From 89916d9520fa753d876b1c3d0b300c91be1eb5d3 Mon Sep 17 00:00:00 2001 From: Karl Berry Date: Sun, 5 May 2019 21:37:48 +0000 Subject: mathfont git-svn-id: svn://tug.org/texlive/trunk@51017 c570f23f-e606-0410-a88d-b1316a301751 --- Master/texmf-dist/doc/latex/mathfont/mathfont.pdf | Bin 297850 -> 302105 bytes .../doc/latex/mathfont/mathfont_example.pdf | Bin 86282 -> 86247 bytes .../doc/latex/mathfont/mathfont_example.tex | 2 +- 3 files changed, 1 insertion(+), 1 deletion(-) (limited to 'Master/texmf-dist/doc/latex/mathfont') diff --git a/Master/texmf-dist/doc/latex/mathfont/mathfont.pdf b/Master/texmf-dist/doc/latex/mathfont/mathfont.pdf index b9945378efe..eef2355f748 100644 Binary files a/Master/texmf-dist/doc/latex/mathfont/mathfont.pdf and b/Master/texmf-dist/doc/latex/mathfont/mathfont.pdf differ diff --git a/Master/texmf-dist/doc/latex/mathfont/mathfont_example.pdf b/Master/texmf-dist/doc/latex/mathfont/mathfont_example.pdf index a441a812a4f..c7bf1445ae0 100644 Binary files a/Master/texmf-dist/doc/latex/mathfont/mathfont_example.pdf and b/Master/texmf-dist/doc/latex/mathfont/mathfont_example.pdf differ diff --git a/Master/texmf-dist/doc/latex/mathfont/mathfont_example.tex b/Master/texmf-dist/doc/latex/mathfont/mathfont_example.tex index d69aace431e..7def38f99f9 100644 --- a/Master/texmf-dist/doc/latex/mathfont/mathfont_example.tex +++ b/Master/texmf-dist/doc/latex/mathfont/mathfont_example.tex @@ -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 \[ \psi_j^{}\circ f\circ\phi_i^{-1} \] -- cgit v1.2.3