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 (M,A) and (N,B) (which may have different dimensions) and are with or without boundary
- A smooth map F:M→N
For each p∈M we define a map
- dFp:Tp(M)→TF(p)Ncalled the differential of F at p[1] as
- (really hard to write - I want a dFp:v↦(something))
Given:
- v∈Tp(M)that is to say v:C∞(M)→R
- f∈C∞(N)
The differential acts on f as follows:
- dFp(v)(f)=v(f∘F)
See also
References
- Jump up ↑ Introduction to smooth manifolds - John M Lee - Second Edition
