Exercise 2184Calculus › Multivariable calculus › Multiple integralsM1Added 3 Nov 2026

Jacobian of a nonlinear transformation

#Jacobian#nonlinear#change of variables

Statement

Show that under the change $(u,v)=(x^2-y^2,2xy)$ , the Jacobian is $4(x^2+y^2)$ .

Hint

Compute the $2\times 2$ determinant of partials.

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Jacobian. $\det\begin{pmatrix}2x&-2y\\2y&2x\end{pmatrix}=4x^2+4y^2$ .

Conclusion. $|J|=4(x^2+y^2)$ .