1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
|
<!DOCTYPE html>
<!--[if lt IE 7]> <html class="no-js lt-ie9 lt-ie8 lt-ie7"> <![endif]-->
<!--[if IE 7]> <html class="no-js lt-ie9 lt-ie8"> <![endif]-->
<!--[if IE 8]> <html class="no-js lt-ie9"> <![endif]-->
<!--[if gt IE 8]><!--> <html class="no-js"> <!--<![endif]-->
<head>
<meta charset="utf-8">
<meta http-equiv="X-UA-Compatible" content="IE=edge,chrome=1">
<title>The Spark Debugger - Spark 1.0.0 Documentation</title>
<meta name="description" content="">
<link rel="stylesheet" href="css/bootstrap.min.css">
<style>
body {
padding-top: 60px;
padding-bottom: 40px;
}
</style>
<meta name="viewport" content="width=device-width">
<link rel="stylesheet" href="css/bootstrap-responsive.min.css">
<link rel="stylesheet" href="css/main.css">
<script src="js/vendor/modernizr-2.6.1-respond-1.1.0.min.js"></script>
<link rel="stylesheet" href="css/pygments-default.css">
<!-- Google analytics script -->
<script type="text/javascript">
var _gaq = _gaq || [];
_gaq.push(['_setAccount', 'UA-32518208-1']);
_gaq.push(['_trackPageview']);
(function() {
var ga = document.createElement('script'); ga.type = 'text/javascript'; ga.async = true;
ga.src = ('https:' == document.location.protocol ? 'https://ssl' : 'http://www') + '.google-analytics.com/ga.js';
var s = document.getElementsByTagName('script')[0]; s.parentNode.insertBefore(ga, s);
})();
</script>
</head>
<body>
<!--[if lt IE 7]>
<p class="chromeframe">You are using an outdated browser. <a href="http://browsehappy.com/">Upgrade your browser today</a> or <a href="http://www.google.com/chromeframe/?redirect=true">install Google Chrome Frame</a> to better experience this site.</p>
<![endif]-->
<!-- This code is taken from http://twitter.github.com/bootstrap/examples/hero.html -->
<div class="navbar navbar-fixed-top" id="topbar">
<div class="navbar-inner">
<div class="container">
<div class="brand"><a href="index.html">
<img src="img/spark-logo-hd.png" style="height:50px;"/></a><span class="version">1.0.0</span>
</div>
<ul class="nav">
<!--TODO(andyk): Add class="active" attribute to li some how.-->
<li><a href="index.html">Overview</a></li>
<li class="dropdown">
<a href="#" class="dropdown-toggle" data-toggle="dropdown">Programming Guides<b class="caret"></b></a>
<ul class="dropdown-menu">
<li><a href="quick-start.html">Quick Start</a></li>
<li><a href="programming-guide.html">Spark Programming Guide</a></li>
<li class="divider"></li>
<li><a href="streaming-programming-guide.html">Spark Streaming</a></li>
<li><a href="sql-programming-guide.html">Spark SQL</a></li>
<li><a href="mllib-guide.html">MLlib (Machine Learning)</a></li>
<li><a href="graphx-programming-guide.html">GraphX (Graph Processing)</a></li>
<li><a href="bagel-programming-guide.html">Bagel (Pregel on Spark)</a></li>
</ul>
</li>
<li class="dropdown">
<a href="#" class="dropdown-toggle" data-toggle="dropdown">API Docs<b class="caret"></b></a>
<ul class="dropdown-menu">
<li><a href="api/scala/index.html#org.apache.spark.package">Scaladoc</a></li>
<li><a href="api/java/index.html">Javadoc</a></li>
<li><a href="api/python/index.html">Python API</a></li>
</ul>
</li>
<li class="dropdown">
<a href="#" class="dropdown-toggle" data-toggle="dropdown">Deploying<b class="caret"></b></a>
<ul class="dropdown-menu">
<li><a href="cluster-overview.html">Overview</a></li>
<li><a href="submitting-applications.html">Submitting Applications</a></li>
<li class="divider"></li>
<li><a href="ec2-scripts.html">Amazon EC2</a></li>
<li><a href="spark-standalone.html">Standalone Mode</a></li>
<li><a href="running-on-mesos.html">Mesos</a></li>
<li><a href="running-on-yarn.html">YARN</a></li>
</ul>
</li>
<li class="dropdown">
<a href="api.html" class="dropdown-toggle" data-toggle="dropdown">More<b class="caret"></b></a>
<ul class="dropdown-menu">
<li><a href="configuration.html">Configuration</a></li>
<li><a href="monitoring.html">Monitoring</a></li>
<li><a href="tuning.html">Tuning Guide</a></li>
<li><a href="job-scheduling.html">Job Scheduling</a></li>
<li><a href="security.html">Security</a></li>
<li><a href="hardware-provisioning.html">Hardware Provisioning</a></li>
<li><a href="hadoop-third-party-distributions.html">3<sup>rd</sup>-Party Hadoop Distros</a></li>
<li class="divider"></li>
<li><a href="building-with-maven.html">Building Spark with Maven</a></li>
<li><a href="https://cwiki.apache.org/confluence/display/SPARK/Contributing+to+Spark">Contributing to Spark</a></li>
</ul>
</li>
</ul>
