Home›Calculus 3›Jacobians and change of variables›Problem 11.329

Jacobians and change of variables

Problem 11.329 · easy

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

\[ \left[\begin{matrix}\frac{\partial}{\partial u} \left(- u - v\right) & \frac{\partial}{\partial v} \left(- u - v\right)\\\frac{\partial}{\partial u} \left(- 2 u + 2 v\right) & \frac{\partial}{\partial v} \left(- 2 u + 2 v\right)\end{matrix}\right] = \left[\begin{matrix}-1 & -1\\-2 & 2\end{matrix}\right] \]
The matrix of partial derivatives.✓ Proved
\[ -4 \]
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 \( -4 \)

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, but the final sentence discusses the absolute value |J| used for area scaling. This distinction is crucial; the Jacobian itself is -4, not 4, and conflating the two can lead to sign errors in change of variables for oriented integrals.

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, but the final sentence discusses the absolute value |J| used for area scaling. This distinction is crucial; the Jacobian itself is -4, not 4, and conflating the two can lead to sign errors in change of variables for oriented integrals.
qwen3.6:27b-mlx: pass 2026-10-10 — The solution correctly computes the Jacobian determinant as -4. The final sentence correctly identifies the absolute value of the Jacobian as the area scaling factor, which is a standard and correct interpretation in the context of change of variables.
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.