Tangent Space Definition (Munkres Analysis on Manifolds)

[mathmonkey]

Hi all,

I'm quite confused concerning the definition of tangent vectors and tangent spaces as presented in Munkres's Analysis on Manifolds. Here is the book's definition:

Given $\textbf{x} \in \mathbb{R}^n$, we define a tangent vector to $\mathbb{R}^n$ at $\textbf{x}$ to be a pair $(\textbf{x};\textbf{v})$, where $\textbf{v} \in \mathbb{R}^n$. The set of all tangent vectors to $\mathbb{R}^n$ at $\textbf{x}$ forms a vector space if we define

$(\textbf{x}; \textbf{v}) + (\textbf{x};\textbf{w}) = (\textbf{x}; \textbf{v + w})$,
$c(\textbf{x};\textbf{v}) = (\textbf{x};c\textbf{v})$.

It is called the tangent space to $\mathbb{R}^n$ at $\textbf{x}$, and is denoted $T_x(\mathbb{R}^n)$.


I'm not quite sure what is meant by the definition. First off, I'm not sure i understand the notation $(\textbf{x};\textbf{v})$, which appears to be an ordered pair of vectors. Next, Munkres goes on to describe $(\textbf{x};\textbf{v})$ as "an arrow with its initial point at $\textbf{x}$, with $T_x(\mathbb{R}^n)$ as the set of all arrows with their initial point at $\textbf{x}$. What I don't understand is from this description, isn't $T_x(\mathbb{R}^n$ just spanning all of $\mathbb{R}^n)$? What is the distinction between the two?

Munkres also describes the set $T_x(\mathbb{R}^n)$ as "just the set $\textbf{x} \times \mathbb{R}^n$. I am also unfamiliar with this notation. If $\textbf{x}$ is also in $\mathbb{R}^n$, then is $T_x(\mathbb{R}^n)$ a subset of $\mathbb{R}^{2n}$? That doesn't seem right to me, although I just don't know how to interpret Munkres's explanation.

I think my problem lies in my not understanding the notation used in this chapter. Perhaps the best way for me to understand is maybe with an example. If $\textbf{x} = (1,1) \in \mathbb{R}^2$, then what is $T_x(\mathbb{R}^2)$ according to Munkres's definition?

Thanks so much for the help all!

 

[tiny-tim]

hi mathmonkey!

I'm not quite sure what is meant by the definition. First off, I'm not sure i understand the notation $(\textbf{x};\textbf{v})$, which appears to be an ordered pair of vectors. Next, Munkres goes on to describe $(\textbf{x};\textbf{v})$ as "an arrow with its initial point at $\textbf{x}$, with $T_x(\mathbb{R}^n)$ as the set of all arrows with their initial point at $\textbf{x}$. What I don't understand is from this description, isn't $T_x(\mathbb{R}^n$ just spanning all of $\mathbb{R}^n)$? What is the distinction between the two?

i think it's easier if instead of ℝⁿ, we use the surface, S, of an egg

then, for each point x on S, T_xS (or T_x(S)) is the tangent plane at x

although S is round, T_xS is flat, and is isomorphic to ℝ²

T_xS and T_yS are isomorphic to each other, even if x and y are at points of the egg of different curvature

S has nothing to do with ℝ² (except that they're there same dimension)

Mukres's example has ℝ² instead of S, so in that particular case the manifold and the tangent space are isomorphic, but they still have nothing to do with each other ​

Munkres also describes the set $T_x(\mathbb{R}^n)$ as "just the set $\textbf{x} \times \mathbb{R}^n$. I am also unfamiliar with this notation. If $\textbf{x}$ is also in $\mathbb{R}^n$, then is $T_x(\mathbb{R}^n)$ a subset of $\mathbb{R}^{2n}$? That doesn't seem right to me, although I just don't know how to interpret Munkres's explanation.

he's just saying that each tangent plane is defined not just as the ℝ² that it's isomorphic to, but also by the point of attachment, x

 

[mathmonkey]

Hi tiny-tim,

Thanks for your help. So am I right in understanding that in my above example where $x = (1,1) \in \mathbb{R}^2$, then $T_x(\mathbb{R}^2)$ is in fact the entire plane $\mathbb{R}^2$, where each of the pairs $(\textbf{x};\textbf{v})$ is a vector $\textbf{v}$ originating from the point $\textbf{x}$? I have not yet gotten to the part of the text relating this definition to manifolds, so perhaps right now this definition seems to me for the time being unmotivated, especially as I already have preconceived notions of what a "tangent" space looks like based on its name. But I just wanted to make sure I understood the formal definition clearly.

Thanks again for your help!

 

[tiny-tim]

… So am I right in understanding that in my above example where $x = (1,1) \in \mathbb{R}^2$, then $T_x(\mathbb{R}^2)$ is in fact the entire plane $\mathbb{R}^2$, where each of the pairs $(\textbf{x};\textbf{v})$ is a vector $\textbf{v}$ originating from the point $\textbf{x}$?

yes, it's an entire plane

(you shouldn't think of all the tangent planes as being the same … they're not the same thing, they're identical different things )

 

[mathmonkey]

Great, thanks for your help! I just wanted to make sure I understood fully before moving on in the chapter!