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Unformatted text preview: 1.4. VECTOR FIELDS AND FLOWS 23 • The integral curves exist only for a small time. • If the vector field is not di ff erentiable, then the integral curve is not unique. 1.4.2 Vector Fields on Manifolds • Let W be an open subset of a mnifold M and v be a smooth vector field on W . • Let ( U α , x α ) be a local chart in W . • If W ⊂ U α , then one can proceed as in R n . • If W is not contained in a single chart, then we choose a cover of W and proceed as follows. Let p ∈ W and ( U α , x α ) and ( U β , x β ) be two charts covering p . • Then the integral curves in both local coordinate systems have the same meaning and define a unique integral curve in M . This defines a local flow on W in M . We just need to check that if the flow equations are satisfied in one coordinate system, then they are satisfied in another coordinate system. 1.4.3 Straightening Flows • Let U be an open set in a manifold M and ϕ t : U → M be a local flow on a M such that ϕ ( p ) =...
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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.
- Spring '10