differential geometry w notes from teacher_Part_12

differential geometry w notes from teacher_Part_12 - 1.4....

Info iconThis preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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 , x ) be a local chart in W . If W U , 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 , x ) and ( U , x ) 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 t : U M be a local flow on a M such that ( p ) =...
View Full Document

Page1 / 2

differential geometry w notes from teacher_Part_12 - 1.4....

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online