First variation

Okay kiddo, let's talk about something called "first variation." Think about when you're playing with play dough and you make a big ball. If you squish it a little bit, it might change shape a little bit too, right? Well, that's kind of like what first variation means in math.

When we do math, sometimes we have a function that tells us how something changes over time, or how something is related to something else. But what if we change a little tiny bit of that function, like squeezing that play dough ball? That's called the first variation. We're just looking at how the function changes a teensy bit when we make a small change to it.

Why do we care about first variation? Well, sometimes we need to figure out how sensitive or responsive something is to change. For example, let's say you're driving a car and you need to turn the steering wheel to avoid hitting something. You want to know how much you need to turn it to make the car go where you want it to. That's where the first variation comes in. We can use first variation to figure out how much a system will change in response to a small change in its inputs.

So, basically, first variation is a way of measuring how much something changes when we make a little change to it. It helps us understand how sensitive or responsive systems are to changes, and can be really useful in a lot of different kinds of math problems. And that's the basic idea of first variation!