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Jacobians and change of variables

Problem 11.213 · easy

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

\[ \left[\begin{matrix}\frac{\partial}{\partial u} \left(u + 4 v\right) & \frac{\partial}{\partial v} \left(u + 4 v\right)\\\frac{\partial}{\partial u} \left(- 3 u + 3 v\right) & \frac{\partial}{\partial v} \left(- 3 u + 3 v\right)\end{matrix}\right] = \left[\begin{matrix}1 & 4\\-3 & 3\end{matrix}\right] \]
The matrix of partial derivatives.✓ Proved
\[ 15 \]
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.
Reviewed

Answer \( 15 \)

Lines: 2 proved, 1 reviewed. 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	Reviewed	reviewers: gpt-oss:20b, qwen3.6:27b-mlx	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: pass — The solution correctly computes the Jacobian determinant as 15. The final sentence correctly identifies the absolute value of the Jacobian as the area scaling factor, which is consistent with the positive result.

Every verdict on record (4)

qwen3.6:27b-mlx: pass 2026-10-04 — The solution correctly computes the Jacobian determinant as 15. The final sentence correctly identifies the absolute value of the Jacobian as the area scaling factor, which is consistent with the positive result.
gpt-oss:20b: pass 2026-10-04
qwen3.6:27b-mlx: fail (error) 2026-10-04 — The problem asks for the Jacobian, which is the determinant of the Jacobian matrix. The calculated determinant is 15, but the solution incorrectly introduces the absolute value |J| in the final sentence. While |J| is used for area scaling, the Jacobian itself is signed (and can be negative), so defining the answer via |J| is conceptually incorrect for finding the Jacobian.
gpt-oss:20b: pass 2026-10-04

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 2026-10-04 with SymPy 1.14.0.