<!--<p class="navbar-text pull-right"><span class="version-text">v1.0.0</span></p>-->
</div>
</div>
</div>
<div class="container" id="content">
<h1 class="title">The Spark Debugger</h1>
<p><strong>Summary:</strong> The Spark debugger provides replay debugging for deterministic (logic) errors in Spark programs. It’s currently in development, but you can try it out in the <a href="https://github.com/apache/spark/tree/arthur">arthur branch</a>.</p>
<h2 id="introduction">Introduction</h2>
<p>From a user’s point of view, debugging a general distributed program can be tedious and confusing. Many distributed programs are nondeterministic; their outcome depends on the interleaving between computation and message passing across multiple machines. Also, the fact that a program is running on a cluster of hundreds or thousands of machines means that it’s hard to understand the program state and pinpoint the location of problems.</p>
<p>In order to tame nondeterminism, a distributed debugger has to log a lot of information, imposing a serious performance penalty on the application being debugged.</p>
<p>But the Spark programming model lets us provide replay debugging for almost zero overhead. Spark programs are a series of RDDs and deterministic transformations, so when debugging a Spark program, we don’t have to debug it all at once – instead, we can debug each transformation individually. Broadly, the debugger lets us do the following two things:</p>
<ul>
<li>Recompute and inspect intermediate RDDs after the program has finished.</li>
<li>Re-run a particular task in a single-threaded debugger to find exactly what went wrong.</li>
</ul>
<p>For deterministic errors, debugging a Spark program is now as easy as debugging a single-threaded one.</p>
<h2 id="approach">Approach</h2>
<p>As your Spark program runs, the slaves report key events back to the master – for example, RDD creations, RDD contents, and uncaught exceptions. (A full list of event types is in <a href="https://github.com/apache/spark/blob/arthur/core/src/main/scala/spark/EventLogging.scala">EventLogging.scala</a>.) The master logs those events, and you can load the event log into the debugger after your program is done running.</p>
<p><em>A note on nondeterminism:</em> For fault recovery, Spark requires RDD transformations (for example, the function passed to <code>RDD.map</code>) to be deterministic. The Spark debugger also relies on this property, and it can also warn you if your transformation is nondeterministic. This works by checksumming the contents of each RDD and comparing the checksums from the original execution to the checksums after recomputing the RDD in the debugger.</p>
<h2 id="usage">Usage</h2>
<h3 id="enabling-the-event-log">Enabling the event log</h3>
<p>To turn on event logging for your program, set <code>$SPARK_JAVA_OPTS</code> in <code>conf/spark-env.sh</code> as follows:</p>
<div class="highlight"><pre><code class="bash"><span class="nb">export </span><span class="nv">SPARK_JAVA_OPTS</span><span class="o">=</span><span class="s1">'-Dspark.arthur.logPath=path/to/event-log'</span>
</code></pre></div>
<p>where <code>path/to/event-log</code> is where you want the event log to go relative to <code>$SPARK_HOME</code>.</p>
<p><strong>Warning:</strong> If <code>path/to/event-log</code> already exists, event logging will be automatically disabled.</p>
<h3 id="loading-the-event-log-into-the-debugger">Loading the event log into the debugger</h3>
<ol>
<li>Run a Spark shell with <code>./bin/spark-shell --master <i>hist</i></code>.</li>
<li>Use <code>EventLogReader</code> to load the event log as follows:</li>
</ol>
<div class="highlight"><pre><code class="scala"><span class="n">spark</span><span class="o">></span> <span class="k">val</span> <span class="n">r</span> <span class="k">=</span> <span class="k">new</span> <span class="n">spark</span><span class="o">.</span><span class="nc">EventLogReader</span><span class="o">(</span><span class="n">sc</span><span class="o">,</span> <span class="nc">Some</span><span class="o">(</span><span class="s">"path/to/event-log"</span><span class="o">))</span>
<span class="n">r</span><span class="k">:</span> <span class="kt">spark.EventLogReader</span> <span class="o">=</span> <span class="n">spark</span><span class="o">.</span><span class="nc">EventLogReader</span><span class="k">@</span><span class="mi">726</span><span class="n">b37ad</span>
</code></pre></div>
<pre><code>**Warning:** If the event log doesn't exist or is unreadable, this will silently fail and `r.events` will be empty.
