Jacobians and change of variables
Problem 11.255 · easy
Find the Jacobian \( \displaystyle \dfrac{\partial(x, y)}{\partial(u, v)} \) of the transformation \( \displaystyle x = - 2 u - 2 v \), \( \displaystyle y = 3 u - v \).
- \[ \left[\begin{matrix}\frac{\partial}{\partial u} \left(- 2 u - 2 v\right) & \frac{\partial}{\partial v} \left(- 2 u - 2 v\right)\\\frac{\partial}{\partial u} \left(3 u - v\right) & \frac{\partial}{\partial v} \left(3 u - v\right)\end{matrix}\right] = \left[\begin{matrix}-2 & -2\\3 & -1\end{matrix}\right] \]The matrix of partial derivatives.✓ Proved
- \[ 8 \]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 \( 8 \)
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: passqwen3.6:27b-mlx: pass — The solution correctly computes the Jacobian determinant as 8. The final sentence correctly interprets the absolute value of the Jacobian as the area scaling factor, which is consistent with the stated answer.
Every verdict on record (4)
gpt-oss:20b: pass 2026-10-06qwen3.6:27b-mlx: pass 2026-10-06 — The solution correctly computes the Jacobian determinant as 8. The final sentence correctly interprets the absolute value of the Jacobian as the area scaling factor, which is consistent with the stated answer.qwen3.6:27b-mlx: fail (error) 2026-10-06 — The problem asks for the Jacobian determinant, which is -11, not its absolute value. The solution incorrectly computes the absolute value (8) and presents it as the answer, failing to distinguish between the Jacobian and the area scaling factor.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.