- #1
mathmonkey
- 34
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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!
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!