</code></pre>
<h3 id="exploring-intermediate-rdds">Exploring intermediate RDDs</h3>
<p>Use <code>r.rdds</code> to get a list of intermediate RDDs generated during your program’s execution. An RDD with id <em>x</em> is located at <code>r.rdds(<i>x</i>)</code>. For example:</p>
<div class="highlight"><pre><code class="scala"><span class="n">scala</span><span class="o">></span> <span class="n">r</span><span class="o">.</span><span class="n">rdds</span>
<span class="n">res8</span><span class="k">:</span> <span class="kt">scala.collection.mutable.ArrayBuffer</span><span class="o">[</span><span class="kt">spark.RDD</span><span class="o">[</span><span class="k">_</span><span class="o">]]</span> <span class="k">=</span> <span class="nc">ArrayBuffer</span><span class="o">(</span><span class="n">spark</span><span class="o">.</span><span class="nc">HadoopRDD</span><span class="nd">@fe85adf</span><span class="o">,</span> <span class="n">spark</span><span class="o">.</span><span class="nc">MappedRDD</span><span class="k">@</span><span class="mi">5</span><span class="n">fa5eea1</span><span class="o">,</span> <span class="n">spark</span><span class="o">.</span><span class="nc">MappedRDD</span><span class="k">@</span><span class="mi">6</span><span class="n">d5bd16</span><span class="o">,</span> <span class="n">spark</span><span class="o">.</span><span class="nc">ShuffledRDD</span><span class="k">@</span><span class="mi">3</span><span class="n">a70f2db</span><span class="o">,</span> <span class="n">spark</span><span class="o">.</span><span class="nc">FlatMappedValuesRDD</span><span class="k">@</span><span class="mi">4</span><span class="n">d5825d6</span><span class="o">,</span> <span class="n">spark</span><span class="o">.</span><span class="nc">MappedValuesRDD</span><span class="k">@</span><span class="mi">561</span><span class="n">c2c45</span><span class="o">,</span> <span class="n">spark</span><span class="o">.</span><span class="nc">CoGroupedRDD</span><span class="k">@</span><span class="mi">539</span><span class="n">e922d</span><span class="o">,</span> <span class="n">spark</span><span class="o">.</span><span class="nc">MappedValuesRDD</span><span class="k">@</span><span class="mi">4</span><span class="n">f8ef33e</span><span class="o">,</span> <span class="n">spark</span><span class="o">.</span><span class="nc">FlatMappedRDD</span><span class="k">@</span><span class="mi">32039440</span><span class="o">,</span> <span class="n">spark</span><span class="o">.</span><span class="nc">ShuffledRDD</span><span class="k">@</span><span class="mi">8</span><span class="n">fa0f67</span><span class="o">,</span> <span class="n">spark</span><span class="o">.</span><span class="nc">MappedValuesRDD</span><span class="k">@</span><span class="mi">590937</span><span class="n">cb</span><span class="o">,</span> <span class="n">spark</span><span class="o">.</span><span class="nc">CoGroupedRDD</span><span class="k">@</span><span class="mi">6</span><span class="n">c2e1e17</span><span class="o">,</span> <span class="n">spark</span><span class="o">.</span><span class="nc">MappedValuesRDD</span><span class="k">@</span><span class="mi">47</span><span class="n">b9af7d</span><span class="o">,</span> <span class="n">spark</span><span class="o">.</span><span class="nc">FlatMappedRDD</span><span class="k">@</span><span class="mi">6</span><span class="n">fb05c54</span><span class="o">,</span> <span class="n">spark</span><span class="o">.</span><span class="nc">ShuffledRDD</span><span class="k">@</span><span class="mi">237</span><span class="n">dc815</span><span class="o">,</span> <span class="n">spark</span><span class="o">.</span><span class="nc">MappedValuesRDD</span><span class="k">@</span><span class="mi">16</span><span class="n">daece7</span><span class="o">,</span> <span class="n">spark</span><span class="o">.