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!

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

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