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authorKarl Berry <karl@freefriends.org>2019-05-05 21:37:48 +0000
committerKarl Berry <karl@freefriends.org>2019-05-05 21:37:48 +0000
commit89916d9520fa753d876b1c3d0b300c91be1eb5d3 (patch)
treefc660d0e59c789d69305213aeec0eb16d633e604 /Master/texmf-dist/doc/latex
parent8d59b9cee353c936139f25239f49f76d4cae8373 (diff)
mathfont
git-svn-id: svn://tug.org/texlive/trunk@51017 c570f23f-e606-0410-a88d-b1316a301751
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@@ -61,7 +61,7 @@ Expanding $e^{iX}$ as a power series gives an expression for $\phi_X$ that we ca
\vfil
-A smooth manifold consists of a topological space $M$ equipped with a smooth maximal atlas $\leftbrace \phi_i\rightbrace$. The maps $\phi_i\colon U_i\longrightarrow\mathbb R$ technically aren't themselves differentiable, but their compositions $\phi_i^{}\circ\phi_j^{-1}$ are diffeomorphisms on subsets of $\mathbb R^n$. If we have a map $f\colon M\longrightarrow N$ between manifolds, this structure allows us to talk about differentiability of $f$. Specifically, we say that $f$ is smooth if for any $i$ and $j$, the composition
+A smooth manifold consists of a topological space $M$ equipped with a smooth maximal atlas $\leftbrace \phi_i\rightbrace$. The maps $\phi_i\colon U_i\longrightarrow\mathbb R$ are continuous, and their compositions $\phi_i^{}\circ\phi_j^{-1}$ are diffeomorphisms on subsets of $\mathbb R^n$. If we have a map $f\colon M\longrightarrow N$ between manifolds, this structure allows us to talk about differentiability of $f$. Specifically, we say that $f$ is smooth if for any $i$ and $j$, the composition
\[
\psi_j^{}\circ f\circ\phi_i^{-1}
\]