How to determine that a given map from $\mathbb{R}^2 \to \mathbb{R}^2$ preserves area or not?

Question

I wanted to show whether the following map preserves area or not, $$f(x,y)=(x+y^2,y+x^2).$$

I tried to draw the graph but as function is not linear, my approach could not work.

Can any one suggest me some approach so that I can work this out?

Answer 1

If it preserved areas, then $f'(a,b)$ would also preserve areas for each $(a,b)\in\mathbb{R}^2$. But $\det f'\left(\frac12,\frac12\right)=0$.

Answer 2

A linear map $A:\>{\mathbb R}^2\to{\mathbb R}^2$ multiplies all areas by the factor $\bigl|{\rm det}(A)\bigr|$. It follows that a $C^1$ map $$f:\>{\mathbb R}^2\to{\mathbb R}^2,\qquad (x,y)\mapsto \bigl(u(x,y),v(x,y)\bigr)$$ possesses a "local area scaling factor" $\bigl|J_f(x,y)\bigr|$ that continuously changes from point to point. The so-called Jacobian determinant $J_f(x,y)$ is given by $$J_f(x,y)={\rm det}\bigl(df(x,y)\bigr)={\rm det}\left[\matrix{u_x&u_y\cr v_x&v_y\cr}\right]\ .$$ In your example we obtain $\bigl|J_f(x,y)\bigr|=|1-4xy|$, which is not constant. For an area preserving map we would need $\bigl|J_f(x,y)\bigr|\equiv1$.

Answer 3

Take a look at the image under $f$ of the unit square $(0,0), (1,0), (1,1), (0,1)$.