@KarlKronenfeld Some people talk math in here sometimes

I don't know how to invite someone

i think i figured it out

anyway, don't pay attention to this if you don't want to, just saying that there is a place to mess around and talk math if you want

Thanks :)

welcome @KarlKronenfeld

1. (physicist's definition) $T_pM = \{(x,v) \mid x : U \ni p \to \tilde U \subseteq \Bbb R^n, v \in \Bbb R^n\}/\sim$ where $(x,v) \sim (y,w) := w = D(y \circ x^{-1})(x(p))(v)$

2. (geometer's definition) $T_pM = \{c \in C^\infty((-\varepsilon,\varepsilon),M) \mid c(0) = p\} / \sim$ where $c \sim c' := \forall x : U \ni p \to \tilde U \subseteq \Bbb R^n, D(x \circ c)(0)(1) = D(x \circ c')(0)(1)$

3. (algebraist's definition) $T_pM = \{D_{p,v} \in C^\infty(M)^\ast \mid \forall f,g, D_{p,v}(fg) = f(p) D_{p,v}(g) + D_{p,v}(f) g(p)\}$

In 2, we only need to verify one chart: if $D(x \circ c)(0)(1) = D(x \circ c')(0)(1)$ then $D(y \circ c)(0)(1) = D(y \circ x^{-1} \circ x \circ c)(0)(1) = D(y \circ x^{-1})(x(p))(D(x \circ c)(0)(1)) = \cdots = D(y \circ c')(0)(1)$