Exterior derivative

The exterior derivative is like a special machine that takes a shape and spits out another shape that tells you information about the original shape. Imagine you have a balloon with a pattern on the surface. You can think of the pattern as a 2-dimensional shape. The exterior derivative machine takes this shape and turns it into a new shape that tells you how the pattern changes when you move around on the balloon. For example, if you move your finger across the pattern, the new shape from the machine will show you how the pattern changes in response to your finger's movement.

The exterior derivative machine works by looking at how the original shape changes as you move in different directions. For example, if you have a curve on the balloon, the machine will look at how the curve changes as you move along it. It will also look at how the curve changes as you move in different directions away from it. This information is used to create the new shape that tells you about the changes in the pattern in response to movement.

The exterior derivative is a really useful tool in mathematics because it can be used to study all sorts of shapes, not just patterns on balloons. Mathematicians use it to study things called differential forms, which are like special types of shapes that are used to describe things like flows of energy or information. By using the exterior derivative machine, they can understand how these forms change and interact with each other in really complex environments.