Key Terms

acceleration vector

the second derivative of the position vector

arc-length function

a function $s\left(t\right)$ that describes the arc length of curve C as a function of t

arc-length parameterization

a reparameterization of a vector-valued function in which the parameter is equal to the arc length

binormal vector

a unit vector orthogonal to the unit tangent vector and the unit normal vector

component functions

the component functions of the vector-valued function $\text{r}\left(t\right)=f\left(t\right)\text{i}+g\left(t\right)\text{j}$ are $f\left(t\right)$ and $g\left(t\right),$ and the component functions of the vector-valued function $\text{r}\left(t\right)=f\left(t\right)\text{i}+g\left(t\right)\text{j}+h\left(t\right)\text{k}$ are $f\left(t\right),$ $g\left(t\right)$ and $h\left(t\right)$

curvature

the derivative of the unit tangent vector with respect to the arc-length parameter

definite integral of a vector-valued function

the vector obtained by calculating the definite integral of each of the component functions of a given vector-valued function, then using the results as the components of the resulting function

derivative of a vector-valued function

the derivative of a vector-valued function $\text{r}\left(t\right)$ is $r^{\prime }\left(t\right)=\underset{\text{Δ}t\to 0}{\text{lim}}\frac{\text{r}\left(t+\text{Δ}t\right)-\text{r}\left(t\right)}{\text{Δ}t},$ provided the limit exists

Frenet frame of reference

(TNB frame) a frame of reference in three-dimensional space formed by the unit tangent vector, the unit normal vector, and the binormal vector

helix

a three-dimensional curve in the shape of a spiral

indefinite integral of a vector-valued function

a vector-valued function with a derivative that is equal to a given vector-valued function

Kepler’s laws of planetary motion

three laws governing the motion of planets, asteroids, and comets in orbit around the Sun

limit of a vector-valued function

a vector-valued function $\text{r}(t)$ has a limit L as t approaches a if $\underset{t\to a}{\text{lim}}\left|\text{r}\left(t\right)-\text{L}\right|=0$

normal component of acceleration

the coefficient of the unit normal vector N when the acceleration vector is written as a linear combination of $\text{T}$ and $\text{N}$

normal plane

a plane that is perpendicular to a curve at any point on the curve

osculating circle

a circle that is tangent to a curve C at a point P and that shares the same curvature

osculating plane

the plane determined by the unit tangent and the unit normal vector

plane curve

the set of ordered pairs $\left(f\left(t\right),g\left(t\right)\right)$ together with their defining parametric equations $x=f\left(t\right)$ and $y=g\left(t\right)$

principal unit normal vector

a vector orthogonal to the unit tangent vector, given by the formula $\frac{T^{\prime }\left(t\right)}{\Vert T^{\prime }\left(t\right)\Vert }$

principal unit tangent vector

a unit vector tangent to a curve C

projectile motion

motion of an object with an initial velocity but no force acting on it other than gravity

radius of curvature

the reciprocal of the curvature

reparameterization

an alternative parameterization of a given vector-valued function

smooth

curves where the vector-valued function $\text{r}(t)$ is differentiable with a non-zero derivative

space curve

the set of ordered triples $\left(f\left(t\right),g\left(t\right),h\left(t\right)\right)$ together with their defining parametric equations $x=f\left(t\right),$ $y=g\left(t\right)$ and $z=h\left(t\right)$

tangent vector

to $\text{r}\left(t\right)$ at $t=t_0$ any vector v such that, when the tail of the vector is placed at point $\text{r}\left(t_0\right)$ on the graph, vector v is tangent to curve C

tangential component of acceleration

the coefficient of the unit tangent vector T when the acceleration vector is written as a linear combination of $\text{T}$ and $\text{N}$

vector parameterization

any representation of a plane or space curve using a vector-valued function

vector-valued function

a function of the form $\text{r}(t)=f\left(t\right)\text{i}+g\left(t\right)\text{j}$ or $\text{r}\left(t\right)=f\left(t\right)\text{i}+g\left(t\right)\text{j}+h\left(t\right)\text{k},$ where the component functions f, g, and h are real-valued functions of the parameter t

velocity vector

the derivative of the position vector