Arc length and curvature of space curves
Problem 9.748 · medium
Find the curvature of \( \displaystyle \mathbf r(t) = \left\langle 4 \cos{\left(t \right)}, 4 \sin{\left(t \right)}, 0 \right\rangle \) at \( \displaystyle t = 0 \).
- \[ \left[\begin{matrix}\left. \frac{d}{d t} 4 \cos{\left(t \right)} \right|_{\substack{ t=0 }}\\\left. \frac{d}{d t} 4 \sin{\left(t \right)} \right|_{\substack{ t=0 }}\\\left. \frac{d}{d t} 0 \right|_{\substack{ t=0 }}\end{matrix}\right] = \left[\begin{matrix}0\\4\\0\end{matrix}\right] \]r′(0).✓ Proved
- \[ \left[\begin{matrix}\left. \frac{d^{2}}{d t^{2}} 4 \cos{\left(t \right)} \right|_{\substack{ t=0 }}\\\left. \frac{d^{2}}{d t^{2}} 4 \sin{\left(t \right)} \right|_{\substack{ t=0 }}\\\left. \frac{d^{2}}{d t^{2}} 0 \right|_{\substack{ t=0 }}\end{matrix}\right] = \left[\begin{matrix}-4\\0\\0\end{matrix}\right] \]r″(0).✓ Proved
- \[ \left[\begin{matrix}4 \cdot 0 + 0 \cdot 0\\0 \left(-4\right) + 0 \cdot 0\\0 \cdot 0 - 4 \left(-4\right)\end{matrix}\right] = \left[\begin{matrix}0\\0\\16\end{matrix}\right] \]r′ × r″.✓ Proved
- \[ \frac{\sqrt{0^{2} + 0^{2} + 16^{2}}}{\left(\sqrt{0^{2} + 0^{2} + 4^{2}}\right)^{3}} = \frac{1}{4} \]κ = ‖r′ × r″‖ / ‖r′‖³.✓ Proved
Answer \( \kappa = \frac{1}{4} \)
✓ 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 — The solution correctly applies the curvature formula for a space curve, computes the derivatives and cross product accurately, and arrives at the correct result.
Every verdict on record (6)
qwen3.6:27b-mlx: pass 2026-10-18 — The solution correctly applies the curvature formula for a space curve, computes the derivatives and cross product accurately, and arrives at the correct result.gpt-oss:20b: pass 2026-10-18qwen3.6:27b-mlx: pass 2026-10-16gpt-oss:20b: pass 2026-10-16qwen3.6:27b-mlx: pass 2026-10-15 — The solution correctly applies the curvature formula for a space curve using the cross product of the first and second derivatives. The calculations for r'(0), r''(0), their cross product, and the final curvature value are all correct.gpt-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.