Mercurial > dylan
diff sicm/deriv.html @ 2:b4de894a1e2e
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| author | Robert McIntyre <rlm@mit.edu> |
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| date | Fri, 28 Oct 2011 00:03:05 -0700 |
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1.122 +/*]]>*///--> 1.123 +</script> 1.124 +</head> 1.125 +<body> 1.126 + 1.127 +<div id="content"> 1.128 + 1.129 + 1.130 + 1.131 +<div class="header"> 1.132 + <div class="float-right"> 1.133 + <!-- 1.134 + <form> 1.135 + <input type="text"/><input type="submit" value="search the blog »"/> 1.136 + </form> 1.137 + --> 1.138 + </div> 1.139 + 1.140 + <h1>aurellem <em>☉</em></h1> 1.141 + <ul class="nav"> 1.142 + <li><a href="/">read the blog »</a></li> 1.143 + <!-- li><a href="#">learn about us »</a></li--> 1.144 + </ul> 1.145 +</div> 1.146 + 1.147 +<h1 class="title">An Unambiguous Notation for Derivatives</h1> 1.148 + 1.149 + 1.150 + 1.151 + 1.152 + 1.153 + 1.154 + 1.155 +<div id="table-of-contents"> 1.156 +<h2>Table of Contents</h2> 1.157 +<div id="text-table-of-contents"> 1.158 +<ul> 1.159 +<li><a href="#sec-1">1 Calculus of Infinitesimals </a> 1.160 +<ul> 1.161 +<li><a href="#sec-1-1">1.1 Differential Objects </a></li> 1.162 +<li><a href="#sec-1-2">1.2 Interactions obey the chain rule </a></li> 1.163 +</ul> 1.164 +</li> 1.165 +</ul> 1.166 +</div> 1.167 +</div> 1.168 + 1.169 +<div id="outline-container-1" class="outline-2"> 1.170 +<h2 id="sec-1"><span class="section-number-2">1</span> Calculus of Infinitesimals </h2> 1.171 +<div class="outline-text-2" id="text-1"> 1.172 + 1.173 + 1.174 +</div> 1.175 + 1.176 +<div id="outline-container-1-1" class="outline-3"> 1.177 +<h3 id="sec-1-1"><span class="section-number-3">1.1</span> Differential Objects </h3> 1.178 +<div class="outline-text-3" id="text-1-1"> 1.179 + 1.180 + 1.181 +<p> 1.182 +A <b>differential object</b> is a pair \([x,\,dx]\) consisting of a variable 1.183 +and an infinitely small increment of it. Differential objects can 1.184 +interact with functions, producing a new differential object as a 1.185 +result; this interaction is for calculating derivatives of functions. 1.186 +</p> 1.187 +<p> 1.188 +Differential objects are for 1.189 +calculating derivatives of functions: the derivative of \(f\) with 1.190 +respect to \(x\) 1.191 +</p> 1.192 +<p> 1.193 +You can “apply” 1.194 +functions to differential objects; the result is: 1.195 +</p> 1.196 + 1.197 + 1.198 +\([x,dx]\xrightarrow{\quad f \quad}[f(x), Df(x)\cdot dx].\) 1.199 + 1.200 +<p> 1.201 +Loosely speaking, the interaction of \(f\) and a differential object 1.202 +of \(x\) is a differential object of \(f\). 1.203 +</p> 1.204 + 1.205 +</div> 1.206 + 1.207 +</div> 1.208 + 1.209 +<div id="outline-container-1-2" class="outline-3"> 1.210 +<h3 id="sec-1-2"><span class="section-number-3">1.2</span> Interactions obey the chain rule </h3> 1.211 +<div class="outline-text-3" id="text-1-2"> 1.212 + 1.213 + 1.214 +<p> 1.215 +The interaction of \(f\) and the differential object \([x, dx]\) is 1.216 +a differential object \([f(x), Df(x)\cdot dx]\). Because of the rule for 1.217 +interactions, if you apply another function \(g\), you get the 1.218 +chain-rule answer you expect: 1.219 +</p> 1.220 + 1.221 + 1.222 +\([f(x), Df(x)\cdot dx]\xrightarrow{\quad g\quad}\left[g(f(x)),\, 1.223 +Dg(f(x))\cdot Df(x)\cdot dx\right]\) 1.224 + 1.225 + 1.226 + 1.227 + 1.228 + 1.229 + 1.230 + 1.231 + 1.232 +</div> 1.233 +</div> 1.234 +</div> 1.235 +<div id="postamble"> 1.236 +<p class="date">Date: 2011-08-08 02:49:24 EDT</p> 1.237 +<p class="author">Author: Dylan Holmes</p> 1.238 +<p class="creator">Org version 7.6 with Emacs version 23</p> 1.239 +<a href="http://validator.w3.org/check?uri=referer">Validate XHTML 1.0</a> 1.240 +</div> 1.241 +</div> 1.242 +</body> 1.243 +</html>