Chapter 9. Differentials as Metrics
In this chapter we'll continue our work analyzing surfaces. We've learned that a function's partials give us a surface's slopes, and higher partials let us calculate bend and warp. Our goal is to apply this material towards identifying low points and high points on a surface. To get there, however, we'll need to introduce a new type of object: metrics.
By studying metrics and their differentials, we'll learn how to identify the shape of a surface by comparing the surface to its tangent plane. A surface will typically peel away from its tangent plane in either a saddle-shaped fashion or a bowl-shaped fashion. Making this distinction is crucial for locating low points and high points on a surface.