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2008-05-18T20:10:33Z

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Vipul: New page: {{basic construct on dm}} ==Definition== Let $M$ be a differential manifold and $p \in M$ . The ''tangent space'' at $p$ to $M$ , denoted
2008-04-05T19:59:30Z

New page: {{basic construct on dm}} ==Definition== Let <math>M</math> be a differential manifold and <math>p \in M</math>. The ''tangent space'' at <math>p</math> to <math>M</math>, denoted <m...

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{{basic construct on dm}}

==Definition==

Let <math>M</math> be a [[differential manifold]] and <math>p \in M</math>. The ''tangent space'' at <math>p</math> to <math>M</math>, denoted <math>T_p(M)</math>, is defined in a number of equivalent ways.

===In the language of derivations===

Consider the algebra <math>C^\infty(M)</math> of infinitely differentiable functions on <math>M</math>. Then, the tangent space <math>T_p(M)</math> is defined as the vector space of all derivations:

<math>d: C^\infty(M) \to \R</math>

By <math>d</math> being a derivation, we mean:

* <math>d</math> is a <math>\R</math>-linear map
* Given functions <math>f,g \in C^\infty(M)</math> we have:

<math>d(fg) = (df)(g(p)) + f(p)(dg)</math>

===In the language of curves===

The tangent space <math>T_p(M)</math> is defined, as a ''set'', as the quotient of the set of all curves <math>\gamma: (-1,1) \to M</math> where <math>\gamma(0) = p</math>, by the following equivalence relation: <math>\gamma_1 = \gamma_2</math> if, under the mapping to an open subset in <math>\R^n</math> via a coordinate chart, <math>\gamma_1'(0) = \gamma_2'(0)</math>.

With this definition, addition is not very clear.

===In the language of local coordinate charts===

In this language, the tangent space is defined as the tangent space to its image in <math>\R^n</math> under a local coordinate chart. {{fillin}}