Home›Calculus 3›Arc length and curvature of space curves›Problem 9.695

Arc length and curvature of space curves

Problem 9.695 · medium

Find the curvature of \( \displaystyle \mathbf r(t) = \left\langle t, t^{2}, t^{3} \right\rangle \) at \( \displaystyle t = 2 \).

\[ \left[\begin{matrix}\left. \frac{d}{d t} t \right|_{\substack{ t=2 }}\\\left. \frac{d}{d t} t^{2} \right|_{\substack{ t=2 }}\\\left. \frac{d}{d t} t^{3} \right|_{\substack{ t=2 }}\end{matrix}\right] = \left[\begin{matrix}1\\4\\12\end{matrix}\right] \]
r′(2).✓ Proved
\[ \left[\begin{matrix}\left. \frac{d^{2}}{d t^{2}} t \right|_{\substack{ t=2 }}\\\left. \frac{d^{2}}{d t^{2}} t^{2} \right|_{\substack{ t=2 }}\\\left. \frac{d^{2}}{d t^{2}} t^{3} \right|_{\substack{ t=2 }}\end{matrix}\right] = \left[\begin{matrix}0\\2\\12\end{matrix}\right] \]
r″(2).✓ Proved
\[ \left[\begin{matrix}2 \left(-12\right) + 4 \cdot 12\\-12 + 12 \cdot 0\\0 \left(-4\right) + 1 \cdot 2\end{matrix}\right] = \left[\begin{matrix}24\\-12\\2\end{matrix}\right] \]
r′ × r″.✓ Proved
\[ \frac{\sqrt{2^{2} + \left(-12\right)^{2} + 24^{2}}}{\left(\sqrt{1^{2} + 4^{2} + 12^{2}}\right)^{3}} = \frac{2 \sqrt{29141}}{25921} \]
κ = ‖r′ × r″‖ / ‖r′‖³.✓ Proved

Answer \( \kappa = \frac{2 \sqrt{29141}}{25921} \)

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: pass
qwen3.6:27b-mlx: pass

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-10-18
gpt-oss:20b: pass 2026-10-18
qwen3.6:27b-mlx: pass 2026-10-15
gpt-oss:20b: pass 2026-10-15
qwen3.6:27b-mlx: inconclusive 2026-10-14 — reviewer returned a non-object
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 2026-10-18 with SymPy 1.14.0.