Introduction

In this chapter, we study vector-valued functions and their derivatives, and we use them to analyze curves and motion in three-space. Although many techniques from single-variable calculus carry over to the vector setting, there is an important new aspect to the derivative. A real-valued function \(f(x)\) can change in just one of two ways: It can increase or decrease. By contrast, a vector-valued function can change not just in magnitude but also in direction, and the rate of change is not a single number but is itself a vector. To develop these new concepts, we begin with an introduction to vector-valued functions.

DNA polymers form helical curves whose spatial orientation influences their biochemical properties.