Cartan's theorems A and B
Okay kiddo, imagine you have a big map of your town with all the streets, buildings, parks and everything else. Now imagine you want to find the shortest route to the candy store. To do that, you need to know something about the roads and how they are connected. That's where Cartan's Theorems A and B come in.
Cartan's Theorems A and B are like secret codes that tell us how the roads, or in math terms, the "vector fields," are connected. Vector fields are arrows that show us which way things are moving, like wind or water currents.
Theorem A says that if you have a vector field and it's not changing too quickly, then you can follow that vector field to get from one point to another. It's kind of like following the road signs to get to the candy store.
But what if there's more than one way to get to the candy store? That's where Theorem B comes in. Theorem B says that if two vector fields start in the same place and end in the same place, and they change at the same rate, then there must be a path that connects those two vector fields. It's like saying that even if there are multiple roads that lead to the candy store, they all connect to each other in some way.
So, in math terms, Cartan's Theorems A and B help us figure out how the roads, or vector fields, are connected so we can find the shortest or most efficient way to get to where we want to go. It's like having a secret map that helps us find the candy store faster!
Cartan's Theorems A and B are like secret codes that tell us how the roads, or in math terms, the "vector fields," are connected. Vector fields are arrows that show us which way things are moving, like wind or water currents.
Theorem A says that if you have a vector field and it's not changing too quickly, then you can follow that vector field to get from one point to another. It's kind of like following the road signs to get to the candy store.
But what if there's more than one way to get to the candy store? That's where Theorem B comes in. Theorem B says that if two vector fields start in the same place and end in the same place, and they change at the same rate, then there must be a path that connects those two vector fields. It's like saying that even if there are multiple roads that lead to the candy store, they all connect to each other in some way.
So, in math terms, Cartan's Theorems A and B help us figure out how the roads, or vector fields, are connected so we can find the shortest or most efficient way to get to where we want to go. It's like having a secret map that helps us find the candy store faster!