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1 #+TITLE:An Unambiguous Notation for Derivatives
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2 #+author: Dylan Holmes
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3 #+EMAIL: rlm@mit.edu
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4 #+MATHJAX: align:"left" mathml:t path:"../MathJax/MathJax.js"
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5 #+STYLE: <link rel="stylesheet" type="text/css" href="../css/aurellem.css" />
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6 #+OPTIONS: H:3 num:t toc:t \n:nil @:t ::t |:t ^:t -:t f:t *:t <:t
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7 #+SETUPFILE: ../templates/level-0.org
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8 #+INCLUDE: ../templates/level-0.org
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9 #+BABEL: :noweb yes
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10
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11 * Calculus of Infinitesimals
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12 ** Differential Objects
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13
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14 A *differential object* is a pair $[x,\,dx]$ consisting of a variable
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15 and an infinitely small increment of it. We want differential objects
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16 to enable us to compute derivatives of functions.
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17
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18 Differential objects are for
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19 calculating derivatives of functions: the derivative of $f$ with
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20 respect to $x$
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21
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22 You can \ldquo{}apply\rdquo{}
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23 functions to differential objects; the result is:
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24
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25 \([x,dx]\xrightarrow{\quad f \quad}[f(x), Df(x)\cdot dx].\)
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26
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27 Loosely speaking, the interaction of $f$ and a differential object
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28 of $x$ is a differential object of $f$.
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29
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30 #As a linguistic convention, we'll call this interaction /applying f
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31 #to the differential object/. This is not to be confused with the
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32 #=apply= function in Clojure.
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33
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34 ** Interactions obey the chain rule
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35
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36 The interaction of $f$ and the differential object $[x, dx]$ is
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37 a differential object $[f(x), Df(x)\cdot dx]$. Because of the rule for
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38 interactions, if you apply another function $g$, you get the
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39 chain-rule answer you expect:
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40
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41 \([f(x), Df(x)\cdot dx]\xrightarrow{\quad g\quad}\left[g(f(x)),\,
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42 Dg(f(x))\cdot Df(x)\cdot dx\right]\)
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43
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44
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45 #+begin_src clojure :tangle deriv.clj
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46
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47 #+end_src
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48
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49 #+results:
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50 : nil
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51
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52
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53
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