Unit 1: Vector-Valued Functions, Space Curves & Arc Length

### 1. Definition and Parametric Representation A **vector-valued function** (or simply **vector function**) is a mapping $\vec{r}: I \subseteq \mathbb{R} \to \mathbb{R}^3$ that assigns to each real number $t$ in an interval $I$ a unique spatial vector: $$\vec{r}(t) = f(t)\hat{i} + g(t)\hat{j} + h(t)\hat{k} = \langle f(t), g(t), h(t) \rangle$$ The functions $f(t), g(t), h(t)$ are real-valued functions of the single real parameter $t$, called the **component functions** of $\vec{r}$. As $t$ varies over the domain $I$, the terminal point of the position vector $\vec{r}(t)$ traces out a one-dimensional geometric locus in three-dimensional space called a **space curve** $C$: $$C = \{ (x, y, z) \in \mathbb{R}^3 : x = f(t), \; y = g(t), \; z = h(t), \; t \in I \}$$ These are the **parametric equations** of the curve $C$, and $t$ is called the **parameter** (often representing physical time). --- ### 2. Canonical Examples of Space Curves 1. **The Circular Helix:** $$\vec{r}(t) = \langle a\cos t, \; a\sin t, \; bt \rangle \quad (a > 0, \; b \neq 0)$$ The projection of this curve onto the $xy$-plane is a circle of radius $a$: $x^2 + y^2 = a^2\cos^2 t + a^2\sin^2 t = a^2$. As $t$ increases, the point winds around the cylinder $x^2 + y^2 = a^2$ while ascending steadily along the $z$-axis at rate $b$. The constant pitch between adjacent coils is $2\pi b$. 2. **The Twisted Cubic:** $$\vec{r}(t) = \langle t, \; t^2, \; t^3 \rangle$$ This curve lies at the intersection of the parabolic cylinder $y = x^2$ and the cubic cylinder $z = x^3$. It is the simplest non-planar algebraic space curve. 3. **The Conical Helix:** $$\vec{r}(t) = \langle t\cos t, \; t\sin t, \; t \rangle$$ Here, $x^2 + y^2 = t^2 = z^2$. The curve winds around the right circular cone $x^2 + y^2 = z^2$, expanding outward as it ascends. --- ### 3. Limits and Continuity The limit of a vector function $\vec{r}(t)$ as $t \to a$ is evaluated componentwise: $$\lim_{t \to a} \vec{r}(t) = \left\langle \lim_{t \to a} f(t), \; \lim_{t \to a} g(t), \; \lim_{t \to a} h(t) \right\rangle$$ provided the limits of all three component functions exist. **Continuity:** A vector function $\vec{r}(t)$ is **continuous at $a$** if: $$\lim_{t \to a} \vec{r}(t) = \vec{r}(a)$$ Equivalently, $\vec{r}(t)$ is continuous at $a$ if and only if each of its component functions $f, g, h$ is continuous at $a$.

### 1. The Derivative of a Vector Function The derivative of a vector-valued function $\vec{r}(t)$ is defined analogously to that of a real-valued function: $$\vec{r}'(t) = \frac{d\vec{r}}{dt} = \lim_{\Delta t \to 0} \frac{\vec{r}(t + \Delta t) - \vec{r}(t)}{\Delta t}$$ In terms of component functions: $$\vec{r}'(t) = \lim_{\Delta t \to 0} \left\langle \frac{f(t+\Delta t) - f(t)}{\Delta t}, \; \frac{g(t+\Delta t) - g(t)}{\Delta t}, \; \frac{h(t+\Delta t) - h(t)}{\Delta t} \right\rangle$$ $$\mathbf{\vec{r}'(t) = \langle f'(t), \; g'(t), \; h'(t) \rangle = f'(t)\hat{i} + g'(t)\hat{j} + h'(t)\hat{k}}$$ --- ### 2. Geometric Interpretation: Tangent Vector and Velocity As $\Delta t \to 0$, the secant vector $\frac{\vec{r}(t+\Delta t) - \vec{r}(t)}{\Delta t}$ approaches a limiting vector that points in the direction of the tangent line to the curve at $P(\vec{r}(t))$. - **Tangent Vector:** $\vec{r}'(t)$ is tangent to the space curve at $\vec{r}(t)$ in the direction of increasing $t$. - **Unit Tangent Vector $\vec{T}(t)$:** For a curve with $\vec{r}'(t) \neq \vec{0}$: $$\mathbf{\vec{T}(t) = \frac{\vec{r}'(t)}{|\vec{r}'(t)|}}$$ - **Smooth Curve:** A curve parameterized by $\vec{r}(t)$ is called **smooth** on an interval $I$ if $\vec{r}'(t)$ is continuous and $\vec{r}'(t) \neq \vec{0}$ for all $t \in I$ (no sharp corners, cusps, or stopping points). - **Tangent Line:** The parametric equation of the tangent line to the space curve at $t = t_0$ is: $$\vec{L}(\tau) = \vec{r}(t_0) + \tau \vec{r}'(t_0) \quad (\tau \in \mathbb{R})$$ --- ### 3. Vector Differentiation Rules Let $\vec{u}(t)$ and $\vec{v}(t)$ be differentiable vector functions, $c$ a scalar constant, and $f(t)$ a differentiable scalar function: 1. $\frac{d}{dt} [\vec{u}(t) + \vec{v}(t)] = \vec{u}'(t) + \vec{v}'(t)$ 2. $\frac{d}{dt} [c\vec{u}(t)] = c\vec{u}'(t)$ 3. $\frac{d}{dt} [f(t)\vec{u}(t)] = f'(t)\vec{u}(t) + f(t)\vec{u}'(t)$ (Scalar-Vector Product Rule) 4. $\frac{d}{dt} [\vec{u}(t) \cdot \vec{v}(t)] = \vec{u}'(t) \cdot \vec{v}(t) + \vec{u}(t) \cdot \vec{v}'(t)$ (Dot Product Rule) 5. $\frac{d}{dt} [\vec{u}(t) \times \vec{v}(t)] = \vec{u}'(t) \times \vec{v}(t) + \vec{u}(t) \times \vec{v}'(t)$ (Cross Product Rule — **order must be strictly preserved!**) 6. $\frac{d}{dt} [\vec{u}(f(t))] = f'(t)\vec{u}'(f(t))$ (Chain Rule) --- ### 4. Fundamental Orthogonality Theorem for Constant Length Vectors **Theorem:** If a vector function $\vec{r}(t)$ has constant magnitude for all $t$, that is, $|\vec{r}(t)| = c$ (a constant), then: $$\mathbf{\vec{r}(t) \cdot \vec{r}'(t) = 0}$$ That is, the derivative vector $\vec{r}'(t)$ is everywhere orthogonal to the position vector $\vec{r}(t)$. #### Proof: Since $|\vec{r}(t)| = c$, the dot product of $\vec{r}(t)$ with itself is constant: $$\vec{r}(t) \cdot \vec{r}(t) = |\vec{r}(t)|^2 = c^2$$ Differentiating both sides with respect to $t$ using the Dot Product Rule: $$\frac{d}{dt} [\vec{r}(t) \cdot \vec{r}(t)] = \frac{d}{dt}[c^2]$$ $$\vec{r}'(t) \cdot \vec{r}(t) + \vec{r}(t) \cdot \vec{r}'(t) = 0$$ $$2 \left( \vec{r}(t) \cdot \vec{r}'(t) \right) = 0 \implies \vec{r}(t) \cdot \vec{r}'(t) = 0 \quad \blacksquare$$ **Geometric Meaning:** If a particle moves on the surface of a sphere centered at the origin ($|\vec{r}(t)| = R$), its velocity vector is always tangent to the sphere, and therefore perpendicular to the radius vector. --- ### 5. Integration of Vector Functions The definite integral of a continuous vector function $\vec{r}(t) = \langle f(t), g(t), h(t) \rangle$ is evaluated componentwise: $$\int_a^b \vec{r}(t) \, dt = \left( \int_a^b f(t) \, dt \right)\hat{i} + \left( \int_a^b g(t) \, dt \right)\hat{j} + \left( \int_a^b h(t) \, dt \right)\hat{k}$$ By the Fundamental Theorem of Calculus, if $\vec{R}'(t) = \vec{r}(t)$, then $\int_a^b \vec{r}(t) \, dt = \vec{R}(b) - \vec{R}(a)$.

