Difference between revisions of "Differential of a smooth map"
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(Created page with " ==Definition== Given: * Two smooth manifolds {{M|(M,\mathcal{A})}} and {{M|(N,\mathcal{B})}} (which may have different dimensions) and are with or without...") |
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* (really hard to write - I want a <math>dF_p:v\mapsto(\text{something})</math>) | * (really hard to write - I want a <math>dF_p:v\mapsto(\text{something})</math>) | ||
'''Given:''' | '''Given:''' | ||
| − | * <math>v\in T_p(M)</math> | + | * <math>v\in T_p(M)</math> that is to say <math>v:C^\infty(M)\rightarrow\mathbb{R}</math> |
* <math>f\in C^\infty(N)</math> | * <math>f\in C^\infty(N)</math> | ||
The differential acts on {{M|f}} as follows: | The differential acts on {{M|f}} as follows: | ||
Latest revision as of 20:58, 13 April 2015
Definition
Given:
- Two smooth manifolds [ilmath](M,\mathcal{A})[/ilmath] and [ilmath](N,\mathcal{B})[/ilmath] (which may have different dimensions) and are with or without boundary
- A smooth map [ilmath]F:M\rightarrow N[/ilmath]
For each [ilmath]p\in M[/ilmath] we define a map
- [math]dF_p:T_p(M)\rightarrow T_{F(p)}N[/math] called the differential of [ilmath]F[/ilmath] at [ilmath]p[/ilmath][1] as
- (really hard to write - I want a [math]dF_p:v\mapsto(\text{something})[/math])
Given:
- [math]v\in T_p(M)[/math] that is to say [math]v:C^\infty(M)\rightarrow\mathbb{R}[/math]
- [math]f\in C^\infty(N)[/math]
The differential acts on [ilmath]f[/ilmath] as follows:
- [math]dF_p(v)(f) = v(f\circ F)[/math]
See also
References
- ↑ Introduction to smooth manifolds - John M Lee - Second Edition
