Chapter 12. Differentials as Rulers
When we studied metrics we learned to interpret the differential as a metric. We're now ready to give an alternative interpretation of the differential: as a ruler.
At first glance this should not actually make that much of a difference. Both metrics and rulers are tools that let us assess change. But rulers will allow us to make more sophisticated measurements. Using vectors and rulers together, we will learn how to definitively tell how steep a surface is at any point.
Moving beyond slope, polymetrics and polyrulers look quite different. Polymetrics allowed us to investigate how a surface bends and warps away from its tangent plane. Polyrulers have an entirely different purpose: they allow us to measure length and area. That said, we won't be measuring any length or area until Chapter 13. Boxes when begin our study of integrals. For now, our focus will be on taking the differential of a polyruler.