What does it mean to say $\partial/\partial x, \partial/\partial y, \partial/\partial z$ are "basis" for $T_p{\mathbb{R}^3}$? I know that $T_p{\mathbb{R}^3}$ is a vector space, but I'm not sure what the exact definition of it is. When we say $\partial/\partial x, \partial/\partial y, \partial/\partial z$ are "basis" for $T_p{\mathbb{R}^3}$, is it intrinsically following from the "definition" of $T_p{\mathbb{R}^3}$? Or do I need to somehow consult geometric interpretation? Is there any concrete definition of $T_p{\mathbb{R}^3}$? Is it linked to the geometric pictures? How do I show, from here, that $\partial/\partial \rho, \partial/\partial \phi, \partial/\partial \theta$, where $\rho,\phi,\theta$ are spherical coordinates also form a "basis" for $T_p{\mathbb{R}^3}$? I guess I can show it by literally calculating the relationship between them, but is there any better way to see it?
 A: Actually the basis of $T_p{\mathbb{R}^3}$ you're looking for is $\partial/\partial x|_p, \partial/\partial y|_p, \partial/\partial z|_p$. It's like this:
$T_p{\mathbb{R}^3}$ consists of tangent vectors to the point $p$ in $\mathbb R^3$. So literally it's a bunch of one-sided arrows that begin at $p$ and then point somewhere in $\mathbb R^3$.
Now, as $\mathbb R$-vector spaces, $T_p{\mathbb{R}^3}$ is $\mathbb R$-isomorphic to $\mathcal D_p{\mathbb{R}^3}$, where the latter consists of derivations at $p$, defined as maps $D: C_p^{\infty}(\mathbb R^3) \ \to \mathbb R$ that are $\mathbb R$-linear and satisfy the Leibniz rule $D(fg) = g(p) D(f)  + f(p) D(g)$, for all $f,g$ s.t. their germs are $[f],[g] \in C_p^{\infty}(\mathbb R^3)$.
What are these mysterious derivations at $p$? What do they look like? How do we describe them? What do we know about them? It's such an intimidating concept! And how do they relate to the arrows earlier?
Actually these derivations at $p$ are $\mathbb R$-linear combinations of :
$$\{\partial/\partial x|_p, \partial/\partial y|_p, \partial/\partial z|_p\}$$
So, the underlying set of  $\mathcal D_p{\mathbb{R}^3}$ is
$$\{a\partial/\partial x|_p+b\partial/\partial y|_p+c\partial/\partial z|_p\}_{a,b,c \in \mathbb R}$$
So for example we have some $p \in \mathbb R^3$ and then $f: \mathbb R^3 \to \mathbb R$, $f \in C^{\infty}(\mathbb R^3)$. Obtain the germ $[f] \in C_p^{\infty}(\mathbb R^3)$. Let's say $(a,b,c)=(1,2,3)$. Then for $D=\partial/\partial x|_p+2\partial/\partial y|_p+3\partial/\partial z|_p$, we can do like $D(f) = \partial/\partial x|_p(f)+2\partial/\partial y|_p(f)+3\partial/\partial z|_p(f)$. But anyway, this operator $D \in \mathcal D_p{\mathbb{R}^3}$ corresponds to the arrow pointing in the direction $(1,2,3)$ and beginning at the point $p$.
So yeah the isomorphism is between a bunch of arrows and a bunch of operators.
