Chapter 13: Vector Functions

13.1 Vector Functions and Space Curves

A vector-valued function is a function whose domain is a set of real numbers, and whose range is a set of vectors. For instance,

r(t)=\langle f(t),g(t),h(t) \rangle = f(t)\textbf{i} + g(t)\textbf{j}+h(t)\textbf{k}

Where $r(t)$ is the vector-valued function and $f,g,h$ are the component functions of $r$ . Often (for 3-dimensions), $r:\mathbb{R}\to \mathbb{R}^{3}$ .

Limits and Continuity

The limit for a vector-valued function is simply:

\lim_{ t \to a } r(t)=\langle \lim_{ t \to a } f(t),\lim_{ t \to a } g(t) ,\lim_{ t \to a } h(t) \rangle

And $r(t)$ is continuous at $t=a$ when

\lim_{ t \to a } r(t)=r(a)

Space Curves

Suppose $f,g,t$ are continuous, real-valued functions defined on an interval $\ell$ . Then the set of $C$ of all points $(x,y,z)$ in space described by

x=f(t)\qquad y=g(t)\qquad z=h(t)

as $t$ varies over $\ell$ is called a space curve.

13.2 Derivatives and Integrals of Vector Functions

Derivatives

The derivative is defined analogously to limits

r'(t)=\langle f'(t),g'(t),h'(t) \rangle

Derivative rules (e.g. Product Rule, Chain Rule, etc.) hold similarly, where it suffices to apply each rule independently for each dimension. There are, however, a few differences.

Let $f(t)$ be a real-valued function, and $u,v$ be vector-valued functions Then,