Calculus Volume 3

# 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
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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