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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!

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# Tangent Space Definition (Munkres Analysis on Manifolds)

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