Chapter 13: Vector Functions

Students learn how to graph space curves and reparameterize them for different motion descriptions. Calculus concepts are extended to vector functions, with derivatives representing velocity vectors and integrals representing displacement and position. The chapter covers tangent, normal, and binormal vectors, along with curvature, to describe the geometry of a curve in space. Arc length and unit tangent vectors are computed, leading to the Frenet–Serret formulas for motion along a curve. Motion in space is analyzed through velocity, speed, and acceleration, including decomposing acceleration into tangential and normal components. Applications include modeling planetary motion, particle trajectories, and objects moving along paths influenced by forces. By the end, students can differentiate and integrate vector functions, analyze curve geometry, and apply these tools to problems in physics, engineering, and computer graphics.