Jacobians and change of variables
Problem 11.352 · easy
Find the Jacobian \( \displaystyle \dfrac{\partial(x, y)}{\partial(u, v)} \) of the transformation \( \displaystyle x = 3 u - 3 v \), \( \displaystyle y = u - 3 v \).
- \[ \left[\begin{matrix}\frac{\partial}{\partial u} \left(3 u - 3 v\right) & \frac{\partial}{\partial v} \left(3 u - 3 v\right)\\\frac{\partial}{\partial u} \left(u - 3 v\right) & \frac{\partial}{\partial v} \left(u - 3 v\right)\end{matrix}\right] = \left[\begin{matrix}3 & -3\\1 & -3\end{matrix}\right] \]The matrix of partial derivatives.✓ Proved
- \[ -6 \]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 \( -6 \)
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 -6. The final sentence correctly interprets the absolute value of the Jacobian as the area scaling factor, which is standard terminology in multivariable calculus contexts involving change of variables.
Every verdict on record (4)
gpt-oss:20b: pass 2026-10-11qwen3.6:27b-mlx: pass 2026-10-11 — The solution correctly computes the Jacobian determinant as -6. The final sentence correctly interprets the absolute value of the Jacobian as the area scaling factor, which is standard terminology in multivariable calculus contexts involving change of variables.qwen3.6:27b-mlx: fail (misleading) 2026-10-11 — The problem asks for the Jacobian determinant, but the solution concludes by defining |J| (the absolute value) as the area scaling factor. While true for area, this obscures the sign of the Jacobian, which is crucial for orientation and change of variables in integrals. The student might incorrectly believe the Jacobian is always positive or that the sign is irrelevant.gpt-oss:20b: pass 2026-10-11
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 2026-10-11 with SymPy 1.14.0.