PGDDSA Study · Semester 1

Core Titles

Key headlines and terms for quick recall

Vector-valued function $\vec{r}(t) = \langle f(t), g(t), h(t) \rangle$
Domain, Range — set of parameter values, image curve
Limit, Continuity — componentwise
Derivative $\vec{r}'(t) = \langle f'(t), g'(t), h'(t) \rangle$ — tangent vector
Velocity, Speed, Acceleration in motion
Integration — componentwise
Product rules (dot, cross, scalar)

Basic Idea

What it is, why it matters, how it works

What is a vector-valued function?

A function that maps a scalar parameter $t$ to a vector: $\vec{r}(t) = f(t)\hat{i} + g(t)\hat{j} + h(t)\hat{k}.$ As $t$ varies, the tip of $\vec{r}(t)$ traces a curve in space.

Examples.

Straight line: $\vec{r}(t) = \vec{r}_0 + t\vec{d}$ .
Circular helix: $\vec{r}(t) = (\cos t, \sin t, t)$ .
Parabola in plane: $\vec{r}(t) = (t, t^2, 0)$ .

Limits and continuity

$\lim_{t \to t_0} \vec{r}(t) = \left\langle \lim f(t), \lim g(t), \lim h(t) \right\rangle.$ $\vec{r}$ is continuous at $t_0$ iff all components are continuous.

Derivative (tangent vector)

$\vec{r}'(t) = \langle f'(t), g'(t), h'(t) \rangle = \lim_{\Delta t \to 0} \frac{\vec{r}(t + \Delta t) - \vec{r}(t)}{\Delta t}.$ Geometrically, $\vec{r}'(t)$ is tangent to the curve at $\vec{r}(t)$ (when non-zero).

Unit tangent $\vec{T}(t) = \dfrac{\vec{r}'(t)}{\|\vec{r}'(t)\|}$ .

Motion

If $\vec{r}(t)$ is position:

Velocity $\vec{v}(t) = \vec{r}'(t)$
Speed $= \|\vec{v}(t)\|$
Acceleration $\vec{a}(t) = \vec{v}'(t) = \vec{r}''(t)$

Product rules

$(\vec{u}(t) + \vec{v}(t))' = \vec{u}'(t) + \vec{v}'(t)$
$(c\vec{u})' = c\vec{u}'$
$(\vec{u}(t) \cdot \vec{v}(t))' = \vec{u}' \cdot \vec{v} + \vec{u} \cdot \vec{v}'$
$(\vec{u}(t) \times \vec{v}(t))' = \vec{u}' \times \vec{v} + \vec{u} \times \vec{v}'$ — order matters!

Integration

$\int \vec{r}(t)\, dt = \langle \textstyle\int f, \int g, \int h \rangle + \vec{C}.$

Why this matters in Data Science

Continuous trajectories in motion-tracking, robotics, time-series of vector embeddings, paths through optimisation landscapes.

Mind Map

Visual structure of the concept

VECTOR-VALUED FUNCTIONS
├── r(t) = ⟨f(t), g(t), h(t)⟩
├── Curve in space
├── Limits / continuity: componentwise
├── Derivative r'(t)
│   ├── Tangent vector
│   ├── Unit tangent T(t) = r'/‖r'‖
│   └── Velocity, speed, acceleration
├── Product rules
│   ├── (u·v)' = u'·v + u·v'
│   └── (u×v)' = u'×v + u×v' (order!)
└── Integration: componentwise

Exam Q&A

Part A (2 marks) and Part B (20 marks) style questions

Part A (2 marks each)

Q1. Define a vector-valued function. A function $\vec{r} : \mathbb{R} \to \mathbb{R}^n$ whose value at each $t$ is a vector — e.g., $\vec{r}(t) = \langle f(t), g(t), h(t) \rangle$ .

Q2. State the derivative of $\vec{r}(t) = (t^2, \sin t, e^t)$ . $\vec{r}'(t) = (2t, \cos t, e^t)$ .

Q3. Differentiate $\vec{u}(t) \cdot \vec{v}(t)$ . $(\vec{u} \cdot \vec{v})' = \vec{u}' \cdot \vec{v} + \vec{u} \cdot \vec{v}'$ .

Q4. What is the unit tangent vector? $\vec{T}(t) = \dfrac{\vec{r}'(t)}{\|\vec{r}'(t)\|}$ .

Part B (20 marks)

Q. Define vector-valued functions and their derivatives. State product rules for vector derivatives. For the curve $\vec{r}(t) = (\cos t,\, \sin t,\, t)$ , find the velocity, speed, acceleration, and unit tangent at $t = \pi/2$ .

Definitions. A vector-valued function $\vec{r} : I \to \mathbb{R}^3$ assigns each scalar $t \in I$ a position vector $\vec{r}(t)$ . Its derivative is $\vec{r}'(t) = \lim_{\Delta t \to 0} \frac{\vec{r}(t + \Delta t) - \vec{r}(t)}{\Delta t} = \langle f', g', h' \rangle.$ It is tangent to the curve at $\vec{r}(t)$ .

Product rules. For differentiable $\vec{u}, \vec{v}$ and scalar $\phi$ :

$(\phi \vec{u})' = \phi' \vec{u} + \phi \vec{u}'$
$(\vec{u} \cdot \vec{v})' = \vec{u}' \cdot \vec{v} + \vec{u} \cdot \vec{v}'$
$(\vec{u} \times \vec{v})' = \vec{u}' \times \vec{v} + \vec{u} \times \vec{v}'$ (preserve order — cross product is non-commutative).

Helix $\vec{r}(t) = (\cos t, \sin t, t)$ .

Velocity: $\vec{v}(t) = \vec{r}'(t) = (-\sin t, \cos t, 1)$ .

Speed: $\|\vec{v}(t)\| = \sqrt{\sin^2 t + \cos^2 t + 1} = \sqrt{2}$ — constant.

Acceleration: $\vec{a}(t) = \vec{v}'(t) = (-\cos t, -\sin t, 0)$ .

Unit tangent: $\vec{T}(t) = \dfrac{\vec{v}(t)}{\|\vec{v}(t)\|} = \dfrac{1}{\sqrt{2}}(-\sin t, \cos t, 1)$ .

At $t = \pi/2$ . $\vec{v}(\pi/2) = (-1, 0, 1)$ , speed $= \sqrt{2}$ . $\vec{a}(\pi/2) = (0, -1, 0)$ . $\vec{T}(\pi/2) = \dfrac{1}{\sqrt{2}}(-1, 0, 1)$ .

Note that $\vec{v} \cdot \vec{a} = 0$ — velocity and acceleration are perpendicular (characteristic of helical motion at constant speed).

Vector-Valued Functions: Differentiation and Integration