Definition 1: Suppose M a differentiable manifold and $$p\in M$$.
A funtion $$f:M \rightarrow \mathbb{R}$$ is differentiable at $$p \in M$$ iff $$\exists U_p \subset M$$ : $$f:U_p \rightarrow \mathbb{R}$$ is differentiable.
Definition 2:Dp ={set of all differentiable functions at p in M}...
Suppose M is a manifold and $$T_{p}M$$ is the tangent space at a point $$p \in M$$. How do i prove that it is indeed a vector space using the axioms:
Suppose that u,v, w $$\in V$$. where u,v, w are vectors and $$\V$$ is a vector space
$$u + v \in V \tag{Closure under addition}$$
$$u + v = v +...
Suppose $$ D=\{ (x,0) \in \mathbb{R}^2 : x \in \mathbb{R}\} \cup \{ (0,y) \in \mathbb{R}^2 : y \in \mathbb{R} \}$$ is a subset of $$\mathbb{R}^2 $$ with subspace topology. Can this be a 1d or 2d manifold?
Thank you!
Suppose you have the map $$\pi : \mathbb{R}^{n+1}-\{0\} \longrightarrow \mathbb{P}^n$$.
I need to prove that the map is differentiable.
But this map is a chart of $$\mathbb{P}^n$$ so by definition is differentiable?
MENTOR NOTE: fixed Latex mistakes double $ signs and backslashes needed for math