I’m trying to show that $S^1={ (x,y) in mathbb{R}^2: x^2+y^2=1}$ is a submanifold of $mathbb{R}^2$ using the following definition:

A subset $M in mathbb{R}^n$ is called a smooth k-dimensional submanifold of $mathbb{R}^n$, $k leq n$, if any point $x in M$ has a neighborhood $O_x$ in $mathbb{R}^n$ and there exists a smooth vector-function $$Phi:O_x to O_0 subset mathbb{R}^{n},$$ onto a neighborhood of the origin $0 in O_0 subset mathbb{R}^n$ with $$text{ } rank frac{d Phi}{dx}|_x=n $$

such that $ Phi(O_x cap M)= mathbb{R}^k cap O_0$.

I know the idea is to make the circle straight locally, but I can’t explain $Phi$, my initial proposal would be $O_x = B_{frac{1}{4}}(x)$ and $O_0=B_0(1)$, $Phi(x,y)=(frac{x}{sqrt{x^2+y^2}+1}, frac{y}{sqrt{x^2+y^2}+1})$, but $Phi(O_x cap M) neq mathbb{R}^k cap O_0$.How can I write to $Phi?$