Sphere as a Level Set

The sphere \(\mathbb{S}^n\) is a regular level set .

To prove this, consider \(f\colon \mathbb{R}^{n+1}\to \mathbb{R}\) with \(f(x)=\lvert x\rvert^2\). Then \(df_x(v)=2\sum_{i=1}^{n+1} v^ix^i\), which is a surjective function except at the origin.