Difference between revisions of "Tangent space"

Revision as of 00:48, 5 April 2015 (view source) Alec (Talk | contribs) m ← Older edit		Revision as of 02:28, 5 April 2015 (view source) Alec (Talk | contribs) m Newer edit →
Line 1:		Line 1:
	I prefer to denote the tangent space (of a set {{M|A}} at a point {{M|p}}) by {{M|T_p(A)}} - as this involves the letter T for tangent however one author<ref>John M. Lee - Introduction to Smooth Manifolds - second edition</ref> uses {{M|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.		I prefer to denote the tangent space (of a set {{M|A}} at a point {{M|p}}) by {{M|T_p(A)}} - as this involves the letter T for tangent however one author<ref>John M. Lee - Introduction to Smooth Manifolds - second edition</ref> uses {{M|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.
−
−
−	{| class="wikitable" border="1"
−	|-
−	!rowspan="2"| Name
−	! Preferred form
−	! Alternate form
−	!rowspan="2"| Definition
−	|-
−	!colspan="2"| 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)
−	|}
	What is defined here may also be called the '''Geometric tangent space'''		What is defined here may also be called the '''Geometric tangent space'''

---

Revision as of 02:28, 5 April 2015

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^[1] 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

References

1. Jump up ↑ John M. Lee - Introduction to Smooth Manifolds - second edition