# first variation of area formula

Okay, so imagine you have a picture of a little pond with some lily pads floating on the surface. You know that the surface of the pond is covered with a green film of tiny plants called algae.

Now, let's say that I give you a special ruler that measures the width of the algae film as you move it around on the surface of the pond. You start at one point and move the ruler a little bit, then measure the width, then move it again, then measure again, and so on until you've gone all the way around the pond.

What you'll notice is that the width of the film changes as you move the ruler in different directions. Sometimes it gets wider, sometimes it gets narrower. But if you add up all the little changes in width for each tiny movement of the ruler, you'll get the total change in the surface area of the pond. This is called the "first variation of area formula."

Basically, what the formula is doing is calculating how much the area of a shape changes when you make small, incremental changes to its boundaries. It's like taking a bunch of little snapshots of the shape and then measuring how much it changes from one snapshot to the next.

This formula is really useful in lots of different fields, from physics to engineering to geometry. And even though it might sound complicated, it's really just a way of measuring the changes in a shape's area as you move it around.