X-Git-Url: http://gitweb.michael.orlitzky.com/?a=blobdiff_plain;f=examples.tex;h=6e1c5007abf16c19d48f8eadb7c1a03b85a49cab;hb=4fd0c97de0370110d6fecab412ec1b5b831611e8;hp=26db4b1a03c81a218f354a6eaa23c038de592352;hpb=2719020929ee56f190a9e0e91083fb70ee086c9c;p=mjotex.git

diff --git a/examples.tex b/examples.tex
index 26db4b1..6e1c500 100644
--- a/examples.tex
+++ b/examples.tex
@@ -32,6 +32,11 @@
     domain, then the preimage under $f$ of $A$ is $\preimage{f}{A}$.
   \end{section}
 
+  \begin{section}{Calculus}
+    The gradient of $f : \Rn \rightarrow \Rn[1]$ is $\gradient{f} :
+    \Rn \rightarrow \Rn$.
+  \end{section}
+
   \begin{section}{Common}
     The function $f$ applied to $x$ is $f\of{x}$. We can group terms
     like $a + \qty{b - c}$ or $a + \qty{b - \sqty{c - d}}$. Here's a
@@ -69,6 +74,20 @@
     \begin{equation*}
       \unionmany{k=1}{\infty}{A_{k}} = \intersectmany{k=1}{\infty}{B_{k}}
     \end{equation*}
+
+    Finally, we have the four standard types of intervals in $\Rn[1]$,
+    %
+    \begin{align*}
+      \intervaloo{a}{b} &= \setc{ x \in \Rn[1]}{ a < x < b },\\
+      \intervaloc{a}{b} &= \setc{ x \in \Rn[1]}{ a < x \le b },\\
+      \intervalco{a}{b} &= \setc{ x \in \Rn[1]}{ a \le x < b }, \text{ and }\\
+      \intervalcc{a}{b} &= \setc{ x \in \Rn[1]}{ a \le x \le b }.
+    \end{align*}
+  \end{section}
+
+  \begin{section}{Complex}
+    We sometimes want to conjugate complex numbers like
+    $\compconj{a+bi} = a - bi$.
   \end{section}
 
   \begin{section}{Cone}
@@ -85,7 +104,9 @@
     The conic hull of a set $X$ is $\cone{X}$; its affine hull is
     $\aff{X}$, and its convex hull is $\conv{X}$. If $K$ is a cone,
     then its lineality space is $\linspace{K}$, its lineality is
-    $\lin{K}$, and its extreme directions are $\Ext{K}$.
+    $\lin{K}$, and its extreme directions are $\Ext{K}$. The fact that
+    $F$ is a face of $K$ is denoted by $F \faceof K$; if $F$ is a
+    proper face, then we write $F \properfaceof K$.
   \end{section}
 
   \begin{section}{Font}
@@ -223,6 +244,10 @@
       fox
     \end{theorem}
 
+    \begin{exercise}
+      jumps
+    \end{exercise}
+
     \begin{definition}
       quod
     \end{definition}
@@ -253,6 +278,10 @@
       fox
     \end{theorem*}
 
+    \begin{exercise*}
+      jumps
+    \end{exercise*}
+
     \begin{definition*}
       quod
     \end{definition*}