Jacobians and change of variables
Problem 11.263 · easy
Find the Jacobian \( \displaystyle \dfrac{\partial(x, y)}{\partial(u, v)} \) of the transformation \( \displaystyle x = 4 u - v \), \( \displaystyle y = - u + 2 v \).
- \[ \left[\begin{matrix}\frac{\partial}{\partial u} \left(4 u - v\right) & \frac{\partial}{\partial v} \left(4 u - v\right)\\\frac{\partial}{\partial u} \left(- u + 2 v\right) & \frac{\partial}{\partial v} \left(- u + 2 v\right)\end{matrix}\right] = \left[\begin{matrix}4 & -1\\-1 & 2\end{matrix}\right] \]The matrix of partial derivatives.✓ Proved
- \[ 7 \]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 \( 7 \)
✓ Nihil obstat Lines: 2 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation.
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 2026-10-06qwen3.6:27b-mlx: pass 2026-10-06 — 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: pass 2026-10-06 — The solution correctly computes the Jacobian determinant as 7. The final sentence correctly identifies the absolute value of the Jacobian as the area scaling factor, which is consistent with the positive result.gpt-oss:20b: pass 2026-10-06
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 2026-10-06 with SymPy 1.14.0.