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\(\ldots\) who by vigor of mind almost divine, the motions and figures of the planets, the paths of comets, and the tides of the seas first demonstrated.
—Newton’s Epitaph
Chapters 2 and 3 focused on real-valued functions. This chapter is largely concerned with vector-valued functions. We begin in the first section of this chapter with a continuation of our study of paths, adding applications of Newton’s second law. Then we study arc length of paths. Following this, we introduce the divergence and curl of a vector field which, in addition to the gradient, are basic operations in vector differential calculus. Then the geometry and calculus of the divergence and curl are studied. The associated integral calculus will be given in Chapter 8.