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Calculus II: Vector Calculus and Multivariable Analysis / Vectors and Curves in Space

Vector-Valued Functions: Limits, Derivatives, Velocity, and Acceleration

A stock price path StS_t, a bond yield curve traced over time, a portfolio's risk-factor trajectory — all are curves in multi-dimensional space. The mathematical object that encodes such a trajectory is a vector-valued function r(t)\mathbf{r}(t), which assigns a vector (a point in R3\mathbb{R}^3 or Rn\mathbb{R}^n) to each instant tt. The derivative r′(t)\mathbf{r}'(t) is the velocity vector at time tt: it points in the instantaneous direction of motion and its magnitude is the instantaneous speed. This derivative, computed componentwise, is the gateway to the curvature and arc length of the next lesson, and to the multivariable differential calculus of Topic 2 — where r′(t)\mathbf{r}'(t) becomes the Jacobian column.
Learning objectives. By the end of this note you should be able to:
  • define and evaluate limits of vector-valued functions componentwise;
  • state and apply the product rules for dot products and cross products of differentiable curves;
  • differentiate a vector-valued function r(t)=⟨r1(t),r2(t),r3(t)⟩\mathbf{r}(t) = \langle r_1(t), r_2(t), r_3(t)\rangle by differentiating each component;
  • compute the velocity vector v(t)=r′(t)\mathbf{v}(t) = \mathbf{r}'(t), the speed v(t)=∥r′(t)∥v(t) = \|\mathbf{r}'(t)\|, and the instantaneous direction of motion;
  • write the parametric equations of the tangent line to a curve at a given point;
  • compute the acceleration vector a(t)=r′′(t)\mathbf{a}(t) = \mathbf{r}''(t) and identify its normal and tangential components.

In this lesson

    1. Parametric Curves and Vector-Valued Functions
    1. Limits and Continuity
    1. Derivatives of Vector-Valued Functions
    1. Velocity, Speed, and Tangent Lines
    1. Acceleration
    1. Integration of Vector-Valued Functions

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