Normal lines and normal vectors to surfaces
Problem 10.656 · medium
Find a normal vector and the normal line to the surface \( \displaystyle x^{2} + 2 y^{2} - z = 0 \) at \( \displaystyle (1, 0, 1) \).
- \[ \left. x^{2} + 2 y^{2} - z \right|_{\substack{ x=1\\ y=0\\ z=1 }} = 0 \]The point is on the surface.✓ Proved
- \[ \left[\begin{matrix}\left. \frac{\partial}{\partial x} \left(x^{2} + 2 y^{2} - z\right) \right|_{\substack{ x=1\\ y=0\\ z=1 }}\\\left. \frac{\partial}{\partial y} \left(x^{2} + 2 y^{2} - z\right) \right|_{\substack{ x=1\\ y=0\\ z=1 }}\\\left. \frac{\partial}{\partial z} \left(x^{2} + 2 y^{2} - z\right) \right|_{\substack{ x=1\\ y=0\\ z=1 }}\end{matrix}\right] = \left[\begin{matrix}2\\0\\-1\end{matrix}\right] \]∇F at the point is normal to the level surface.✓ Proved
- \[ \left[\begin{matrix}\left. 2 t + 1 \right|_{\substack{ t=0 }}\\\left. 0 \right|_{\substack{ t=0 }}\\\left. 1 - t \right|_{\substack{ t=0 }}\end{matrix}\right] = \left[\begin{matrix}1\\0\\1\end{matrix}\right] \]The line r(t) = P + t∇F passes through P at t = 0.✓ Proved
Answer \( \mathbf n = \left\langle 2, 0, -1 \right\rangle,\quad \mathbf r(t) = \left\langle 2 t + 1, 0, 1 - t \right\rangle \)
✓ 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 |
| answer, a second way | ✓ Checked independently | sympy 1.14.0 + mpmath 1.3.0 | the claimed normal is parallel to a finite-difference gradient |
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 2026-10-16gpt-oss:20b: pass 2026-10-16qwen3.6:27b-mlx: pass 2026-10-14gpt-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/normal_line, checked 2026-10-18 with SymPy 1.14.0.