I have got stuck in showing whether this particular subsets are submanifolds(smooth):
Let $f:\mathbb{R}\to\mathbb{R}$ be the function $f(x)=|x|$. Then the graph of the function f is smooth $1-$dimensional manifold, but it is not a smooth submanifold of $\mathbb{R}^2$.
That it is an $1-$dimensional manifold I have been able to show but I not getting the later part at all.
Is the spiral $X$ given as $$X=\{(e^t \cos t,e^t \sin t):\ t \in \mathbb{R}\}$$ a submanifold of $\mathbb{R}^2$?
I have no idea to show this part as well. I think it would be so. Thanks in advance for any help.