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| author | Karl Berry <karl@freefriends.org> | 2019-05-05 21:37:48 +0000 |
|---|---|---|
| committer | Karl Berry <karl@freefriends.org> | 2019-05-05 21:37:48 +0000 |
| commit | 89916d9520fa753d876b1c3d0b300c91be1eb5d3 (patch) | |
| tree | fc660d0e59c789d69305213aeec0eb16d633e604 /Master/texmf-dist/doc/latex | |
| parent | 8d59b9cee353c936139f25239f49f76d4cae8373 (diff) | |
mathfont
git-svn-id: svn://tug.org/texlive/trunk@51017 c570f23f-e606-0410-a88d-b1316a301751
Diffstat (limited to 'Master/texmf-dist/doc/latex')
| -rw-r--r-- | Master/texmf-dist/doc/latex/mathfont/mathfont.pdf | bin | 297850 -> 302105 bytes | |||
| -rw-r--r-- | Master/texmf-dist/doc/latex/mathfont/mathfont_example.pdf | bin | 86282 -> 86247 bytes | |||
| -rw-r--r-- | Master/texmf-dist/doc/latex/mathfont/mathfont_example.tex | 2 |
3 files changed, 1 insertions, 1 deletions
diff --git a/Master/texmf-dist/doc/latex/mathfont/mathfont.pdf b/Master/texmf-dist/doc/latex/mathfont/mathfont.pdf Binary files differindex b9945378efe..eef2355f748 100644 --- a/Master/texmf-dist/doc/latex/mathfont/mathfont.pdf +++ b/Master/texmf-dist/doc/latex/mathfont/mathfont.pdf diff --git a/Master/texmf-dist/doc/latex/mathfont/mathfont_example.pdf b/Master/texmf-dist/doc/latex/mathfont/mathfont_example.pdf Binary files differindex a441a812a4f..c7bf1445ae0 100644 --- a/Master/texmf-dist/doc/latex/mathfont/mathfont_example.pdf +++ b/Master/texmf-dist/doc/latex/mathfont/mathfont_example.pdf 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 \[ \psi_j^{}\circ f\circ\phi_i^{-1} \] |