</span><span class="nc">CoGroupedRDD</span><span class="k">@</span><span class="mi">7</span><span class="n">ef73d69</span><span class="o">,</span> <span class="n">spark</span><span class="o">.</span><span class="nc">MappedValuesRDD</span><span class="k">@</span><span class="mi">19</span><span class="n">e0f99e</span><span class="o">,</span> <span class="n">spark</span><span class="o">.</span><span class="nc">FlatMappedRDD</span><span class="k">@</span><span class="mi">1240158</span><span class="o">,</span> <span class="n">spark</span><span class="o">.</span><span class="nc">ShuffledRDD</span><span class="k">@</span><span class="mi">62</span><span class="n">d438fd</span><span class="o">,</span> <span class="n">spark</span><span class="o">.</span><span class="nc">MappedValuesRDD</span><span class="k">@</span><span class="mi">5</span><span class="n">ae99cbb</span><span class="o">,</span> <span class="n">spark</span><span class="o">.</span><span class="nc">FilteredRDD</span><span class="k">@</span><span class="mi">1</span><span class="n">f30e79e</span><span class="o">,</span> <span class="n">spark</span><span class="o">.</span><span class="nc">MappedRDD</span><span class="k">@</span><span class="mi">43</span><span class="n">b64611</span><span class="o">)</span>
</code></pre></div>
<p>Use <code>r.printRDDs()</code> to get a formatted list of intermediate RDDs, along with the source location where they were created. For example:</p>
<div class="highlight"><pre><code class="scala"><span class="n">scala</span><span class="o">></span> <span class="n">r</span><span class="o">.</span><span class="n">printRDDs</span>
<span class="k">#</span><span class="mi">00</span><span class="k">:</span> <span class="kt">HadoopRDD</span> <span class="kt">spark.bagel.examples.WikipediaPageRankStandalone$.main</span><span class="o">(</span><span class="kt">WikipediaPageRankStandalone.scala:</span><span class="err">31</span><span class="o">)</span>
<span class="k">#</span><span class="err">01</span><span class="kt">:</span> <span class="kt">MappedRDD</span> <span class="kt">spark.bagel.examples.WikipediaPageRankStandalone$.main</span><span class="o">(</span><span class="kt">WikipediaPageRankStandalone.scala:</span><span class="err">31</span><span class="o">)</span>
<span class="k">#</span><span class="err">02</span><span class="kt">:</span> <span class="kt">MappedRDD</span> <span class="kt">spark.bagel.examples.WikipediaPageRankStandalone$.main</span><span class="o">(</span><span class="kt">WikipediaPageRankStandalone.scala:</span><span class="err">35</span><span class="o">)</span>
<span class="k">#</span><span class="err">03</span><span class="kt">:</span> <span class="kt">ShuffledRDD</span> <span class="kt">spark.bagel.examples.WikipediaPageRankStandalone$.main</span><span class="o">(</span><span class="kt">WikipediaPageRankStandalone.scala:</span><span class="err">35</span><span class="o">)</span>
<span class="k">#</span><span class="err">04</span><span class="kt">:</span> <span class="kt">FlatMappedValuesRDD</span> <span class="kt">spark.bagel.examples.WikipediaPageRankStandalone$.main</span><span class="o">(</span><span class="kt">WikipediaPageRankStandalone.scala:</span><span class="err">35</span><span class="o">)</span>
<span class="k">#</span><span class="err">05</span><span class="kt">:</span> <span class="kt">MappedValuesRDD</span> <span class="kt">spark.bagel.examples.WikipediaPageRankStandalone$.pageRank</span><span class="o">(</span><span class="kt">WikipediaPageRankStandalone.scala:</span><span class="err">91</span><span class="o">)</span>
