Jacobians and change of variables
Problem 11.357 · easy
Find the Jacobian \( \displaystyle \dfrac{\partial(x, y)}{\partial(u, v)} \) of the transformation \( \displaystyle x = - 3 u - 2 v \), \( \displaystyle y = 4 u - v \).
- \[ \left[\begin{matrix}\frac{\partial}{\partial u} \left(- 3 u - 2 v\right) & \frac{\partial}{\partial v} \left(- 3 u - 2 v\right)\\\frac{\partial}{\partial u} \left(4 u - v\right) & \frac{\partial}{\partial v} \left(4 u - v\right)\end{matrix}\right] = \left[\begin{matrix}-3 & -2\\4 & -1\end{matrix}\right] \]The matrix of partial derivatives.✓ Proved
- \[ 11 \]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 \( 11 \)
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 and provides a standard, correct interpretation of its geometric meaning in the context of 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 and provides a standard, correct interpretation of its geometric meaning in the context of change of variables.qwen3.6:27b-mlx: fail (misleading) 2026-10-11 — The problem asks for the Jacobian determinant, but the solution concludes with a statement about the absolute value |J| used for area scaling. While the numerical value 11 is correct here, the distinction between the signed Jacobian and its absolute value is crucial in calculus, and the final sentence does not answer the specific question asked.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.