Calculus is a branch of mathematics that, simply enough, lets people make calculations! Most of the time the calculations we'll make using Calculus will have something geometric to say. We'll learn how to calculate whether a curve is steep or shallow. We'll learn to analyze how bent and warped a surface is. And we'll learn to measure length and area in powerful new ways. Remarkably, all of these geometric ideas can be linked to a single algebraic idea: the differential.
In Part I, we'll learn to calculate the differential, and we'll see how we can apply the differential to study curves. Before we can define the differential, however, we'll need to know how to decompose functions. By learning about dependence, we'll see how complicated functions can be built from simpler functions.
In Part II, we'll look at metrics as a way study surfaces. Metrics will help us understand the shape of a surface by comparing the surface to a plane. The differential lives comfortably in the world of dependent metrics.
In Part III, we'll look at a few more types of object: rulers, vectors, and boxes. We'll use rulers to measure vectors, and we'll use dependent rulers to measure boxes. The differential also lives comfortably in the world of dependent rulers.
To get the most out of this text, you'll need to keep pen and paper nearby as you read. Following another person's calculations can get tedious quite quickly. Math is much more fun when you're sorting things out for yourself! Whether you have some background in Calculus already or you're entirely new to the subject, I hope that you find a few ideas that spark your interest and make you want to continue on exploring.