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Jacobians and change of variables

Problem 11.327 · easy

Find the Jacobian \( \displaystyle \dfrac{\partial(x, y)}{\partial(u, v)} \) of the transformation \( \displaystyle x = - u - 4 v \), \( \displaystyle y = 2 u + 3 v \).

\[ \left[\begin{matrix}\frac{\partial}{\partial u} \left(- u - 4 v\right) & \frac{\partial}{\partial v} \left(- u - 4 v\right)\\\frac{\partial}{\partial u} \left(2 u + 3 v\right) & \frac{\partial}{\partial v} \left(2 u + 3 v\right)\end{matrix}\right] = \left[\begin{matrix}-1 & -4\\2 & 3\end{matrix}\right] \]
The matrix of partial derivatives.✓ Proved
\[ 5 \]
Its determinant: x_u·y_v − x_v·y_u.✓ Proved
|J| is the factor by which the transformation stretches small areas: dA = |J| du dv.

Answer \( 5 \)

Lines: 2 proved, 1 not checked. 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.

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	Not checked	—	a sentence; read, not computed
answer, a second way	✓ Checked independently	sympy 1.14.0 + mpmath 1.3.0	the image of a tiny square measured by the shoelace formula

Reviewers

gpt-oss:20b: pass
qwen3.6:27b-mlx: fail (misleading) — The problem asks for the Jacobian determinant, which is -5, but the solution states the answer is 5. Line 3 incorrectly conflates the Jacobian with its absolute value (the area scaling factor), leading to the wrong sign in the final answer.

Every verdict on record (4)

gpt-oss:20b: pass 2026-10-10
qwen3.6:27b-mlx: fail (misleading) 2026-10-10 — The problem asks for the Jacobian determinant, which is -5, but the solution states the answer is 5. Line 3 incorrectly conflates the Jacobian with its absolute value (the area scaling factor), leading to the wrong sign in the final answer.
qwen3.6:27b-mlx: fail (error) 2026-10-10 — The problem asks for the Jacobian determinant, which is -5, but the solution calculates the absolute value (5). The final sentence incorrectly defines the Jacobian as the absolute value used for area scaling, whereas the Jacobian itself is the signed determinant.
gpt-oss:20b: pass 2026-10-10

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/jacobian, checked 2026-10-10 with SymPy 1.14.0.