I am given the map $(x,y)\mapsto (x+y^2,y+x^2)$. I am unable to find the Jacobian by making the substitution $u=x+y^2$ and $v=y+x^2$. Any hints would be appreciated.
(I am trying to find whether the map is area preserving? I know "the map $f:\mathbb{R}^n\to\mathbb{R}^n$ is area and orientation preserving iff the determinant of the Jacobian is $\pm1$".)
The Jacobian of a multivariable function $f:\mathbb{R}^2 \rightarrow \mathbb{R}^2$ such that $f(x,y)=(u(x,y),v(x,y))$ is:
$$\textbf{J}(f(x,y)) = det\begin{pmatrix} \frac{\partial u}{\partial x} & \frac{\partial u}{\partial y} \\ \frac{\partial v}{\partial x} & \frac{\partial v}{\partial y} \\ \end{pmatrix}$$
where $det$ denotes the determinant of the matrix.
Calculating the partial derivatives of $u(x,y)=x+y^2$ and $v(x,y)=y+x^2$ we have that the Jacobian of your given function is:
$$det\begin{pmatrix} 1 & 2y \\ 2x & 1 \\ \end{pmatrix}=1-4xy$$