An area preserving diffeomorphism between a disk and an ellipse

Question

This is a self-answered question. I post it here since (embarrassingly) it took me some time to realize that the solution is obvious.
Let $D subseteq mathbb R^2$ be the closed unit disk and let $E$ be an ellipse with the same area, i.e.  with minor and major axes of lengths $a<b$ and $ab=1$.
$$
E={(x,y) , | , frac{x^2}{a^2} + frac{y^2}{b^2} le 1 }
$$

Question: Can we construct explicitly an area preserving diffeomorphism $f:D to E$?

(i.e. $Jf=1$ identically on $D$).

Asaf Shachar · Answer

We have $A(E)=pi ab$, so $A(E)=A(D)=pi$ if and only if $ab=1$.
The linear map $f:(x,y) to (tilde x ,tilde y)=(ax,by)$ is an area preserving diffeomorphism $D to E$.
Clearly $Jf=ab=1$. We just need to make sure that $f$ maps $D$ onto $E$.
Indeed,
$$
(tilde x ,tilde y) in E iff (frac{tilde x}{a})^2 + (frac{tilde y}{b})^2 le 1 iff x^2+y^2 le 1 iff (x.y) in D.
$$

An area preserving diffeomorphism between a disk and an ellipse

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