Arc length and curvature of space curves
Problem 9.747 · medium
Find the curvature of \( \displaystyle \mathbf r(t) = \left\langle \cos{\left(t \right)}, \sin{\left(t \right)}, t \right\rangle \) at \( \displaystyle t = \frac{\pi}{2} \).
- \[ \left[\begin{matrix}\left. \frac{d}{d t} \cos{\left(t \right)} \right|_{\substack{ t=\frac{\pi}{2} }}\\\left. \frac{d}{d t} \sin{\left(t \right)} \right|_{\substack{ t=\frac{\pi}{2} }}\\\left. \frac{d}{d t} t \right|_{\substack{ t=\frac{\pi}{2} }}\end{matrix}\right] = \left[\begin{matrix}-1\\0\\1\end{matrix}\right] \]r′(pi/2).✓ Proved
- \[ \left[\begin{matrix}\left. \frac{d^{2}}{d t^{2}} \cos{\left(t \right)} \right|_{\substack{ t=\frac{\pi}{2} }}\\\left. \frac{d^{2}}{d t^{2}} \sin{\left(t \right)} \right|_{\substack{ t=\frac{\pi}{2} }}\\\left. \frac{d^{2}}{d t^{2}} t \right|_{\substack{ t=\frac{\pi}{2} }}\end{matrix}\right] = \left[\begin{matrix}0\\-1\\0\end{matrix}\right] \]r″(pi/2).✓ Proved
- \[ \left[\begin{matrix}0 \cdot 0 + 1\\1 \cdot 0 + 0\\0 \cdot 0 - 1 \left(-1\right)\end{matrix}\right] = \left[\begin{matrix}1\\0\\1\end{matrix}\right] \]r′ × r″.✓ Proved
- \[ \frac{\sqrt{0^{2} + 1^{2} + 1^{2}}}{\left(\sqrt{0^{2} + \left(-1\right)^{2} + 1^{2}}\right)^{3}} = \frac{1}{2} \]κ = ‖r′ × r″‖ / ‖r′‖³.✓ Proved
Answer \( \kappa = \frac{1}{2} \)
✓ Nihil obstat 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. Reviewers found nothing wrong with the explanation. 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 2026-10-18gpt-oss:20b: pass 2026-10-18qwen3.6:27b-mlx: pass 2026-10-16 — The solution correctly applies the curvature formula using the cross product of the first and second derivatives evaluated at the specified point. All algebraic steps are verified and the final result is correct.gpt-oss:20b: pass 2026-10-16qwen3.6:27b-mlx: pass 2026-10-15gpt-oss:20b: pass 2026-10-15
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.