math214:02-07
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| It is an exercise to check that $[ [X, Y], Z] + [ [Y, Z], X ] + [ [ Z, X], Y ] = 0$. Hence the space of vector fields forms a ** Lie algebra **. | It is an exercise to check that $[ [X, Y], Z] + [ [Y, Z], X ] + [ [ Z, X], Y ] = 0$. Hence the space of vector fields forms a ** Lie algebra **. | ||
| - | ==== Integral Curve ==== | + | ===== Integral Curve ===== |
| Let $X$ be a smooth vector field on $M$. Let $p \in M$ be any points. And integral curve of $X$ through $p$ is a map | Let $X$ be a smooth vector field on $M$. Let $p \in M$ be any points. And integral curve of $X$ through $p$ is a map | ||
| $$ \gamma: (a, b) \mapsto M $$ | $$ \gamma: (a, b) \mapsto M $$ | ||
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| If we work in coordinate near $p$, then finding an integral curve through $p$ is equivalent to solving an ODE. By the fundamental theorem of ODE, there exists an $\epsilon> | If we work in coordinate near $p$, then finding an integral curve through $p$ is equivalent to solving an ODE. By the fundamental theorem of ODE, there exists an $\epsilon> | ||
| - | ==== Flow ==== | + | ===== Flow ===== |
| Given an integral curve through a point $p$, we can define the motion of $p$ for some small interval of time $t$. If we consider the motion of all the points, we get a flow on $M$. However, there is subtlety that the flow may not exist for arbitrary long time. | Given an integral curve through a point $p$, we can define the motion of $p$ for some small interval of time $t$. If we consider the motion of all the points, we get a flow on $M$. However, there is subtlety that the flow may not exist for arbitrary long time. | ||
math214/02-07.1581060541.txt.gz · Last modified: 2020/02/06 23:29 by pzhou
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