Problem Metadata

Date November 16, 2025 (optional — no deadline)
Topic Unit Tangent Vector of a Parametric Curve in $R^3$
Sub-Topic Curvature, torsion, Frenet-Serret Frame
Curator @Anonymous

Main Problem

Daily Math Problem - DAY 6

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Difficulty

Problem 1 — Unit Tangent Vector

Consider the curve

$$ G(t) = [e^t, \sin t, t^2] \subset R^3 $$

Compute the unit tangent vector

$$ T(t) = \frac{G'(t)}{\|G'(t)\|}. $$

Problem 2 — Curvature of a Space Curve

Consider the curve

$$ H(t) = [t, t^2, \cos t] $$

  1. Compute $H'(t)$ and $H^H(t)$.
  2. Compute the curvature using

$$ \kappa(t) = \frac{\|H'(t) \times H''(t)\|}{\|H'(t)\|^{3}}. $$

Simplify your final expression for $k(t)$.

Problem 3 — Torsion of a Curve