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VECTOR Calc 1–4
4d Streak
120 XP
Mode A: Guided Applet
x₀ = 1.00| f(x₀) = 0.750| f'(x₀) = 0.250
Dynamic Parameters
Target Point x₀ (x0x_0)1.0
Epsilon Corridor ε (ε\varepsilon)0.50
Delta Window δ (δ\delta)0.35
Visual Overlays & Layers
Chapter 2 Theory

Vector-Valued Functions, Space Curves & Curvature

Frenet-Serret TNB frame, arc length parameterization, and 3D kinematics

Watch a point travel along a 3D helix r(t) = ⟨cos(t), sin(t), 0.3t⟩ while its unit tangent T, principal normal N, and binormal B vectors guide motion.

1. The Frenet-Serret Moving Frame

As a particle moves in 3D space, the unit tangent T points along velocity, the principal normal N points toward the center of curvature, and binormal B = T × N completes the right-handed orthonormal triad.

T=r′∣r′∣,N=T′∣T′∣,B=T×N\mathbf{T} = \frac{\mathbf{r}'}{|\mathbf{r}'|}, \quad \mathbf{N} = \frac{\mathbf{T}'}{|\mathbf{T}'|}, \quad \mathbf{B} = \mathbf{T} \times \mathbf{N}

Quick Reference & Cheat Sheet

Curvature Formula
κ=∣T′(s)∣1=∣r′(t)×r′′(t)∣∣r′(t)∣3\kappa = \frac{|\mathbf{T}'(s)|}{1} = \frac{|\mathbf{r}'(t) \times \mathbf{r}''(t)|}{|\mathbf{r}'(t)|^3}
Rate of direction turning per unit distance traveled (1/R_osculating).