider our manifold to be $\mathbb{R}^n$ with the Euclidean metric.
In several texts that I've been reading, $\{\partial/\partial x_i\}$ evaluated at $p\in U \subset \mathbb{R}^n$ is given as the basis set for the tangent space at p so that any $v\in T_pM$ can be written is terms of them. The texts further state that a $\partial/\partial x_k$ is a unit basis vector.
My question is why are the $\partial/\partial x_i$ unit vectors? I can see how they would give the canonical basis and be unit vectors if they acted on the standard coordinate curves $(x_1, …, x_n)$, but don't they act on arbitrary functions? Or do they only act on the coordinate curves?
I have a feeling that I'm missing something fundamental in my understanding.
Best Answer
It depends on the notation you are using. Let $(e_1,...,e_n)$ be the natural basis for $\mathbb{R}^n$.
One way to understand $\partial/\partial x_i$ is to imagine it as a derivation on the direction $e_i$ calculated at a point $p$. In this case, you can interpret $\partial/\partial x_i$ as the unit vector $e_i$.
To understand it better you should show that any given derivation on $C^\infty(V,\mathbb{R})$ (i.e. an element of $T_p \mathbb{R}^n$) is actually a linear combination of $\partial/\partial x_1, ..., \partial/\partial x_n$.