Higher-order partial derivatives
Problem 10.619 · medium
For \( \displaystyle f(x, y) = x^{2} + 2 x y - y^{2} \), find \( \displaystyle f_{xx} \), \( \displaystyle f_{yy} \) and \( \displaystyle f_{xy} \), check that \( \displaystyle f_{xy} = f_{yx} \), and decide whether \( \displaystyle f \) is harmonic (\( \displaystyle f_{xx} + f_{yy} = 0 \)).
- \[ \frac{\partial^{2}}{\partial x^{2}} \left(x^{2} + 2 x y - y^{2}\right) = 2 \]f_xx.✓ Proved
- \[ \frac{\partial^{2}}{\partial y^{2}} \left(x^{2} + 2 x y - y^{2}\right) = -2 \]f_yy.✓ Proved
- \[ \frac{\partial^{2}}{\partial y\partial x} \left(x^{2} + 2 x y - y^{2}\right) = 2 \]f_xy.✓ Proved
- \[ \frac{\partial^{2}}{\partial x\partial y} \left(x^{2} + 2 x y - y^{2}\right) = 2 \]f_yx, the same (Clairaut).✓ Proved
- \[ \frac{\partial^{2}}{\partial x^{2}} \left(x^{2} + 2 x y - y^{2}\right) + \frac{\partial^{2}}{\partial y^{2}} \left(x^{2} + 2 x y - y^{2}\right) = 0 \]f_xx + f_yy: 0, so f is harmonic.✓ Proved
Answer \( f_{xx} = 2,\ f_{yy} = -2,\ f_{xy} = 2;\ \text{harmonic} \)
✓ 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 |
| 5 | ✓ 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 | second differences of f at two points |
Reviewers
gpt-oss:20b: passqwen3.6:27b-mlx: pass — The solution correctly computes all required partial derivatives and verifies the harmonic condition. The reference to Clairaut's theorem is appropriate given the continuity of the mixed partials.
Every verdict on record (6)
qwen3.6:27b-mlx: pass 2026-10-17 — The solution correctly computes all required partial derivatives and verifies the harmonic condition. The reference to Clairaut's theorem is appropriate given the continuity of the mixed partials.gpt-oss:20b: pass 2026-10-17qwen3.6:27b-mlx: pass 2026-10-15 — The solution correctly computes the partial derivatives, verifies the equality of mixed partials, and correctly concludes that the function is harmonic based on the Laplacian being zero.gpt-oss:20b: pass 2026-10-15qwen3.6:27b-mlx: pass 2026-10-14 — The solution correctly computes the partial derivatives, verifies Clairaut's theorem, and confirms the function is harmonic. The logic is sound and the steps are clear.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/higher_order_partials, checked 2026-10-17 with SymPy 1.14.0.