Arc length and curvature of space curves
Problem 9.667 · medium
Find the curvature of \( \displaystyle \mathbf r(t) = \left\langle t, t^{2}, t^{3} \right\rangle \) at \( \displaystyle t = -1 \).
- \[ \left[\begin{matrix}\left. \frac{d}{d t} t \right|_{\substack{ t=-1 }}\\\left. \frac{d}{d t} t^{2} \right|_{\substack{ t=-1 }}\\\left. \frac{d}{d t} t^{3} \right|_{\substack{ t=-1 }}\end{matrix}\right] = \left[\begin{matrix}1\\-2\\3\end{matrix}\right] \]r′(-1).✓ Proved
- \[ \left[\begin{matrix}\left. \frac{d^{2}}{d t^{2}} t \right|_{\substack{ t=-1 }}\\\left. \frac{d^{2}}{d t^{2}} t^{2} \right|_{\substack{ t=-1 }}\\\left. \frac{d^{2}}{d t^{2}} t^{3} \right|_{\substack{ t=-1 }}\end{matrix}\right] = \left[\begin{matrix}0\\2\\-6\end{matrix}\right] \]r″(-1).✓ Proved
- \[ \left[\begin{matrix}2 \left(-3\right) - 2 \left(-6\right)\\3 \cdot 0 + 6\\0 \cdot 2 + 1 \cdot 2\end{matrix}\right] = \left[\begin{matrix}6\\6\\2\end{matrix}\right] \]r′ × r″.✓ Proved
- \[ \frac{\sqrt{2^{2} + 6^{2} + 6^{2}}}{\left(\sqrt{1^{2} + \left(-2\right)^{2} + 3^{2}}\right)^{3}} = \frac{\sqrt{266}}{98} \]κ = ‖r′ × r″‖ / ‖r′‖³.✓ Proved
Answer \( \kappa = \frac{\sqrt{266}}{98} \)
Every line of this solution was proved by the computer algebra system SymPy. The answer was also checked a second way, without looking at the solution. The reviewers disagree about how one step is explained; every verdict is in the receipt. How we verify
The full receipt
| Line | Status | Checked by | Detail |
|---|---|---|---|
| 1 | ✓ Proved | sympy 1.14.0 | simplify(lhs - rhs) reduced to 0 |
| 2 | ✓ Proved | sympy 1.14.0 | simplify(lhs - rhs) reduced to 0 |
| 3 | ✓ Proved | sympy 1.14.0 | simplify(lhs - rhs) reduced to 0 |
| 4 | ✓ Proved | sympy 1.14.0 | simplify(lhs - rhs) reduced to 0 |
| answer, a second way | ✓ Checked independently | sympy 1.14.0 + mpmath 1.3.0 | the circle through three points 10⁻⁵ apart on the curve |
Reviewers
gpt-oss:20b: passqwen3.6:27b-mlx: pass
Every verdict on record (6)
qwen3.6:27b-mlx: pass 2026-10-17gpt-oss:20b: pass 2026-10-17qwen3.6:27b-mlx: pass 2026-10-15gpt-oss:20b: pass 2026-10-15qwen3.6:27b-mlx: pass 2026-10-14 — The solution correctly applies the curvature formula for a space curve, with accurate derivative evaluations and cross product calculations.gpt-oss:20b: fail (misleading) 2026-10-14 — The cross product in step 3 is computed incorrectly: the j‑component should be –6, not +6. Although the magnitude (and thus the final curvature) is unchanged, this error could mislead a student about the correct cross‑product formula.
Proved: SymPy reduced the difference between the two sides to zero. Checked independently: a separate
method, named above, confirmed it. Checked numerically: the two sides agree at every sampled point, which is evidence,
not proof. Reviewed: a model or a person read it; that is all a sentence can have. Solution by
generator:structured/curvature, checked 2026-10-17 with SymPy 1.14.0.