Lecture Note: §13 Vector-Valued Functions and Motion in Space

Last Update: 2026-04-16

§13.1 Curves in Space and Their Tangents

• Vector-Valued Function
  • A vector-valued function, or vector function, on a domain set $D$ is a rule that assigns a vector in space to each element in $D$.\(\vec{r}(t) = \langle f(t),g(t),h(t) \rangle = f(t) \vec{i} + g(t)\vec{j} + h(t)\vec{k}\)
  • Real-valued functions are often called scalar functions.
• Limits: $\displaystyle \lim_{t \to t_0} \vec{r}(t) = \vec{L}$
  • For every number $\varepsilon > 0$, there exists a corresponding number $\delta > 0$ such that, for all $t \in D$, \(\lvert \vec{r}(t) - \vec{L}\rvert < \varepsilon \quad \text{whenever} \quad 0 < \lvert t - t_0\rvert < \delta.\)
  • $\displaystyle \lim_{t \to t_0} \vec{r}(t) = \langle \lim_{t \to t_0} f(t), \lim_{t \to t_0} g(t), \lim_{t \to t_0}h(t) \rangle$
  • $\displaystyle \lim_{t \to t_0} \vec{r}(t) = \vec{L} \Longleftrightarrow \lim_{t \to t_0} f(t) = L_1, \lim_{t \to t_0} g(t) = L_2, \lim_{t \to t_0}h(t) = L_3$
• Continuity
  • A vector function $\vec{r}(t)$ is continuous at a point $t = t_0$ in its domain if \(\lim_{t \to t_0} \vec{r}(t) = \vec{r}(t_0).\)
  • The function is continuous if it is continuous at every point in its domain.
• Derivative
  • The vector function $\vec{r}(t) = f(t)\vec{i} + g(t)\vec{j} + h(t)\vec{k}$ has a derivative (is differentiable) at $t$ if $f$, $g$, and $h$ have derivatives at $t$. The derivative is the vector function \(\vec{r}'(t) = \frac{d\vec{r}}{dt} = \lim_{\Delta t \to 0} \frac{\vec{r}(t + \Delta t) - \vec{r}(t)}{\Delta t} = \frac{df}{dt}\vec{i} + \frac{dg}{dt}\vec{j} + \frac{dh}{dt}\vec{k}.\)
  • A vector function $\vec{r}$ is differentiable if it is differentiable at every point of its domain.
  • The curve traced by $\vec{r}$ is smooth if $d\vec{r}/dt$ is continuous and never $\vec{0}$.
• If $\mathbf{r}$ is the position vector of a particle moving along a smooth curve in space, then
  1. Velocity is the derivative of position: $\quad \vec{v} = \dfrac{d\vec{r}}{dt}$.
  2. Speed is the magnitude of velocity: $\quad \text{Speed} = \lvert\vec{v}\rvert$.
  3. Acceleration is the derivative of velocity: $\quad \vec{a} = \dfrac{d\vec{v}}{dt} = \dfrac{d^2\vec{r}}{dt^2}$.
  4. The unit vector $\vec{v}/\lvert\vec{v}\rvert$ is the direction of motion at time $t$.
• Differentiation Rules for Vector Functions. Let $\vec{u}$ and $\vec{v}$ be differentiable vector functions of $t$, $\vec{C}$ a constant vector, $c$ any scalar, and $f$ any differentiable scalar function.
  1. Constant Function Rule: $\dfrac{d}{dt}(\vec{C}) = \vec{0}$
  2. Scalar Multiple Rules: $\dfrac{d}{dt}[c\vec{u}(t)] = c\vec{u}’(t)$, $\dfrac{d}{dt}[f(t)\vec{u}(t)] = f’(t)\vec{u}(t) + f(t)\vec{u}’(t)$
  3. Sum Rule: $\dfrac{d}{dt}[\vec{u}(t) + \vec{v}(t)] = \vec{u}’(t) + \vec{v}’(t)$
  4. Difference Rule: $\dfrac{d}{dt}[\vec{u}(t) - \vec{v}(t)] = \vec{u}’(t) - \vec{v}’(t)$
  5. Dot Product Rule: $\dfrac{d}{dt}[\vec{u}(t) \cdot \vec{v}(t)] = \vec{u}’(t) \cdot \vec{v}(t) + \vec{u}(t) \cdot \vec{v}’(t)$
  6. Cross Product Rule: $\dfrac{d}{dt}[\vec{u}(t) \times \vec{v}(t)] = \vec{u}’(t) \times \vec{v}(t) + \vec{u}(t) \times \vec{v}’(t)$
  7. Chain Rule: $\dfrac{d}{dt}[\vec{u}(f(t))] = f’(t)\vec{u}’(f(t))$
• Theorem. Vector Functions of Constant Length
  • If $\vec{r}$ is a differentiable vector function of $t$ and the length of $\vec{r}(t)$ is constant, then \(\vec{r} \cdot \dfrac{d\vec{r}}{dt} = 0.\)

§13.2 Integrals of Vector Functions; Projectile Motion

• Indefinite Integral
  • The indefinite integral of $\vec{r}$ with respect to $t$ is the set of all antiderivatives of $\vec{r}$, denoted by \(\int \vec{r}(t)\,dt.\)
• Definite Integral
  • If the components of $\vec{r}(t) = f(t)\vec{i} + g(t)\vec{j} + h(t)\vec{k}$ are integrable over $[a, b]$, then so is $\vec{r}$, and the definite integral of $\vec{r}$ from $a$ to $b$ is \(\int_{a}^{b} \vec{r}(t)\,dt = \left( \int_{a}^{b} f(t)\,dt \right)\vec{i} + \left( \int_{a}^{b} g(t)\,dt \right)\vec{j} + \left( \int_{a}^{b} h(t)\,dt \right)\vec{k}.\)

