Essential Concepts

Introduction to Vector-Valued Functions

• A vector-valued function is a function of the form [latex]{\bf{r}}(t)=f(t){\bf{i}}+g(t){\bf{j}}[/latex] or [latex]{\bf{r}}(t)=f(t){\bf{i}}+g(t){\bf{j}}+h(t){\bf{k}}[/latex] where the component functions [latex]f[/latex], [latex]g[/latex], and [latex]h[/latex] are real-valued functions of the parameter [latex]t[/latex].
• The graph of a vector-valued function of the form [latex]{\bf{r}}(t)=f(t){\bf{i}}+g(t){\bf{j}}[/latex] is called a plane curve. The graph of a vector-valued function of the form [latex]{\bf{r}}(t)=f(t){\bf{i}}+g(t){\bf{j}}+h(t){\bf{k}}[/latex] is called a space curve.
• It is possible to represent an arbitrary plane curve by a vector-valued function.
• To calculate the limit of a vector-valued function, calculate the limits of the component functions separately.

Calculus of Vector-Valued Functions

• To calculate the derivative of a vector-valued function, calculate the derivatives of the component functions, then put them back into a new vector-valued function.
• Many of the properties of differentiation from the Calculus I: Derivatives also apply to vector-valued functions.
• The derivative of a vector-valued function [latex]{\bf{r}}(t)[/latex] is also a tangent vector to the curve. The unit tangent vector [latex]{\bf{T}}(t)[/latex] is calculated by dividing the derivative of a vector-valued function by its magnitude.
• The antiderivative of a vector-valued function is found by finding the antiderivatives of the component functions, then putting them back together in a vector-valued function.
• The definite integral of a vector-valued function is found by finding the definite integrals of the component functions, then putting them back together in a vector-valued function.

Motion in Space

• If [latex]{\bf{r}}(t)[/latex] represents the position of an object at time [latex]t[/latex], then [latex]{\bf{r}}^{\prime}(t)[/latex] represents the velocity and [latex]{\bf{r}}^{\prime\prime}(t)[/latex] represents the acceleration of the object at time [latex]t[/latex]. The magnitude of the velocity vector is speed.
• The acceleration vector always points toward the concave side of the curve defined by [latex]{\bf{r}}(t)[/latex]. The tangential and normal components of acceleration [latex]a_{\bf{T}}[/latex] and [latex]a_{\bf{N}}[/latex] are the projections of the acceleration vector onto the unit tangent and unit normal vectors to the curve.
• Kepler’s three laws of planetary motion describe the motion of objects in orbit around the Sun. His third law can be modified to describe motion of objects in orbit around other celestial objects as well.
• Newton was able to use his law of universal gravitation in conjunction with his second law of motion and calculus to prove Kepler’s three laws.

Arc Length and Curvature

• The arc-length function for a vector-valued function is calculated using the integral formula [latex]s(t)=\displaystyle\int_{a}^{t} \parallel{\bf{r}}^{\prime}(u)\parallel{du}[/latex]. This formula is valid in both two and three dimensions.
• The curvature of a curve at a point in either two or three dimensions is defined to be the curvature of the inscribed circle at that point. The arc-length parameterization is used in the definition of curvature.
• There are several different formulas for curvature. The curvature of a circle is equal to the reciprocal of its radius.
• The principal unit normal vector at [latex]t[/latex] is defined to be [latex]{\bf{N}}(t)=\dfrac{{\bf{T}}^{\prime}(t)}{\parallel{\bf{T}}^{\prime}(t)\parallel}[/latex].
• The binormal vector at [latex]t[/latex] is defined as [latex]{\bf{B}}(t)={\bf{T}}(t)\times{\bf{N}}(t)[/latex], where [latex]{\bf{T}}(t)[/latex] is the unit tangent vector.
• The Frenet frame of reference is formed by the unit tangent vector, the principal unit normal vector, and the binormal vector.
• The osculating circle is tangent to a curve at a point and has the same curvature as the tangent curve at that point.

