Jacobians and change of variables
Problem 11.383 · easy
Find the Jacobian \( \displaystyle \dfrac{\partial(x, y)}{\partial(u, v)} \) of the transformation \( \displaystyle x = - 3 u + v \), \( \displaystyle y = 3 u - 2 v \).
- \[ \left[\begin{matrix}\frac{\partial}{\partial u} \left(- 3 u + v\right) & \frac{\partial}{\partial v} \left(- 3 u + v\right)\\\frac{\partial}{\partial u} \left(3 u - 2 v\right) & \frac{\partial}{\partial v} \left(3 u - 2 v\right)\end{matrix}\right] = \left[\begin{matrix}-3 & 1\\3 & -2\end{matrix}\right] \]The matrix of partial derivatives.✓ Proved
- \[ 3 \]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 \( 3 \)
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 3, but the final sentence discusses |J| (the absolute value) in the context of area scaling. While |3| = 3, distinguishing between the Jacobian determinant and its absolute value is a critical conceptual distinction in multivariable calculus; conflating them teaches that the Jacobian is always positive or that the sign is irrelevant, which is false for orientation-preserving vs. reversing 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 3, but the final sentence discusses |J| (the absolute value) in the context of area scaling. While |3| = 3, distinguishing between the Jacobian determinant and its absolute value is a critical conceptual distinction in multivariable calculus; conflating them teaches that the Jacobian is always positive or that the sign is irrelevant, which is false for orientation-preserving vs. reversing transformations.qwen3.6:27b-mlx: fail (error) 2026-10-13 — The problem asks for the Jacobian determinant, which is -3, but the solution incorrectly calculates it as 3. Additionally, the final sentence discusses the absolute value |J|, which is irrelevant to finding the signed Jacobian requested.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.