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author Robert McIntyre <rlm@mit.edu>
date Fri, 28 Oct 2011 04:56:48 -0700
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rlm@2 136
rlm@2 137 <h1>aurellem <em>&#x2609;</em></h1>
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rlm@2 142 </div>
rlm@2 143
rlm@2 144 <h1 class="title">An Unambiguous Notation for Derivatives</h1>
rlm@2 145
rlm@2 146
rlm@2 147
rlm@2 148
rlm@2 149
rlm@2 150
rlm@2 151
rlm@2 152 <div id="table-of-contents">
rlm@2 153 <h2>Table of Contents</h2>
rlm@2 154 <div id="text-table-of-contents">
rlm@2 155 <ul>
rlm@2 156 <li><a href="#sec-1">1 Calculus of Infinitesimals </a>
rlm@2 157 <ul>
rlm@2 158 <li><a href="#sec-1-1">1.1 Differential Objects </a></li>
rlm@2 159 <li><a href="#sec-1-2">1.2 Interactions obey the chain rule </a></li>
rlm@2 160 </ul>
rlm@2 161 </li>
rlm@2 162 </ul>
rlm@2 163 </div>
rlm@2 164 </div>
rlm@2 165
rlm@2 166 <div id="outline-container-1" class="outline-2">
rlm@2 167 <h2 id="sec-1"><span class="section-number-2">1</span> Calculus of Infinitesimals </h2>
rlm@2 168 <div class="outline-text-2" id="text-1">
rlm@2 169
rlm@2 170
rlm@2 171 </div>
rlm@2 172
rlm@2 173 <div id="outline-container-1-1" class="outline-3">
rlm@2 174 <h3 id="sec-1-1"><span class="section-number-3">1.1</span> Differential Objects </h3>
rlm@2 175 <div class="outline-text-3" id="text-1-1">
rlm@2 176
rlm@2 177
rlm@2 178 <p>
rlm@2 179 A <b>differential object</b> is a pair \([x,\,dx]\) consisting of a variable
rlm@2 180 and an infinitely small increment of it. Differential objects can
rlm@2 181 interact with functions, producing a new differential object as a
rlm@2 182 result; this interaction is for calculating derivatives of functions.
rlm@2 183 </p>
rlm@2 184 <p>
rlm@2 185 Differential objects are for
rlm@2 186 calculating derivatives of functions: the derivative of \(f\) with
rlm@2 187 respect to \(x\)
rlm@2 188 </p>
rlm@2 189 <p>
rlm@2 190 You can &ldquo;apply&rdquo;
rlm@2 191 functions to differential objects; the result is:
rlm@2 192 </p>
rlm@2 193
rlm@2 194
rlm@2 195 \([x,dx]\xrightarrow{\quad f \quad}[f(x), Df(x)\cdot dx].\)
rlm@2 196
rlm@2 197 <p>
rlm@2 198 Loosely speaking, the interaction of \(f\) and a differential object
rlm@2 199 of \(x\) is a differential object of \(f\).
rlm@2 200 </p>
rlm@2 201
rlm@2 202 </div>
rlm@2 203
rlm@2 204 </div>
rlm@2 205
rlm@2 206 <div id="outline-container-1-2" class="outline-3">
rlm@2 207 <h3 id="sec-1-2"><span class="section-number-3">1.2</span> Interactions obey the chain rule </h3>
rlm@2 208 <div class="outline-text-3" id="text-1-2">
rlm@2 209
rlm@2 210
rlm@2 211 <p>
rlm@2 212 The interaction of \(f\) and the differential object \([x, dx]\) is
rlm@2 213 a differential object \([f(x), Df(x)\cdot dx]\). Because of the rule for
rlm@2 214 interactions, if you apply another function \(g\), you get the
rlm@2 215 chain-rule answer you expect:
rlm@2 216 </p>
rlm@2 217
rlm@2 218
rlm@2 219 \([f(x), Df(x)\cdot dx]\xrightarrow{\quad g\quad}\left[g(f(x)),\,
rlm@2 220 Dg(f(x))\cdot Df(x)\cdot dx\right]\)
rlm@2 221
rlm@2 222
rlm@2 223
rlm@2 224
rlm@2 225
rlm@2 226
rlm@2 227
rlm@2 228
rlm@2 229 </div>
rlm@2 230 </div>
rlm@2 231 </div>
rlm@2 232 <div id="postamble">
rlm@2 233 <p class="date">Date: 2011-08-08 02:49:24 EDT</p>
rlm@2 234 <p class="author">Author: Dylan Holmes</p>
rlm@2 235 <p class="creator">Org version 7.6 with Emacs version 23</p>
rlm@2 236 <a href="http://validator.w3.org/check?uri=referer">Validate XHTML 1.0</a>
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