Home›Calculus 3›Jacobians and change of variables›Problem 11.206

Jacobians and change of variables

Problem 11.206 · easy

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

\[ \left[\begin{matrix}\frac{\partial}{\partial u} \left(u + v^{2}\right) & \frac{\partial}{\partial v} \left(u + v^{2}\right)\\\frac{d}{d u} v & \frac{d}{d v} v\end{matrix}\right] = \left[\begin{matrix}1 & 2 v\\0 & 1\end{matrix}\right] \]
The matrix of partial derivatives.✓ Proved
\[ 1 \]
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 \( 1 \)

Lines: 2 proved, 1 not checked. 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	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 (style) — [domain objection, downgraded to style] The problem asks for the Jacobian determinant, but the solution concludes with the absolute value |J|. While |J| = 1 in this specific case, the Jacobian itself is defined as the determinant (which can be negative), and distinguishing between the two is crucial for change of variables in integrals.

Every verdict on record (4)

qwen3.6:27b-mlx: fail (style) 2026-10-04 — [domain objection, downgraded to style] The problem asks for the Jacobian determinant, but the solution concludes with the absolute value |J|. While |J| = 1 in this specific case, the Jacobian itself is defined as the determinant (which can be negative), and distinguishing between the two is crucial for change of variables in integrals.
gpt-oss:20b: pass 2026-10-04
qwen3.6:27b-mlx: pass 2026-10-04 — The solution correctly computes the Jacobian determinant as 1. The final sentence correctly interprets the absolute value of the Jacobian as the area scaling factor for 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.