Sard's lemma

Sard's Lemma is a rule that tells us that most of the time, if you look at a map from one place to another place (we can call it a function), the places where the function doesn't change very much are not very important.

Let's imagine you have a map of your town, with all the streets and buildings drawn on it. You can use this map to get from your house to your school. Along the way, you may notice that some parts of the map are very detailed and show every little street and alley, while other parts don't have as much detail and just show the main roads.

Now, let's imagine that you make a rule that says every time you come to a part of the map where the streets start to look really similar, you'll just skip over it and only look at the parts of the map where the streets are very different.

This is kind of what Sard's Lemma does for us. If we take a look at a function (a map from one place to another), we can use Sard's Lemma to ignore the parts where the function doesn't change very much and only focus on the parts where it changes a lot. This helps us simplify the function and make it easier to work with.

So, in short: Sard's Lemma tells us that we can ignore the unimportant parts of a function and focus on the parts where it changes the most.