Let $M$ submanifold of $\mathbb{R}^n$. Show that $T^1M = \{(p,v) \in TM; \| v\| = 1 \}$ is a submanifold of $TM$ of dimension $2n-1$.
Comments: I'm trying to build the function $$f: TM \longrightarrow \mathbb{R}, \ \ \ f(p,v) = \| v \|^2 $$
This way is correct or need to take charts of submanifolds? I'm having trouble proving that $1$ is a regular value.
Hint: is enough (why?) to prove that $1$ is a regular value of $$N: v\longmapsto\|v\|^2 = \langle v,v\rangle.$$ Write it as a composition of functions, $N = B\circ\Delta$: $$\Delta(v) = \pmatrix{v\cr v}$$ (the diagonal, linear) with $$B\pmatrix{v\cr w} = \langle v,w\rangle $$ (scalar product, bilinear) and apply the chain rule.