differential geometry w notes from teacher_Part_11

differential geometry w notes from teacher_Part_11 - 1.4...

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Unformatted text preview: 1.4. VECTOR FIELDS AND FLOWS 21 1.4 Vector Fields and Flows 1.4.1 Vector Fields in R n • Let x = ( x j ) ∈ R n be a point in R n and v ( x ) ∈ T x R n be a vector at x given by v = n X j = 1 v j ( x ) ∂ j , where ∂ j = ∂/∂ x j and the components v j ( x ) are smooth (or just di ff eren- tiable) functions of x . Then v ( x ) is called a vector field in R n . • Let t ∈ (- ε, ε ) and ϕ t : R n → R n be a family of di ff eomorphisms such that ϕ = Id and for any t , t 1 , t 2 ∈ (- ε, ε ) such that- t , t 1 + t 2 ∈ (- ε, ε ) there holds ϕ t 1 ◦ ϕ t 2 = ϕ t 1 + t 2 and ϕ- t = ϕ- 1 t . Such a one-parameter group of di ff eomorphisms is called a flow on R n . • A flow ϕ t defines a vector field v by v ( x ) = d dt ϕ t ( x ) t = with the components v j ( x ) = dx j dt . • The corresponding di ff erential operator v ( f )( x ) = d dt f ( ϕ t ( x )) t = = n X j = 1 v j ( x ) ∂ f ∂ x j is the derivative along the streamline of the flow through the point...
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This note was uploaded on 11/26/2011 for the course MAT 4821 taught by Professor Wong during the Spring '10 term at FSU.

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differential geometry w notes from teacher_Part_11 - 1.4...

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