Jacobians and change of variables
Problem 11.257 · easy
Find the Jacobian \( \displaystyle \dfrac{\partial(x, y)}{\partial(u, v)} \) of the transformation \( \displaystyle x = - u + 2 v \), \( \displaystyle y = - 3 u + v \).
- \[ \left[\begin{matrix}\frac{\partial}{\partial u} \left(- u + 2 v\right) & \frac{\partial}{\partial v} \left(- u + 2 v\right)\\\frac{\partial}{\partial u} \left(- 3 u + v\right) & \frac{\partial}{\partial v} \left(- 3 u + v\right)\end{matrix}\right] = \left[\begin{matrix}-1 & 2\\-3 & 1\end{matrix}\right] \]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: passqwen3.6:27b-mlx: fail (error) — The problem asks for the Jacobian determinant, which is -5, but the solution incorrectly calculates the absolute value (5). The final sentence also incorrectly defines the Jacobian determinant as the absolute value used for area scaling, whereas the Jacobian itself is the signed determinant.
Every verdict on record (4)
gpt-oss:20b: pass 2026-10-06qwen3.6:27b-mlx: fail (error) 2026-10-06 — The problem asks for the Jacobian determinant, which is -5, but the solution incorrectly calculates the absolute value (5). The final sentence also incorrectly defines the Jacobian determinant as the absolute value used for area scaling, whereas the Jacobian itself is the signed determinant.qwen3.6:27b-mlx: fail (error) 2026-10-06 — The problem asks for the Jacobian determinant, which is -5, but the solution incorrectly calculates the determinant as 5. Additionally, the final sentence discusses the absolute value |J| for area scaling, which is irrelevant to finding the signed Jacobian determinant requested.gpt-oss:20b: pass 2026-10-06
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 2026-10-06 with SymPy 1.14.0.