I'm a bit confused by a notation in Spivak's Calculus on Manifolds p. 137. In problem 5-35 he says let $F(x)=x_x$ (see the image below) or $F(x)=x_{\chi}$. I have no idea what this notation is, and I'm not even sure which is the correct one.


What does $x_x$ or $x_{\chi}$ mean?

  • 1
    $\begingroup$ Is $x_x$ a tangent vector to $\mathbb R^n$ at the point $x$? Look at Spivak's notation for tangent vectors introduced earlier in the book. $\endgroup$
    – littleO
    Commented Nov 14, 2019 at 23:59
  • $\begingroup$ @littleO Yup, seems as though it is. That's the only thing that makes sense given the context. Where's the notation introduced earlier? Don't remember passing by it, but it's very possible I've forgotten. $\endgroup$ Commented Nov 15, 2019 at 15:56

1 Answer 1


As mentioned in the comments, $x_x$ is indeed the vector $x \in \Bbb{R}^n$ thought of as being an element of the tangent space $T_x(\Bbb{R}^n)$. This notation is intorduced in Spivak's Chapter 4, in the section on Fields and forms (there he uses the notation $\Bbb{R}^n_p$ and $(p,v)$ or equivalently $v_p$).


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