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Higher-order partial derivatives

Problem 10.624 · medium

For \( \displaystyle f(x, y) = e^{x} \sin{\left(y \right)} \), 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 (\( \displaystyle f_{xx} + f_{yy} = 0 \)).

\[ \frac{\partial^{2}}{\partial x^{2}} e^{x} \sin{\left(y \right)} = e^{x} \sin{\left(y \right)} \]
f_xx.✓ Proved
\[ \frac{\partial^{2}}{\partial y^{2}} e^{x} \sin{\left(y \right)} = - e^{x} \sin{\left(y \right)} \]
f_yy.✓ Proved
\[ \frac{\partial^{2}}{\partial y\partial x} e^{x} \sin{\left(y \right)} = e^{x} \cos{\left(y \right)} \]
f_xy.✓ Proved
\[ \frac{\partial^{2}}{\partial x\partial y} e^{x} \sin{\left(y \right)} = e^{x} \cos{\left(y \right)} \]
f_yx, the same (Clairaut).✓ Proved
\[ \frac{\partial^{2}}{\partial x^{2}} e^{x} \sin{\left(y \right)} + \frac{\partial^{2}}{\partial y^{2}} e^{x} \sin{\left(y \right)} = 0 \]
f_xx + f_yy: 0, so f is harmonic.✓ Proved

Answer \( f_{xx} = e^{x} \sin{\left(y \right)},\ f_{yy} = - e^{x} \sin{\left(y \right)},\ f_{xy} = e^{x} \cos{\left(y \right)};\ \text{harmonic} \)

✓ 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: pass
qwen3.6:27b-mlx: pass — The solution correctly computes the partial derivatives, verifies the equality of mixed partials, and correctly concludes that the function is harmonic.

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-10-17 — The solution correctly computes the partial derivatives, verifies the equality of mixed partials, and correctly concludes that the function is harmonic.
gpt-oss:20b: pass 2026-10-17
qwen3.6:27b-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 2026-10-15
qwen3.6:27b-mlx: pass 2026-10-14 — The solution correctly computes the partial derivatives, verifies the equality of mixed partials, and correctly concludes that the function is harmonic because the sum of the second partials is 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.