Chapter 15. Angles
Upon completing the previous chapter on paths, we've accomplished the goals we set for ourselves! We learned to calculate the differential, and we've seen many geometric interpretations. We can analyze a graph's slope, bend, warp, gradient, etc. We can measure length and area using integrals. And we can understand the position, velocity, and speed of a point moving along a path. Sounds like it's about time to pack up and head home.
But wait! Before we call it quits, let's see what Calculus has to say about angles. So far, all of our tools have used standard \( (x, y) \)-coordinates. Metrics, vectors, rulers, and boxes, have all been written using \( x \) and \( y \) standards. But these tools are actually quite flexible. They can be used with non-standard coordinates, too.
In this last chapter we'll study angles in order to define polar coordinates. We'll investigate polar rulers and use them to measure the area of polar regions. The differential is a powerful idea that has applications in many contexts!