Key Equations

• Vector-valued function
  [latex]{\bf{r}}(t)=f(t){\bf{i}}+g(t){\bf{j}}[/latex] or [latex]{\bf{r}}(t)=f(t){\bf{i}}+g(t){\bf{j}}+h(t){\bf{k}}[/latex], or [latex]{\bf{r}}(t)=\langle{f(t),g(t)}\rangle[/latex] or [latex]{\bf{r}}(t)=\langle{f(t),g(t),h(t)}\rangle[/latex]
• Limit of a vector-valued function
  [latex]\underset{t\to{a}}{\lim}{\bf{r}}(t)=\left[\underset{t\to{a}}{\lim}f(t)\right]{\bf{i}}+\left[\underset{t\to{a}}{\lim}g(t)\right]{\bf{j}}[/latex] or [latex]\underset{t\to{a}}{\lim}{\bf{r}}(t)=\left[\underset{t\to{a}}{\lim}f(t)\right]{\bf{i}}+\left[\underset{t\to{a}}{\lim}g(t)\right]{\bf{j}}+\left[\underset{t\to{a}}{\lim}h(t)\right]{\bf{k}}[/latex]
• Derivative of a vector-valued function
  [latex]{\bf{r}}^{\prime}(t)=\underset{\Delta{t}\to{0}}{\lim}\frac{{\bf{r}}(t+\Delta{t})-{\bf{r}}(t)}{\Delta{t}}[/latex]
• Principal unit tangent vector
  [latex]{\bf{T}}(t)=\dfrac{{\bf{r}}^{\prime}(t)}{\parallel{\bf{r}}^{\prime}(t)\parallel}[/latex]
• Indefinite integral of a vector-valued function
  [latex]\displaystyle\int [f(t){\bf{i}}+g(t){\bf{j}}+h(t){\bf{k}}]dt=\left[\displaystyle\int f(t)dt\right]{\bf{i}}+\left[\displaystyle\int g(t)dt\right]{\bf{j}}+\left[\displaystyle\int h(t)dt\right]{\bf{k}}[/latex]
• Definite integral of a vector-valued function
  [latex]\displaystyle\int_{a}^{b} [f(t){\bf{i}}+g(t){\bf{j}}+h(t){\bf{k}}]dt=\left[\displaystyle\int_{a}^{b} f(t)dt\right]{\bf{i}}+\left[\displaystyle\int_{a}^{b} g(t)dt\right]{\bf{j}}+\left[\displaystyle\int_{a}^{b} h(t)dt\right]{\bf{k}}[/latex]
• Velocity
  [latex]{\bf{v}}(t)={\bf{r}}^{\prime}(t)[/latex]
• Acceleration
  [latex]{\bf{a}}(t)={\bf{v}}^{\prime}(t)={\bf{r}}^{\prime\prime}(t)[/latex]
• Speed
  [latex]v(t)=\parallel{\bf{v}}(t)\parallel=\parallel{\bf{r}}^{\prime}(t)\parallel=\frac{ds}{dt}[/latex]
• Tangential component of acceleration
  [latex]a_{\bf{T}}={\bf{a}}\cdot{\bf{T}}=\frac{\bf{v}\cdot\bf{a}}{\parallel{\bf{v}}\parallel}[/latex]
• Normal component of acceleration
  [latex]a_{\bf{N}}={\bf{a}}\cdot{\bf{N}}=\frac{\parallel{\bf{v}}\times{\bf{a}}\parallel}{\parallel{\bf{v}}\parallel}=\sqrt{\parallel{\bf{a}}\parallel^{2}-{a_{\bf{T}}}^{2}}[/latex]
• Arc length of space curve
  [latex]s={\displaystyle\int_{a}^{b}} \sqrt{\left[f^{\prime}(t)\right]^{2}+\left[g^{\prime}(t)\right]^{2}+\left[h^{\prime}(t)\right]^{2}} dt =\displaystyle\int_{a}^{b} \parallel{\bf{r}}^{\prime}(t)\parallel{dt}[/latex]
• Arc-length function
  [latex]s(t)={\displaystyle\int_{a}^{t}} \sqrt{\left(f^{\prime}(u)\right)^{2}+\left(g^{\prime}(u)\right)^{2}+\left(h^{\prime}(u)\right)^{2}} du[/latex] or [latex]s(t)=\displaystyle\int_{a}^{t} \parallel{\bf{r}}^{\prime}(u)\parallel{du}[/latex]
• Curvature
  [latex]\kappa=\dfrac{\parallel{\bf{T}}^{\prime}(t)\parallel}{\parallel{\bf{r}}^{\prime}(t)\parallel}[/latex] or [latex]\kappa=\dfrac{\parallel{\bf{r}}^{\prime}(t)\times{\bf{r}}^{\prime\prime}(t)\parallel}{\parallel{\bf{r}}^{\prime}(t)\parallel^{3}}[/latex] or [latex]\kappa=\dfrac{|y^{\sigma}|}{[1+(y^{\prime})^{2}]^{3{/}2}}[/latex]
• Principal unit normal vector
  [latex]{\bf{N}}(t)=\dfrac{{\bf{T}}^{\prime}(t)}{\parallel{\bf{T}}^{\prime}(t)\parallel}[/latex]
• Binormal vector
  [latex]{\bf{B}}(t)={\bf{T}}(t)\times{\bf{N}}(t)[/latex]

