Jacobians and change of variables
Problem 11.209 · medium
Find the Jacobian \( \displaystyle \dfrac{\partial(x, y)}{\partial(u, v)} \) of the transformation \( \displaystyle x = u^{2} - v^{2} \), \( \displaystyle y = 2 u v \).
- \[ \left[\begin{matrix}\frac{\partial}{\partial u} \left(u^{2} - v^{2}\right) & \frac{\partial}{\partial v} \left(u^{2} - v^{2}\right)\\\frac{\partial}{\partial u} 2 u v & \frac{\partial}{\partial v} 2 u v\end{matrix}\right] = \left[\begin{matrix}2 u & - 2 v\\2 v & 2 u\end{matrix}\right] \]The matrix of partial derivatives.✓ Proved
- \[ 4 u^{2} + 4 v^{2} \]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 \( 4 u^{2} + 4 v^{2} \)
✓ 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
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-10-04gpt-oss:20b: pass 2026-10-04qwen3.6:27b-mlx: pass 2026-10-04 — The solution correctly computes the Jacobian determinant and provides a standard, correct interpretation of its geometric meaning in the context of change of variables.gpt-oss:20b: pass 2026-10-04
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 2026-10-04 with SymPy 1.14.0.