Morse–Smale system
Okay kiddo, so imagine you have a big mountain with a lot of different paths on it. Some paths might go up to the top of the mountain, some might go partially up and then come back down, and some might just lead to dead ends. But, we are interested in finding the easiest and quickest way to get to the top of the mountain.
That's where Morse-Smale systems come in. They are like a map that helps us figure out the best path to take.
First, we look at every point on the mountain and we give them each a different color based on their height. The lowest points are the darkest color and the highest points are the lightest color.
Next, we find all the paths that start at the dark points and end at the light points. These paths are called "flow lines." They represent how water would flow down the mountain if it rained.
Now, we look for spots on the mountain where two different flow lines meet. These spots are called "critical points." They are very important because they show us where the flow lines change direction.
Finally, we link all the critical points together into groups called "index pairs." Each index pair has a lower critical point and a higher critical point.
So, what does all of this mean? Well, by using Morse-Smale systems, we can figure out the easiest and quickest path to climb up the mountain. We just follow the path from the lower critical point to the higher critical point in each index pair.
It's like a treasure map that shows us where the gold is hidden. And now, we know exactly how to get there!
That's where Morse-Smale systems come in. They are like a map that helps us figure out the best path to take.
First, we look at every point on the mountain and we give them each a different color based on their height. The lowest points are the darkest color and the highest points are the lightest color.
Next, we find all the paths that start at the dark points and end at the light points. These paths are called "flow lines." They represent how water would flow down the mountain if it rained.
Now, we look for spots on the mountain where two different flow lines meet. These spots are called "critical points." They are very important because they show us where the flow lines change direction.
Finally, we link all the critical points together into groups called "index pairs." Each index pair has a lower critical point and a higher critical point.
So, what does all of this mean? Well, by using Morse-Smale systems, we can figure out the easiest and quickest path to climb up the mountain. We just follow the path from the lower critical point to the higher critical point in each index pair.
It's like a treasure map that shows us where the gold is hidden. And now, we know exactly how to get there!