Higher-order partial derivatives
Problem 10.622 · medium
For \( \displaystyle f(x, y) = x^{3} y^{2} - 3 x y \), 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 (\( \displaystyle f_{xx} + f_{yy} = 0 \)).
- \[ \frac{\partial^{2}}{\partial x^{2}} \left(x^{3} y^{2} - 3 x y\right) = 6 x y^{2} \]f_xx.✓ Proved
- \[ \frac{\partial^{2}}{\partial y^{2}} \left(x^{3} y^{2} - 3 x y\right) = 2 x^{3} \]f_yy.✓ Proved
- \[ \frac{\partial^{2}}{\partial y\partial x} \left(x^{3} y^{2} - 3 x y\right) = 6 x^{2} y - 3 \]f_xy.✓ Proved
- \[ \frac{\partial^{2}}{\partial x\partial y} \left(x^{3} y^{2} - 3 x y\right) = 6 x^{2} y - 3 \]f_yx, the same (Clairaut).✓ Proved
- \[ \frac{\partial^{2}}{\partial x^{2}} \left(x^{3} y^{2} - 3 x y\right) + \frac{\partial^{2}}{\partial y^{2}} \left(x^{3} y^{2} - 3 x y\right) = 2 x \left(x^{2} + 3 y^{2}\right) \]f_xx + f_yy: not 0, so f is not harmonic.✓ Proved
Answer \( f_{xx} = 6 x y^{2},\ f_{yy} = 2 x^{3},\ f_{xy} = 6 x^{2} y - 3;\ \text{not harmonic} \)
✓ 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 equality of mixed partials. The conclusion that the function is not harmonic follows logically from the non-zero Laplacian.
Every verdict on record (6)
qwen3.6:27b-mlx: pass 2026-10-17 — The solution correctly computes all required partial derivatives and verifies the equality of mixed partials. The conclusion that the function is not harmonic follows logically from the non-zero Laplacian.gpt-oss:20b: pass 2026-10-17qwen3.6:27b-mlx: pass 2026-10-15 — The solution correctly computes the partial derivatives, verifies Clairaut's theorem, and correctly determines that the function is not harmonic by showing the Laplacian is non-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 correctly concludes that the function is not harmonic because the Laplacian is not identically zero.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.