Are the following functions [itex]\mathbb{R}^2\rightarrow\mathbb{R}^2 [/itex] diffeomorphisms. If not is there an open set containing the origin on which the function is a diffeomorphism to its image?(adsbygoogle = window.adsbygoogle || []).push({});

1. [tex](x,y)\mapsto(x+y^3, y) [/tex]

2. [tex](x,y)\mapsto(x+x^3,x) [/tex]

I have the definition of a diffeomorphism: [itex]f:X\rightarrow Y [/itex] is a diffeomorphism if f^{-1}exists, and f and f^{-1}are [itex]C^{\infty}[/itex] maps.

Where f is a [itex]C^{\infty}[/itex] map iff coordinates of a point [itex]y \in Y[/itex] are infinitely differentiable functions of [itex]f^{-1}(y)\in X[/itex].

So, for 1. I have tried the following:

[tex]f:\mathbb{R}^2\rightarrow \mathbb{R}^2: (x,y)\mapsto(x+y^3, y) =(u,v) \\

\Rightarrow x+y^3=u , y=v \Rightarrow x=u-v^3, y=v [/tex]

So [itex]f^{-1}: (u,v)\mapsto (u-v^3,v)=(x,y) [/itex], and thus f^{-1}exists and both f and f^{-1}are [itex]C^{\infty}[/itex], therefore f is a diffeomorphism.

Is this right? I've tried a similar method for Q2, but to no avail!

Any help would be much appreciated!

**Physics Forums | Science Articles, Homework Help, Discussion**

Dismiss Notice

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# On which the function is a diffeomorphism to its image?

**Physics Forums | Science Articles, Homework Help, Discussion**