Tangent space

I prefer to denote the tangent space (of a set $A$ at a point $p$) by $T_p(A)$ - as this involves the letter T for tangent however one author uses $T_p(A)$ 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	$$T_p(A)$$ $$T_p(\mathbb{R}^n)$$	$$A_p$$ $$\mathbb{R}^n_p$$	$$=\left\{(p,v)|v\in A\right\}$$
Set of all derivations at a point	$$\mathcal{D}_p(A)$$	$$T_p(A)$$	(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 $$T_p(\mathbb{R}^n)=\left\{(p,v)|v\in\mathbb{R}^n\right\}$$

References

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TODO:

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