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