<span class="k">#</span><span class="err">06</span><span class="kt">:</span> <span class="kt">CoGroupedRDD</span> <span class="kt">spark.bagel.examples.WikipediaPageRankStandalone$.pageRank</span><span class="o">(</span><span class="kt">WikipediaPageRankStandalone.scala:</span><span class="err">92</span><span class="o">)</span>
<span class="err">[</span><span class="kt">...</span><span class="err">]</span>
</code></pre></div>
<p>Use <code>r.visualizeRDDs()</code> to visualize the RDDs as a dependency graph. For example:</p>
<div class="highlight"><pre><code class="scala"><span class="n">scala</span><span class="o">></span> <span class="n">r</span><span class="o">.</span><span class="n">visualizeRDDs</span>
<span class="o">/</span><span class="n">tmp</span><span class="o">/</span><span class="n">spark</span><span class="o">-</span><span class="n">rdds</span><span class="o">-</span><span class="mf">3758182885839775712.</span><span class="n">pdf</span>
</code></pre></div>
<p><img src="http://www.ankurdave.com/images/rdd-dep-graph.png" alt="Example RDD dependency graph" /></p>
<p>Iterate over the <code>RDDCreation</code> entries in <code>r.events</code> (e.g. <code>for (RDDCreation(rdd, location) <- events)</code>) to access the RDD creation locations as well as the RDDs themselves.</p>
<h3 id="debugging-a-particular-task">Debugging a particular task</h3>
<ol>
<li>Find the task you want to debug. If the task threw an exception, the <code>ExceptionEvent</code> that was created will have a reference to the task. For example:</li>
</ol>
<div class="highlight"><pre><code class="scala"><span class="n">spark</span><span class="o">></span> <span class="k">val</span> <span class="n">task</span> <span class="k">=</span> <span class="n">r</span><span class="o">.</span><span class="n">events</span><span class="o">.</span><span class="n">collect</span> <span class="o">{</span> <span class="k">case</span> <span class="n">e</span><span class="k">:</span> <span class="kt">ExceptionEvent</span> <span class="o">=></span> <span class="n">e</span> <span class="o">}.</span><span class="n">head</span><span class="o">.</span><span class="n">task</span>
</code></pre></div>
<pre><code>Otherwise, look through the list of all tasks in `r.tasks`, or browse tasks by RDD using <code>r.tasksForRDD(<i>rdd</i>)</code>, which returns a list of tasks whose input is the given RDD.
</code></pre>
<ol>
<li>Run the task by calling <code>r.debugTask(<i>taskStageId</i>, <i>taskPartition</i>)</code>. The task should contain these two values; you can extract them as follows:</li>
</ol>
<div class="highlight"><pre><code class="scala"><span class="k">val</span> <span class="o">(</span><span class="n">taskStageId</span><span class="o">,</span> <span class="n">taskPartition</span><span class="o">)</span> <span class="k">=</span> <span class="n">task</span> <span class="k">match</span> <span class="o">{</span>
<span class="k">case</span> <span class="n">rt</span><span class="k">:</span> <span class="kt">ResultTask</span><span class="o">[</span><span class="k">_</span>, <span class="k">_</span><span class="o">]</span> <span class="k">=></span> <span class="o">(</span><span class="n">rt</span><span class="o">.</span><span class="n">stageId</span><span class="o">,</span> <span class="n">rt</span><span class="o">.</span><span class="n">partition</span><span class="o">)</span>
<span class="k">case</span> <span class="n">smt</span><span class="k">:</span> <span class="kt">ShuffleMapTask</span> <span class="o">=></span> <span class="o">(</span><span class="n">smt</span><span class="o">.</span><span class="n">stageId</span><span class="o">,</span> <span class="n">smt</span><span class="o">.</span><span class="n">partition</span><span class="o">)</span>
<span class="k">case</span> <span class="k">_</span> <span class="k">=></span> <span class="k">throw</span> <span class="k">new</span> <span class="nc">UnsupportedOperationException</span>
