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Calculus Volume 3

# Key Terms

### Key Terms

acceleration vector
the second derivative of the position vector
arc-length function
a function $s(t)s(t)$ 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 $r(t)=f(t)i+g(t)jr(t)=f(t)i+g(t)j$ are $f(t)f(t)$ and $g(t),g(t),$ and the component functions of the vector-valued function $r(t)=f(t)i+g(t)j+h(t)kr(t)=f(t)i+g(t)j+h(t)k$ are $f(t),f(t),$ $g(t)g(t)$ and $h(t)h(t)$
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 $r(t)r(t)$ is $r′(t)=limΔt→0r(t+Δt)−r(t)Δt,r′(t)=limΔt→0r(t+Δt)−r(t)Δt,$ 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 $r(t)r(t)$ has a limit L as t approaches a if $limt→a|r(t)−L|=0limt→a|r(t)−L|=0$
normal component of acceleration
the coefficient of the unit normal vector N when the acceleration vector is written as a linear combination of $TT$ and $NN$
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 $(f(t),g(t))(f(t),g(t))$ together with their defining parametric equations $x=f(t)x=f(t)$ and $y=g(t)y=g(t)$
principal unit normal vector
a vector orthogonal to the unit tangent vector, given by the formula $T′(t)‖T′(t)‖T′(t)‖T′(t)‖$
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 $r(t)r(t)$ is differentiable with a non-zero derivative
space curve
the set of ordered triples $(f(t),g(t),h(t))(f(t),g(t),h(t))$ together with their defining parametric equations $x=f(t),x=f(t),$ $y=g(t)y=g(t)$ and $z=h(t)z=h(t)$
tangent vector
to $r(t)r(t)$ at $t=t0t=t0$ any vector v such that, when the tail of the vector is placed at point $r(t0)r(t0)$ 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 $TT$ and $NN$
vector parameterization
any representation of a plane or space curve using a vector-valued function
vector-valued function
a function of the form $r(t)=f(t)i+g(t)jr(t)=f(t)i+g(t)j$ or $r(t)=f(t)i+g(t)j+h(t)k,r(t)=f(t)i+g(t)j+h(t)k,$ where the component functions f, g, and h are real-valued functions of the parameter t
velocity vector
the derivative of the position vector
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