Tangent space

I prefer to denote the tangent space (of a set [ilmath]A[/ilmath] at a point [ilmath]p[/ilmath]) by [ilmath]T_p(A)[/ilmath] - as this involves the letter T for tangent however one author uses [ilmath]T_p(A)[/ilmath] as Set of all derivations at a point - the two are indeed isomorphic but as readers will know - I do not see this as an excuse.

Name	Preferred form	Alternate form	Definition
	example
Tangent space	[math]T_p(A)[/math] [math]T_p(\mathbb{R}^n)[/math]	[math]A_p[/math] [math]\mathbb{R}^n_p[/math]	[math]=\left\{(p,v)|v\in A\right\}[/math]
Set of all derivations at a point	[math]\mathcal{D}_p(A)[/math]	[math]T_p(A)[/math]	(see page)

Definition

It is the set of arrows at a point, the set of all directions essentially. As the reader knows, a vector is usually just a direction, we keep track of tangent vectors and know them to be "tangent vectors at t" or something similar. A tangent vector is actually a point with an associated direction.

Euclidean (motivating) definition

We define [math]T_p(\mathbb{R}^n)=\left\{(p,v)|v\in\mathbb{R}^n\right\}[/math]

References

---

TODO:

---