<span class="o">})</span>
</code></pre></div>
<pre><code>The Spark debugger will launch the task in a separate JVM, but you will see the task's stdout and stderr inline with the Spark shell. If you want to pass custom debugging arguments to the task's JVM (for example, to change the debugging port), set the optional `debugOpts` argument to `r.debugTask`. When `debugOpts` is left unset, it defaults to:
</code></pre>
<div class="highlight"><pre><code class="scala"><span class="o">-</span><span class="nc">Xdebug</span> <span class="o">-</span><span class="n">agentlib</span><span class="k">:</span><span class="kt">jdwp</span><span class="o">=</span><span class="n">transport</span><span class="k">=</span><span class="n">dt_socket</span><span class="o">,</span><span class="n">server</span><span class="k">=</span><span class="n">y</span><span class="o">,</span><span class="n">suspend</span><span class="k">=</span><span class="n">y</span><span class="o">,</span><span class="n">address</span><span class="k">=</span><span class="mi">8000</span>
</code></pre></div>
<ol>
<li>
<p>In another terminal, attach your favorite conventional debugger to the Spark shell. For example, if you want to use jdb, run <code>jdb -attach 8000</code>.</p>
</li>
<li>
<p>Debug the task as you would debug a normal program. For example, to break when an exception is thrown:</p>
</li>
</ol>
<div class="highlight"><pre><code class="scala"><span class="o">></span> <span class="k">catch</span> <span class="n">org</span><span class="o">.</span><span class="n">xml</span><span class="o">.</span><span class="n">sax</span><span class="o">.</span><span class="nc">SAXParseException</span>
</code></pre></div>
<ol>
<li>When the task ends, its JVM will quit and control will return to the main Spark shell. To stop it prematurely, you can kill it from the debugger, or interrupt it from the terminal with Ctrl-C.</li>
</ol>
<h3 id="detecting-nondeterminism-in-your-transformations">Detecting nondeterminism in your transformations</h3>
<p>When a task gets run more than once, Arthur is able to compare the checksums of the task’s output. If they are different, Arthur will insert a <code>ChecksumEvent</code> into <code>r.checksumMismatches</code> and print a warning like the following:</p>
<div class="highlight"><pre><code class="scala"><span class="mi">12</span><span class="o">/</span><span class="mi">04</span><span class="o">/</span><span class="mi">07</span> <span class="mi">11</span><span class="k">:</span><span class="err">42</span><span class="kt">:</span><span class="err">44</span> <span class="kt">WARN</span> <span class="kt">spark.EventLogWriter:</span> <span class="kt">Nondeterminism</span> <span class="kt">detected</span> <span class="kt">in</span> <span class="kt">shuffle</span> <span class="kt">output</span> <span class="kt">on</span> <span class="kt">RDD</span> <span class="err">2</span><span class="o">,</span> <span class="n">partition</span> <span class="mi">3</span><span class="o">,</span> <span class="n">output</span> <span class="n">split</span> <span class="mi">0</span>
</code></pre></div>
</div> <!-- /container -->
<script src="js/vendor/jquery-1.8.0.min.js"></script>
<script src="js/vendor/bootstrap.min.js"></script>
<script src="js/main.js"></script>
<!-- MathJax Section -->
<script type="text/x-mathjax-config">
MathJax.Hub.Config({
TeX: { equationNumbers: { autoNumber: "AMS" } }
});
</script>
<script type="text/javascript"
src="http://cdn.mathjax.org/mathjax/latest/MathJax.js?config=TeX-AMS-MML_HTMLorMML"></script>
<script>
MathJax.Hub.Config({
tex2jax: {
inlineMath: [ ["$", "$"], ["\\\\(","\\\\)"] ],
displayMath: [ ["$$","$$"], ["\\[", "\\]"] ],
processEscapes: true,
skipTags: ['script', 'noscript', 'style', 'textarea', 'pre']
}
});
</script>
</body>
</html>
|
