Arc length and curvature of space curves
Problem 9.666 · medium
Find the curvature of \( \displaystyle \mathbf r(t) = \left\langle t, \sin{\left(t \right)}, 0 \right\rangle \) at \( \displaystyle t = 0 \).
- \[ \left[\begin{matrix}\left. \frac{d}{d t} t \right|_{\substack{ t=0 }}\\\left. \frac{d}{d t} \sin{\left(t \right)} \right|_{\substack{ t=0 }}\\\left. \frac{d}{d t} 0 \right|_{\substack{ t=0 }}\end{matrix}\right] = \left[\begin{matrix}1\\1\\0\end{matrix}\right] \]r′(0).✓ Proved
- \[ \left[\begin{matrix}\left. \frac{d^{2}}{d t^{2}} t \right|_{\substack{ t=0 }}\\\left. \frac{d^{2}}{d t^{2}} \sin{\left(t \right)} \right|_{\substack{ t=0 }}\\\left. \frac{d^{2}}{d t^{2}} 0 \right|_{\substack{ t=0 }}\end{matrix}\right] = \left[\begin{matrix}0\\0\\0\end{matrix}\right] \]r″(0).✓ Proved
- \[ \left[\begin{matrix}1 \cdot 0 + 0 \cdot 0\\0 \cdot 0 + 0\\1 \cdot 0 + 0\end{matrix}\right] = \left[\begin{matrix}0\\0\\0\end{matrix}\right] \]r′ × r″.✓ Proved
- \[ \frac{\sqrt{0^{2} + 0^{2} + 0^{2}}}{\left(\sqrt{0^{2} + 1^{2} + 1^{2}}\right)^{3}} = 0 \]κ = ‖r′ × r″‖ / ‖r′‖³.✓ Proved
Answer \( \kappa = 0 \)
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 — The solution correctly applies the curvature formula using the cross product of the first and second derivatives. The calculations for r'(0), r''(0), and their cross product are correct, leading to the correct result.
Every verdict on record (6)
qwen3.6:27b-mlx: pass 2026-10-17 — The solution correctly applies the curvature formula using the cross product of the first and second derivatives. The calculations for r'(0), r''(0), and their cross product are correct, leading to the correct result.gpt-oss:20b: pass 2026-10-17qwen3.6:27b-mlx: fail (error) 2026-10-15 — The second derivative of r(t) at t=0 is calculated as the zero vector, but r''(t) = <0, -sin(t), 0>, so r''(0) should be <0, 0, 0>. Wait, sin(0)=0, so r''(0) IS <0,0,0>. Let me re-check. r(t) = <t, sin(t), 0>. r'(t) = <1, cos(t), 0>. r'(0) = <1, 1, 0>. r''(t) = <0, -sin(t), 0>. r''(0) = <0, 0, 0>. The cross product is indeed 0. The curvature is 0. The solution is actually correct. Why did I think it was wrong? Let me look closer at the provided solution text. Line 2 says r''(0) is <0,0,0>. This is correct. Line 3 says cross product is <0,0,0>. Correct. Line 4 calculates kappa = 0. Correct. Is there a subtle error? The problem asks for curvature. The formula used is correct. The values are correct. The result is correct. Wait, let me double check the derivative of sin(t). d/dt sin(t) = cos(t). d^2/dt^2 sin(t) = -sin(t). At t=0, -sin(0) = 0. So r''(0) is indeed <0,0,0>. The curvature of a straight line is 0. Near t=0, sin(t) ~ t, so the curve looks like y=x, which is a straight line. So curvature 0 is correct. Is there any issue with the notation or the steps? The steps are logically sound and algebraically correct. The 'proved' lines are correct. The sentences are just labels. Let me re-read the prompt constraints. 'Find what is wrong'. If nothing is wrong, I should say pass. Let me check if I missed a sign or a value. r'(0) = <1, 1, 0>. Norm is sqrt(2). r''(0) = <0, 0, 0>. Cross product is 0. Kappa = 0 / (sqrt(2))^3 = 0. Everything seems correct. I will mark it as pass.gpt-oss:20b: pass 2026-10-15qwen3.6:27b-mlx: fail (error) 2026-10-14 — The second derivative of r(t) is calculated incorrectly; r''(0) should be <0, -1, 0>, not <0, 0, 0>. This leads to an incorrect curvature of 0 instead of 1/sqrt(2).gpt-oss:20b: pass 2026-10-14
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.