@TedShifrin well... depending on level of works, some things define things as having properties so as to handwave certain proofs away. Bad practice, sure. But I don't know the authors intent.
@Danu It's basically pigeonholing. Multiply all those partial sums by $1 - x$ so you end up with $1 - x^{n+1}$. Now as you say $x^{n+1}$ hits everything else except $0$, so you just miss $1 - 0 = 1$
So if$X$ and $Y$ are smooth manifolds of dimension $k$ and $l$ respectively, and $x \in X$ and $y \in Y$ and we are given parameterizations $\phi : U \to \phi[U]$ and $\psi : V \to \psi[V]$ of neighbourhoods $\phi[U]$ of $x$ and $\psi[V]$ of $y$ with $psi(v) =y$ and $\phi(u) = x$, then $d\phi_u : \mathbb{R}^k \to TX_x$ is an isomorphism and $d\psi_v : \mathbb{R}^l \to TY_y$ is an isomorphism, and $d(\phi \times \psi)_{(u, v)} : \mathbb{R}^{k+l} \to TX_x \times TY_y$ is an isomorphism
@Perturbative I don't know, but I saw something about block matrices, so maybe $\begin{bmatrix}A&\\&B\end{bmatrix}{}^{-1}= \begin{bmatrix}A^{-1}&\\&B^{-1}\end{bmatrix}$ is relevant
You have to interpret things appropriately, @Perturbative. We have $d\phi_u\colon\Bbb R^k\to\Bbb R^n$, so it's not invertible. But it is an isomorphism to its image, $TxX$, and then there's a smooth map $g$ defined on a neighborhood of $X$ extending $\phi^{-1}$ as a map from a neighborhood of $x$ to $\Bbb R^k$. By definition, $d(\phi^{-1})_x$ is the restriction of $dg_x$ to $T_xX$.
@Semiclassical But I think the "obviously $1+a+a^2+\dotsb$ is fixed under that, and it equals $1/(1+a)^{-1}$ if you handwave enough" argument should work
@TedShifrin I understand that, but I'm not seeing exactly how the fact the inverse of the derivative of a product of maps equals the product of the derivative of maps (if that makes some sense)
I want to try and identify a geometric structure I thought up while doing some weird stuff with making things walk on the surface of a 3D model and trying to incorporate backface culling into the surface geometry itself. See, in computer graphics each side of a polygon or triangle are considered ...