Vector-Valued Functions

The central idea

A vector-valued function $\mathbf r:I\subset\mathbb R\to\mathbb R^n$ assigns a vector to each parameter value. Limits, continuity, and derivatives are defined componentwise in finite dimensions. Its image is a parametrized curve, while the vector $\mathbf r(t)$ may represent position. Input dimension one distinguishes it from a vector field on space.

Definitions, hypotheses, and notation

Continuity requires all components to approach their corresponding vector entries; a failure in one component prevents the vector limit. Norm identities such as $\|\mathbf r\|^2=\mathbf r\cdot\mathbf r$ connect scalar and vector calculus. A constant norm constrains motion to a sphere but does not make the vector constant.

Different parametrizations can share the same image yet encode different velocities. For geometric quantities like arc length, regular monotone reparametrizations preserve the result. For physical motion, the time parametrization is essential because it determines velocity and acceleration.

Conceptual meaning

Several coordinate quantities evolve in synchrony under one parameter. Component graphs show how each coordinate changes with time; the combined image shows the trajectory. The parameter interval determines which portion and orientation are included.

A dependable method and decision rule

1. State the common domain of all components.
2. Evaluate or take limits componentwise.
3. Plot key parameter points in output space.
4. Record orientation from increasing parameter and detect repeated values.
5. Differentiate componentwise for velocity or tangent analysis.

Fully worked example

Common mistakes and why they fail