1 Definition If $r(t)=<f(t),g(t),h(t)>$, then $\lim_{t\to a}{r(t)}=$
$<\lim_{t\to a}{f(t)},\lim_{t\to a}{g(t)},\lim_{t\to a}{h(t)}>$
2A vector function r is continuous at a if
$\lim_{t \rightarrow a} r(t) = r(a)$
3 Theorem If $\textbf{r}(t) = <f(t),g(t),h(t)>$, where f, g, and h are differentiable functions, then $\textbf{r}'(t)=$
$<f'(t),g'(t),h'(t)>$
4 Differentiation Rules $\frac{d}{dt}[\textbf{u}(t)+\textbf{v}(t)]=$
$\textbf{u}'(t)+\textbf{v}'(t)$
5 Differentiation Rules $\frac{d}{dt}[c\textbf{u}(t)]=$
$c\textbf{u}'(t)$
6 Differentiation Rules $\frac{d}{dt}[f(t)\textbf{u}(t)]=$
$f'(t)\textbf{u}(t)+f(t)\textbf{u}'(t)$
7 Differentiation Rules $\frac{d}{dt}[\textbf{u}(t)\cdot\textbf{v}(t)]=$
$\textbf{u}'(t)\cdot\textbf{v}(t)+\textbf{u}(t)\cdot\textbf{v}'(t)$
8 Differentiation Rules $\frac{d}{dt}[\textbf{u}(t)\times\textbf{v}(t)]=$
$\textbf{u}'(t)\times\textbf{v}(t)+\textbf{u}(t)\times\textbf{v}'(t)$
9 Differentiation Rules $\frac{d}{dt}[\textbf{u}(f(t))]=$
$f'(t)\textbf{u}'(f(t))$
10If $|\textbf{r}(t) = c|$ (a constant), then $\textbf{r}'(t)$ is
orthogonal to $\textbf{r}(t)$ for all t
11Arc Length Formula
$L = \int_a^b{|\textbf{r}'(t)|} dt$
12 Definition Arc Length Function
$s(t)=\int_a^t{|\textbf{r}'(t)|}dt$
13 Definition Curvature of C is how quickly the curve changes direction at a given point
$\kappa(t) = \frac{\textbf{T}'(t)}{|\textbf{r}'(t)|}$
14 Theorem The curvature of the curve given by the vector function r is
$\kappa(t)=\frac {|\textbf{r}'(t) \times \textbf{r}''(t)|} { |\textbf{r}'(t)|^3} $
15Unit Tangent Vector
$\textbf{T}(t)=\frac{\textbf{r}'(t)}{|\textbf{r}'(t)|}$
16Unit Normal Vector
$N(t)=\frac {|\textbf{T}'(t)}{|T'(t)|} $
17 Definition Binormal Vector is perpendicular to both T and N
$\textbf{B}(t)= \textbf{T}(t)\times\textbf{N}(t)$
18 Definition The radius of the circle of curviture of ocilating plane is
$\frac{1}{\kappa}$
19 Definition The torsion of a curve is
$\tau(t)=-\frac{\textbf{B}'(t)\cdot\textbf{N}(t)}{|\textbf{r}'(t)|}$
20 Theorem The torsion of the curve given by the vector function r is
$\tau(t)=\frac{[\textbf{r}'(t)\times\textbf{r}''(t)]\textbf{r}'''(t)}{|\textbf{r}'(t)\times\textbf{r}''(t)|^2}$