Mercurial > dylan

     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

    51 

    52 

    53
author	Robert McIntyre <rlm@mit.edu>
date	Fri, 28 Oct 2011 00:03:05 -0700
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