Glossary

acceleration vector

the second derivative of the position vector

arc-length function

a function [latex]s(t)[/latex] that describes the arc length of curve [latex]C[/latex] as a function of [latex]t[/latex]

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 [latex]{\bf{r}}(t)=f(t){\bf{i}}+g(t){\bf{j}}[/latex] are [latex]f(t)[/latex] and [latex]g(t)[/latex], and the component functions of the vector-valued function [latex]{\bf{r}}(t)=f(t){\bf{i}}+g(t){\bf{j}}+h(t){\bf{k}}[/latex] are [latex]f(t)[/latex], [latex]g(t)[/latex] and [latex]h(t)[/latex]

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 [latex]{\bf{r}}(t)[/latex] is [latex]{\bf{r}}^{\prime}(t)=\underset{\Delta{t}\to{0}}{\lim}\frac{{\bf{r}}(t+\Delta{t})-{\bf{r}}(t)}{\Delta{t}}[/latex], 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 [latex]{\bf{r}}(t)[/latex] has a limit [latex]{\bf{L}}[/latex] as [latex]t[/latex] approaches [latex]a[/latex] if [latex]\underset{t\to{a}}{\lim}|{\bf{r}}(t)-{\bf{L}}|=0[/latex]

normal component of acceleration

the coefficient of the unit normal vector [latex]{\bf{N}}[/latex] when the acceleration vector is written as a linear combination of [latex]{\bf{T}}[/latex] and [latex]{\bf{N}}[/latex]

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 [latex]C[/latex] at a point [latex]P[/latex] 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 [latex]\left(f(t),g(t)\right)[/latex] together with their defining parametric equations [latex]x=f(t)[/latex] and [latex]y=g(t)[/latex]

principal unit normal vector

a vector orthogonal to the unit tangent vector, given by the formula [latex]\frac{{\bf{T}}^{\prime}(t)}{\parallel{\bf{T}}^{\prime}(t)\parallel}[/latex]

principal unit tangent vector

a unit vector tangent to a curve [latex]C[/latex]

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 [latex]{\bf{r}}(t)[/latex] is differentiable with a non-zero derivative

space curve

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

tangent vector

to [latex]{\bf{r}}(t)[/latex] at [latex]t=t_{0}[/latex] any vector [latex]{\bf{v}}[/latex] such that, when the tail of the vector is placed at point [latex]{\bf{r}}(t_{0})[/latex] on the graph, vector [latex]{\bf{v}}[/latex] is tangent to curve [latex]C[/latex]

tangential component of acceleration

the coefficient of the unit tangent vector [latex]{\bf{T}}[/latex] when the acceleration vector is written as a linear combination of [latex]{\bf{T}}[/latex] and [latex]{\bf{N}}[/latex]

vector parameterization

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

vector-valued function

a function of the form [latex]{\bf{r}}(t)=f(t){\bf{i}}+g(t){\bf{j}}[/latex] or [latex]{\bf{r}}(t)=f(t){\bf{i}}+g(t){\bf{j}}+h(t){\bf{k}}[/latex], where the component functions [latex]f[/latex], [latex]g[/latex], and [latex]h[/latex] are real-valued functions of the parameter [latex]t[/latex]

velocity vector

the derivative of the position vector