math214:02-07
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| The zero of $\nabla f$ are called critical points of $f$. **Warning: ** for general smooth manifold, we do not have the notion of a gradient vector field for $f$, unless we are given a metric tensor. | The zero of $\nabla f$ are called critical points of $f$. **Warning: ** for general smooth manifold, we do not have the notion of a gradient vector field for $f$, unless we are given a metric tensor. | ||
| - | ==== Integral Curve ==== | + | ===== Commutator of Vector fields ===== |
| + | Given two vector fields, $X, Y$, we can define its commutator as follows, for any smooth function $f$, we define | ||
| + | $$ [X, Y]_p f = X_p Y_p (f) - Y_p X_p (f) $$ | ||
| + | It is an exercise to check that this $[X, Y]_p$ is indeed a derivation, hence $[X, Y]_p \in T_p M$. | ||
| + | Concretely, if we have coordinates $u_i$, we have $[\d_i, \d_j]=0$. | ||
| + | $$ [ \sum_i X^i \d_i , \sum_j Y^j \d_j ] = \sum_{i,j} X^i \d_i Y_j \d_j - Y_j \d_j X^i \d_i. $$ | ||
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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 **. | ||
| + | |||
| + | ===== 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. | ||
math214/02-07.1581060106.txt.gz · Last modified: 2020/02/06 23:21 by pzhou
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