Home›Calculus 3›Jacobians and change of variables›Problem 11.330

Jacobians and change of variables

Problem 11.330 · easy

Find the Jacobian \( \displaystyle \dfrac{\partial(x, y)}{\partial(u, v)} \) of the transformation \( \displaystyle x = 2 u - 3 v \), \( \displaystyle y = - u - 4 v \).

\[ \left[\begin{matrix}\frac{\partial}{\partial u} \left(2 u - 3 v\right) & \frac{\partial}{\partial v} \left(2 u - 3 v\right)\\\frac{\partial}{\partial u} \left(- u - 4 v\right) & \frac{\partial}{\partial v} \left(- u - 4 v\right)\end{matrix}\right] = \left[\begin{matrix}2 & -3\\-1 & -4\end{matrix}\right] \]
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.

Answer \( -11 \)

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: pass
qwen3.6:27b-mlx: fail (misleading) — The problem asks for the Jacobian determinant, which is -11. The final sentence discusses |J| (the absolute value), which is 11. This distinction is crucial: the Jacobian can be negative, but area scaling factors are positive. Stating that |J| is the factor without clarifying that the requested answer is the signed determinant J, not |J|, is misleading and potentially confusing for a student trying to reconcile the answer -11 with the geometric interpretation.

Every verdict on record (4)

gpt-oss:20b: pass 2026-10-10
qwen3.6:27b-mlx: fail (misleading) 2026-10-10 — The problem asks for the Jacobian determinant, which is -11. The final sentence discusses |J| (the absolute value), which is 11. This distinction is crucial: the Jacobian can be negative, but area scaling factors are positive. Stating that |J| is the factor without clarifying that the requested answer is the signed determinant J, not |J|, is misleading and potentially confusing for a student trying to reconcile the answer -11 with the geometric interpretation.
qwen3.6:27b-mlx: fail (misleading) 2026-10-10 — The problem asks for the Jacobian determinant, which is -11. The solution concludes by defining |J| as the area scaling factor, which is 11. This distinction is crucial: the Jacobian can be negative, but the area element uses the absolute value. Stating the final result in terms of |J| without clarifying that the Jacobian itself is -11 is misleading.
gpt-oss:20b: pass 2026-10-10

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 2026-10-10 with SymPy 1.14.0.