Home›Calculus 3›Jacobians and change of variables›Problem 11.204

Jacobians and change of variables

Problem 11.204 · easy

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

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

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 and provides a standard, correct interpretation of its absolute value in the context of change of variables.

Every verdict on record (4)

qwen3.6:27b-mlx: pass 2026-10-04 — The solution correctly computes the Jacobian determinant and provides a standard, correct interpretation of its absolute value in the context of change of variables.
gpt-oss:20b: pass 2026-10-04
qwen3.6:27b-mlx: fail (error) 2026-10-04 — The problem asks for the Jacobian determinant, but line 3 incorrectly defines |J| as the area scaling factor, implying the absolute value is required for the Jacobian itself or conflating the Jacobian with its absolute value. The Jacobian is signed; the area element uses the absolute value. Furthermore, the stated answer is just the determinant, so introducing |J| in the explanation is conceptually confusing and potentially misleading regarding the sign.
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.