§13.3 Arc Length in Space

• Arc Length \(L = \int_{a}^{b} \sqrt{\left( \frac{dx}{dt} \right)^2 + \left( \frac{dy}{dt} \right)^2 + \left( \frac{dz}{dt} \right)^2}\,dt.\)

• Arc Length Parameter \(s(t) = \int_{t_0}^{t} \sqrt{[x'(\tau)]^2 + [y'(\tau)]^2 + [z'(\tau)]^2}\,d\tau = \int_{t_0}^{t} \lvert\vec{v}(\tau)\rvert\,d\tau\)
  • It is often useful to parametrize a curve with respect to arc length, since the arc length arises naturally from the shape of the curve and does not depend on a particular coordinate system or a particular parametrization.
• Unit Tangent Vector
  • $\vec{T} = \dfrac{d\vec{r}}{ds} = \dfrac{d\vec{r}/dt}{ds/dt}= \dfrac{\vec{v}}{\lvert\vec{v}\rvert}$

§13.4 Curvature and the Normal Vector of a Curve

• Curvature
  • $\displaystyle \kappa = \left\lvert \frac{d\vec{T}}{ds}\right\rvert = \frac{1}{\lvert\vec{v}\rvert}\left\lvert\frac{d\vec{T}}{dt}\right\rvert$
  • The curvature shows how points “curve” in the plane defined by $\vec{T}$ and $\vec{N}$.
• Principal Unit Normal Vector
  • $\displaystyle \vec{N} = \frac{1}{\kappa}\frac{d\vec{T}}{ds} = \frac{d\vec{T}/dt}{\lvert d\vec{T}/dt\rvert}$
  • The vector $d\vec{T}/ds$ is orthogonal to $\vec{T}$

§13.5 Tangential and Normal Components of Acceleration

• Binormal Vector
  • $\vec{B} = \vec{T} \times \vec{N}$
• Torsion
  • $\displaystyle \tau = -\frac{d\vec{B}}{ds} \cdot \vec{N}$
  • The torsion show how the plane defined by $\vec{T}$ and $\vec{N}$ “twists”.

• TNB Frame Together $\vec{T}$, $\vec{N}$, and $\vec{B}$ define a moving right-handed vector frame that plays a significant role in analyzing the paths of particles moving through space. It is called the Frenet (“fre-nay”) frame, or the TNB frame.

  \[\begin{bmatrix} \dfrac{d\vec{T}}{ds} \\ \dfrac{d\vec{N}}{ds} \\ \dfrac{d\vec{B}}{ds}\end{bmatrix} = \begin{bmatrix} 0 & \kappa & 0 \\ -\kappa & 0 &\tau \\ 0 & -\tau & 0\end{bmatrix} \begin{bmatrix} \vec{T} \\ \vec{N} \\ \vec{B}\end{bmatrix}\]
• Tangential and the Normal scalar components of acceleration
  • $\vec{a} = a_{\vec{T}} \vec{T} + a_{\vec{N}} \vec{N}$
  • $\displaystyle a_{\vec{T}} = \frac{d^{2} s}{dt^{2}} = \frac{d}{dt} \lvert\vec{v}\rvert$ and $\displaystyle a_{\vec{N}} = \kappa \left( \frac{ds}{dt} \right)^{2} = \kappa \lvert \vec{v}\rvert^{2}$
• Computation Formulas for Curves in Space
  • Unit tangent vector: $\vec{T} = \dfrac{\vec{v}}{\lvert\vec{v}\rvert}$
  • Principal unit normal vector: $\vec{N} = \dfrac{d\vec{T}/dt}{\lvert d\vec{T}/dt\rvert}$
  • Binormal vector: $\vec{B} = \vec{T} \times \vec{N}$
  • Curvature: $\displaystyle \kappa = \left\lvert\frac{d\vec{T}}{ds}\right\rvert = \frac{\lvert\vec{v} \times \vec{a}\rvert}{\lvert\vec{v}\rvert^{3}}$
  • Torsion: \(\displaystyle \tau = -\frac{d\vec{B}}{ds} \cdot \vec{N} = \frac{(\vec{v}\times \vec{a})\cdot d^{3}\vec{r}/dt^{3}}{\lvert \vec{v} \times \vec{a}\rvert^{2}}= \frac{\begin{vmatrix} \dot{x} & \dot{y} & \dot{z} \\ \ddot{x} & \ddot{y} & \ddot{z} \\ \dddot{x} & \dddot{y} & \dddot{z} \end{vmatrix}}{\lvert \vec{v} \times \vec{a}\rvert^{2}}\)
  • Tangential and normal scalar components of acceleration:
    • $\vec{a} = a_{\vec{T}} \vec{T} + a_{\vec{N}} \vec{N}$
    • $\displaystyle a_{\vec{T}} = \frac{d}{dt} \lvert\vec{v}\rvert$
    • $\displaystyle a_{\vec{N}} = \kappa \lvert\vec{v}\rvert^{2} = \sqrt{\lvert\vec{a}\rvert^{2} - a_{\vec{T}}^{2}}$