### 1. Arc Length of a Space Curve Let $C$ be a smooth space curve parameterized by $\vec{r}(t) = \langle f(t), g(t), h(t) \rangle$ for $a \le t \le b$. Partition the parameter interval $[a, b]$ into $n$ subintervals by $a = t_0 < t_1 < \dots < t_n = b$. The length of the polygonal secant connecting $\vec{r}(t_{i-1})$ to $\vec{r}(t_i)$ is: $$\Delta L_i = |\vec{r}(t_i) - \vec{r}(t_{i-1})| = |\Delta \vec{r}_i| \approx |\vec{r}'(t_i^*)| \Delta t_i$$ Taking the limit as the partition mesh goes to zero ($\Delta t \to 0$): $$\mathbf{L = \int_a^b |\vec{r}'(t)| \, dt = \int_a^b \sqrt{[f'(t)]^2 + [g'(t)]^2 + [h'(t)]^2} \, dt}$$ In differential notation: $$ds = |\vec{r}'(t)| \, dt = \sqrt{dx^2 + dy^2 + dz^2}$$ --- ### 2. The Arc Length Function Let $t_0$ be a chosen reference starting point on the curve. The **arc length function** $s(t)$ measures the directed distance along the curve from $\vec{r}(t_0)$ to $\vec{r}(t)$: $$\mathbf{s(t) = \int_{t_0}^t |\vec{r}'(u)| \, du}$$ By the Fundamental Theorem of Calculus, the rate of change of arc length with respect to the parameter $t$ is: $$\frac{ds}{dt} = |\vec{r}'(t)| = v(t)$$ where $v(t)$ is the **speed** of the trajectory. Since the curve is smooth ($|\vec{r}'(t)| > 0$), $\frac{ds}{dt} > 0$ strictly, meaning $s(t)$ is a strictly monotonically increasing function of $t$. --- ### 3. Arc Length Parameterization (Natural Parameter) Because $s(t)$ is strictly increasing, it has a unique inverse function $t = t(s)$. We can reparameterize the curve in terms of the arc length $s$ by composing: $$\vec{r}_1(s) = \vec{r}(t(s))$$ **Theorem (Unit Speed Property of Arc Length Parameterization):** If a curve is parameterized by its arc length $s$, then its tangent vector has unit length everywhere: $$\left| \frac{d\vec{r}}{ds} \right| = 1 \quad \forall s$$ #### Proof: By the Chain Rule: $$\frac{d\vec{r}}{ds} = \frac{d\vec{r}}{dt} \frac{dt}{ds} = \vec{r}'(t) \frac{1}{ds/dt} = \frac{\vec{r}'(t)}{|\vec{r}'(t)|} = \vec{T}(t)$$ Taking the norm: $$\left| \frac{d\vec{r}}{ds} \right| = |\vec{T}(t)| = 1 \quad \blacksquare$$ Arc length parameterization is termed the **intrinsic** or **natural parameterization** of the curve because it depends solely on the curve's geometric shape and is entirely independent of any artificial choice of time or velocity.