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author | Dylan Holmes <ocsenave@gmail.com> |
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date | Thu, 27 Oct 2011 22:38:23 -0500 |
parents | b4de894a1e2e |
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}55 /*]]>*/-->56 </style>57 <link rel="stylesheet" type="text/css" href="../css/aurellem.css" />58 <script type="text/javascript">59 <!--/*--><![CDATA[/*><!--*/60 function CodeHighlightOn(elem, id)61 {62 var target = document.getElementById(id);63 if(null != target) {64 elem.cacheClassElem = elem.className;65 elem.cacheClassTarget = target.className;66 target.className = "code-highlighted";67 elem.className = "code-highlighted";68 }69 }70 function CodeHighlightOff(elem, id)71 {72 var target = document.getElementById(id);73 if(elem.cacheClassElem)74 elem.className = elem.cacheClassElem;75 if(elem.cacheClassTarget)76 target.className = elem.cacheClassTarget;77 }78 /*]]>*///-->79 </script>80 <script type="text/javascript" src="../MathJax/MathJax.js">81 <!--/*--><![CDATA[/*><!--*/82 MathJax.Hub.Config({83 // Only one of the two following lines, depending on user settings84 // First allows browser-native MathML display, second forces HTML/CSS85 config: ["MMLorHTML.js"], jax: ["input/TeX"],86 // jax: ["input/TeX", "output/HTML-CSS"],87 extensions: ["tex2jax.js","TeX/AMSmath.js","TeX/AMSsymbols.js",88 "TeX/noUndefined.js"],89 tex2jax: {90 inlineMath: [ ["\\(","\\)"] ],91 displayMath: [ ['$$','$$'], ["\\[","\\]"], ["\\begin{displaymath}","\\end{displaymath}"] ],92 skipTags: ["script","noscript","style","textarea","pre","code"],93 ignoreClass: "tex2jax_ignore",94 processEscapes: false,95 processEnvironments: true,96 preview: "TeX"97 },98 showProcessingMessages: true,99 displayAlign: "left",100 displayIndent: "2em",102 "HTML-CSS": {103 scale: 100,104 availableFonts: ["STIX","TeX"],105 preferredFont: "TeX",106 webFont: "TeX",107 imageFont: "TeX",108 showMathMenu: true,109 },110 MMLorHTML: {111 prefer: {112 MSIE: "MML",113 Firefox: "MML",114 Opera: "HTML",115 other: "HTML"116 }117 }118 });119 /*]]>*///-->120 </script>121 </head>122 <body>124 <div id="content">128 <div class="header">129 <div class="float-right">130 <!--131 <form>132 <input type="text"/><input type="submit" value="search the blog »"/>133 </form>134 -->135 </div>137 <h1>aurellem <em>☉</em></h1>138 <ul class="nav">139 <li><a href="/">read the blog »</a></li>140 <!-- li><a href="#">learn about us »</a></li-->141 </ul>142 </div>144 <h1 class="title">An Unambiguous Notation for Derivatives</h1>152 <div id="table-of-contents">153 <h2>Table of Contents</h2>154 <div id="text-table-of-contents">155 <ul>156 <li><a href="#sec-1">1 Calculus of Infinitesimals </a>157 <ul>158 <li><a href="#sec-1-1">1.1 Differential Objects </a></li>159 <li><a href="#sec-1-2">1.2 Interactions obey the chain rule </a></li>160 </ul>161 </li>162 </ul>163 </div>164 </div>166 <div id="outline-container-1" class="outline-2">167 <h2 id="sec-1"><span class="section-number-2">1</span> Calculus of Infinitesimals </h2>168 <div class="outline-text-2" id="text-1">171 </div>173 <div id="outline-container-1-1" class="outline-3">174 <h3 id="sec-1-1"><span class="section-number-3">1.1</span> Differential Objects </h3>175 <div class="outline-text-3" id="text-1-1">178 <p>179 A <b>differential object</b> is a pair \([x,\,dx]\) consisting of a variable180 and an infinitely small increment of it. Differential objects can181 interact with functions, producing a new differential object as a182 result; this interaction is for calculating derivatives of functions.183 </p>184 <p>185 Differential objects are for186 calculating derivatives of functions: the derivative of \(f\) with187 respect to \(x\)188 </p>189 <p>190 You can “apply”191 functions to differential objects; the result is:192 </p>195 \([x,dx]\xrightarrow{\quad f \quad}[f(x), Df(x)\cdot dx].\)197 <p>198 Loosely speaking, the interaction of \(f\) and a differential object199 of \(x\) is a differential object of \(f\).200 </p>202 </div>204 </div>206 <div id="outline-container-1-2" class="outline-3">207 <h3 id="sec-1-2"><span class="section-number-3">1.2</span> Interactions obey the chain rule </h3>208 <div class="outline-text-3" id="text-1-2">211 <p>212 The interaction of \(f\) and the differential object \([x, dx]\) is213 a differential object \([f(x), Df(x)\cdot dx]\). Because of the rule for214 interactions, if you apply another function \(g\), you get the215 chain-rule answer you expect:216 </p>219 \([f(x), Df(x)\cdot dx]\xrightarrow{\quad g\quad}\left[g(f(x)),\,220 Dg(f(x))\cdot Df(x)\cdot dx\right]\)229 </div>230 </div>231 </div>232 <div id="postamble">233 <p class="date">Date: 2011-08-08 02:49:24 EDT</p>234 <p class="author">Author: Dylan Holmes</p>235 <p class="creator">Org version 7.6 with Emacs version 23</p>236 <a href="http://validator.w3.org/check?uri=referer">Validate XHTML 1.0</a>237 </div>238 </div>239 </body>240 </html>