Jacobians and change of variables
Problem 11.380 · easy
Find the Jacobian \( \displaystyle \dfrac{\partial(x, y)}{\partial(u, v)} \) of the transformation \( \displaystyle x = 3 u + 4 v \), \( \displaystyle y = - 3 u - 2 v \).
- \[ \left[\begin{matrix}\frac{\partial}{\partial u} \left(3 u + 4 v\right) & \frac{\partial}{\partial v} \left(3 u + 4 v\right)\\\frac{\partial}{\partial u} \left(- 3 u - 2 v\right) & \frac{\partial}{\partial v} \left(- 3 u - 2 v\right)\end{matrix}\right] = \left[\begin{matrix}3 & 4\\-3 & -2\end{matrix}\right] \]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.
Answer \( 6 \)
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 (misleading) — The problem asks for the Jacobian determinant, which is -6, but the solution states the answer is 6. Line 3 incorrectly defines the Jacobian as the absolute value |J|, which is only true for the area scaling factor in integrals, not the Jacobian determinant itself. This distinction is crucial for change of variables involving orientation or inverse transformations.
Every verdict on record (4)
gpt-oss:20b: pass 2026-10-13qwen3.6:27b-mlx: fail (misleading) 2026-10-13 — The problem asks for the Jacobian determinant, which is -6, but the solution states the answer is 6. Line 3 incorrectly defines the Jacobian as the absolute value |J|, which is only true for the area scaling factor in integrals, not the Jacobian determinant itself. This distinction is crucial for change of variables involving orientation or inverse transformations.qwen3.6:27b-mlx: pass 2026-10-13 — 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 for change of variables.gpt-oss:20b: pass 2026-10-13
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 2026-10-13 with SymPy 1.14.0.