Consider a set
$$M = \{ (s\cos t, s\sin t, t) \colon s,t\in \mathbb{R}\}\subset \mathbb{R}^3.$$
I am asked to show from the definition that $M$ is a 2-dimensional submanifold of $\mathbb{R}^3$ which is say for each $x\in M$ there is an open set $U$ (relative to $M$), an open set $V\subset \mathbb{R}^2$ together with a diffeomorphism $\varphi\colon U\to M$.
Building $\varphi$ (subject to $x$) seems to be difficult. I tried to employ some version of theorem about local dipheomorphisms but no version seems to be applicable here.
May I ask for help?
Best Answer
This is relatively straightforward, computing the derivative of the parametrization, which I will call $\phi$, gives
$$d\phi=\begin{pmatrix}\cos t \\ \sin t \\ 0\end{pmatrix}ds+\begin{pmatrix} -s\sin t \\ s\cos t \\ 1\end{pmatrix}\,dt$$
clearly the two vectors are linearly independent since the latter has component $dt$ and the former has component $0$ in the third, so that you get two linearly independent vectors at every point, hence the matrix is always rank $2$, and you have a local diffeomorphism.