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.

What is defined here may also be called the Geometric tangent space

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 $$T_p(\mathbb{R}^n)=\left\{(p,v)|v\in\mathbb{R}^n\right\}$$

Generally then we may say: $$T_p(A)=\left\{(p,v)|v\in A\right\}$$

Notation

A tangent vector (often $v$ is used) shall be left as just $v$ if the point to which it is a tangent to is implicit (ie "$v$ is a tangent at $p$")

Rather than writing $(p,v)$ we may write:

• $v$ (if it is implicitly understood that this is a tangent to the point $p$)
• $v_a$
• $$v|_a$$

Why ordered pairs

Ordered pairs are used because now the tangent space at two distinct points are disjoint sets, that is $$\alpha\ne\beta\implies T_\alpha(A)\cap T_\beta(A)=\emptyset$$

Vector space

$$T_p(A)$$ is a vector space when equipped with the following definitions:

• $v_a+w_a=(v+w)_a$
• $c(v_a)=(cv)_a$

It is easily seen that the basis for this is the standard basis $$\{e_1|_p,\cdots, e_n|_p\}$$ and that the tangent space $T_p(A)$ is basically just a copy of $A$

See also

• Set of all derivations at a point
• The tangent space and derivations at a point are isomorphic