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?
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 $v_p$).