Calc-3 Discussion 03

Discussion Topics covered: Vector-valued functions, 3D space curve domains, component-wise set intersection, velocity and acceleration vectors, scalar speed, principal unit tangent vector, vector chain rule, indefinite and definite vector integrals, constant integration vectors, kinematic initial value problems, radical integration, u-substitution, trigonometric substitution, completing the square, arc length parametrization inversion

Part A: Review This Week's Lecture Topics (~30 mins)

Vector-Valued Functions, Space Curves & Domain Analysis

• Evaluating 3D vector-valued functions $\vec{r}(t) = \langle f(t), g(t), h(t) \rangle = f(t)\hat{i} + g(t)\hat{j} + h(t)\hat{k}$ mapping scalar parameter $t \in \mathbb{R}$ to continuous spatial trajectories
• Determining component domains by systematically enforcing algebraic restriction rules: even roots ($u(t) \ge 0$), rational denominators ($d(t) \ne 0$), radical denominators ($u(t) > 0$), and logarithms ($u(t) > 0$)
• Calculating global curve domains via strict set intersection $\text{Domain}(\vec{r}) = \text{Domain}(f) \cap \text{Domain}(g) \cap \text{Domain}(h)$, identifying redundant interval bounds, and classifying contradictory constraints as empty-set ($\emptyset$) domains
• Modeling physical actuator stroke envelopes and robotic toolpath operating domains where real-parameter cutoffs prevent physical boundary breaches

Space Curve Differentiation, Kinematics & Tangent Headings

• Differentiating vector-valued functions component-wise ($\vec{r}'(t) = \langle f'(t), g'(t), h'(t) \rangle$) applying single-variable power, product, quotient, and chain rules
• Computing kinematic velocity $\vec{v}(t) = \vec{r}'(t)$, acceleration $\vec{a}(t) = \vec{r}''(t)$, and scalar traversal speed $\|\vec{v}(t)\| = \sqrt{[x'(t)]^2 + [y'(t)]^2 + [z'(t)]^2}$
• Constructing the dimensionless principal unit tangent vector $\hat{T}(t) = \frac{\vec{r}'(t)}{\|\vec{r}'(t)\|}$ to isolate pure directional heading independent of throttle speed
• Applying the Vector Chain Rule $\frac{d}{dt}[\vec{r}(f(t))] = \vec{r}'(f(t)) \cdot f'(t)$ to analyze composite trajectories and reparameterized flight paths

Vector Integration & Kinematic Initial Value Problems (IVPs)

• Evaluating indefinite vector integrals component-by-component, recognizing that three independent scalar integration constants consolidate into a single constant vector $\vec{C} = \langle C_1, C_2, C_3 \rangle$
• Solving two-stage kinematic Initial Value Problems (IVPs) from acceleration $\vec{a}(t)$ to velocity $\vec{v}(t)$ and position $\vec{r}(t)$ using initial calibration vectors $\vec{v}(0)$ and $\vec{r}(0)$ (inertial dead reckoning in GPS-denied navigation)
• Evaluating definite vector integrals $\Delta \vec{r} = \int_a^b \vec{v}(t)\,dt = \vec{r}(b) - \vec{r}(a)$ to calculate exact net displacement vectors across finite operational intervals

Part B: Next Week's Prerequisite Refresher (~30 mins)

Review the following prerequisite skills:

• Integration by Substitution ($u$-Substitution with Radical Forms)
• Trigonometric Substitution (Arc Length Radicals)
• Completing the Square (Radical Restructuring)
• Inverting Functions & Domain Mapping (Solving $s(t)$ for $t(s)$)
• Arc Length Parametrization ($\vec{r}(s)$ & Unit Speed Verification)

Part C: General Q/A

Open the floor to questions from students regarding any topic