Unit 2 of 6

Vector-Valued Functions

  • 5 lessons
  • Parts 15 to 24

A vector function traces a curve in space as its parameter runs, the way a moving particle traces its path. This unit takes limits, derivatives and integrals of vector functions one component at a time and reads them as tangent vectors and accumulated displacement.

Arc length makes it possible to measure a curve and to parametrize it by distance traveled. Curvature and torsion describe how sharply a curve bends and twists, and the unit ends with velocity and acceleration split into tangential and normal parts.

Every lesson will be a pair of videos, a theory part that explains the idea and a practice part that works through problems. The videos are coming soon, and each lesson page already has the definitions and an example for every key term.

Curves and Their Calculus

  1. Vector Functions and Space Curves

    What a vector function is, how its domain, limits and continuity reduce to its components, and how it traces an oriented curve in space such as a helix or the intersection of two surfaces.

    Theory part 15Practice part 16

  2. Derivatives and Integrals of Vector Functions

    How to differentiate and integrate vector functions one component at a time, what the tangent vector and unit tangent of a regular curve are, and how the dot and cross product rules work.

    Theory part 17Practice part 18

  3. Arc Length and Reparametrization

    How the length of a space curve comes from its speed, why that length does not depend on how fast or in which direction the curve is traced, and how to switch to the arc length parameter.

    Theory part 19Practice part 20

Curvature and Motion

  1. Curvature, Torsion, and the Moving Frame

    How curvature measures turning per unit length, how the tangent, normal and binormal vectors form a moving frame, and how torsion measures twisting out of a plane, with the conditions each one needs.

    Theory part 21Practice part 22

  2. Velocity and Acceleration in Space

    How velocity and acceleration describe motion in space, how acceleration splits into a part that changes speed and a part that turns, and what central forces and the inverse-square law imply for orbits.

    Theory part 23Practice part 24