Arc length and curvature of space curves
Problem 9.694 · medium
Find the curvature of \( \displaystyle \mathbf r(t) = \left\langle \cos{\left(t \right)}, \sin{\left(t \right)}, t \right\rangle \) at \( \displaystyle t = 1 \).
- \[ \left[\begin{matrix}\left. \frac{d}{d t} \cos{\left(t \right)} \right|_{\substack{ t=1 }}\\\left. \frac{d}{d t} \sin{\left(t \right)} \right|_{\substack{ t=1 }}\\\left. \frac{d}{d t} t \right|_{\substack{ t=1 }}\end{matrix}\right] = \left[\begin{matrix}- \sin{\left(1 \right)}\\\cos{\left(1 \right)}\\1\end{matrix}\right] \]r′(1).✓ Proved
- \[ \left[\begin{matrix}\left. \frac{d^{2}}{d t^{2}} \cos{\left(t \right)} \right|_{\substack{ t=1 }}\\\left. \frac{d^{2}}{d t^{2}} \sin{\left(t \right)} \right|_{\substack{ t=1 }}\\\left. \frac{d^{2}}{d t^{2}} t \right|_{\substack{ t=1 }}\end{matrix}\right] = \left[\begin{matrix}- \cos{\left(1 \right)}\\- \sin{\left(1 \right)}\\0\end{matrix}\right] \]r″(1).✓ Proved
- \[ \left[\begin{matrix}\cos{\left(1 \right)} 0 - - \sin{\left(1 \right)}\\1 \left(- \cos{\left(1 \right)}\right) - 1 \left(- \sin{\left(1 \right)}\right) 0\\\cos{\left(1 \right)} \cos{\left(1 \right)} + - \sin{\left(1 \right)} \left(- \sin{\left(1 \right)}\right)\end{matrix}\right] = \left[\begin{matrix}\sin{\left(1 \right)}\\- \cos{\left(1 \right)}\\\cos^{2}{\left(1 \right)} + \sin^{2}{\left(1 \right)}\end{matrix}\right] \]r′ × r″.✓ Proved
- \[ \frac{\sqrt{\cos^{2}{\left(1 \right)} + \sin^{2}{\left(1 \right)} + \left(\cos^{2}{\left(1 \right)} + \sin^{2}{\left(1 \right)}\right)^{2}}}{\left(\cos^{2}{\left(1 \right)} + \sin^{2}{\left(1 \right)} + 1\right)^{\frac{3}{2}}} = \frac{1}{2} \]κ = ‖r′ × r″‖ / ‖r′‖³.✓ Proved
Answer \( \kappa = \frac{1}{2} \)
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: inconclusive — reviewer response could not be parsed: {"verdict":"fail","severity":"error","notes":"The cross product in step 3 is incorrect: the k‑component should be \(2\sin(1)\cos(1)\), not \( \cos^2(1)+\sin^2(1)=1\). This leads to an incorrect curvatqwen3.6:27b-mlx: pass — The solution correctly applies the curvature formula using the cross product of the first and second derivatives. The algebraic simplifications and final evaluation are correct.
Every verdict on record (6)
qwen3.6:27b-mlx: pass 2026-10-18 — The solution correctly applies the curvature formula using the cross product of the first and second derivatives. The algebraic simplifications and final evaluation are correct.gpt-oss:20b: inconclusive 2026-10-18 — reviewer response could not be parsed: {"verdict":"fail","severity":"error","notes":"The cross product in step 3 is incorrect: the k‑component should be \(2\sin(1)\cos(1)\), not \( \cos^2(1)+\sin^2(1)=1\). This leads to an incorrect curvatqwen3.6:27b-mlx: pass 2026-10-15 — The solution correctly applies the curvature formula using the cross product of the first and second derivatives. The algebraic simplifications and final evaluation are correct.gpt-oss:20b: pass 2026-10-15qwen3.6:27b-mlx: pass 2026-10-14gpt-oss:20b: inconclusive 2026-10-14 — reviewer response could not be parsed: {"verdict":"fail","severity":"error","notes":"The cross product in step 3 is computed incorrectly. The k‑component should be \(2\sin(1)\cos(1)\), not \(\cos^2(1)+\sin^2(1)\). This mistake propagates t
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 2026-10-18 with SymPy 1.14.0.