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author Robert McIntyre <rlm@mit.edu>
date Fri, 28 Oct 2011 00:03:05 -0700
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1:8d8278e09888 2:b4de894a1e2e
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7 <title>An Unambiguous Notation for Derivatives</title>
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137 <h1>aurellem <em>&#x2609;</em></h1>
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143
144 <h1 class="title">An Unambiguous Notation for Derivatives</h1>
145
146
147
148
149
150
151
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>
165
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">
169
170
171 </div>
172
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">
176
177
178 <p>
179 A <b>differential object</b> is a pair \([x,\,dx]\) consisting of a variable
180 and an infinitely small increment of it. Differential objects can
181 interact with functions, producing a new differential object as a
182 result; this interaction is for calculating derivatives of functions.
183 </p>
184 <p>
185 Differential objects are for
186 calculating derivatives of functions: the derivative of \(f\) with
187 respect to \(x\)
188 </p>
189 <p>
190 You can &ldquo;apply&rdquo;
191 functions to differential objects; the result is:
192 </p>
193
194
195 \([x,dx]\xrightarrow{\quad f \quad}[f(x), Df(x)\cdot dx].\)
196
197 <p>
198 Loosely speaking, the interaction of \(f\) and a differential object
199 of \(x\) is a differential object of \(f\).
200 </p>
201
202 </div>
203
204 </div>
205
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">
209
210
211 <p>
212 The interaction of \(f\) and the differential object \([x, dx]\) is
213 a differential object \([f(x), Df(x)\cdot dx]\). Because of the rule for
214 interactions, if you apply another function \(g\), you get the
215 chain-rule answer you expect:
216 </p>
217
218
219 \([f(x), Df(x)\cdot dx]\xrightarrow{\quad g\quad}\left[g(f(x)),\,
220 Dg(f(x))\cdot Df(x)\cdot dx\right]\)
221
222
223
224
225
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227
228
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>
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