1.1 --- /dev/null	Thu Jan 01 00:00:00 1970 +0000
     1.2 +++ b/sicm/deriv.org	Fri Oct 28 00:03:05 2011 -0700
     1.3 @@ -0,0 +1,53 @@
     1.4 +#+TITLE:An Unambiguous Notation for Derivatives
     1.5 +#+author: Dylan Holmes
     1.6 +#+EMAIL: rlm@mit.edu
     1.7 +#+MATHJAX: align:"left" mathml:t path:"../MathJax/MathJax.js"
     1.8 +#+STYLE: <link rel="stylesheet" type="text/css" href="../css/aurellem.css" />
     1.9 +#+OPTIONS:   H:3 num:t toc:t \n:nil @:t ::t |:t ^:t -:t f:t *:t <:t
    1.10 +#+SETUPFILE: ../templates/level-0.org
    1.11 +#+INCLUDE: ../templates/level-0.org
    1.12 +#+BABEL: :noweb yes
    1.13 +
    1.14 +* Calculus of Infinitesimals
    1.15 +** Differential Objects
    1.16 +
    1.17 +A *differential object* is a pair $[x,\,dx]$ consisting of a variable
    1.18 +and an infinitely small increment of it. We want differential objects
    1.19 +to enable us to compute derivatives of functions. 
    1.20 +
    1.21 +Differential objects are for
    1.22 +calculating derivatives of functions: the derivative of $f$ with
    1.23 +respect to $x$ 
    1.24 +
    1.25 +You can \ldquo{}apply\rdquo{}
    1.26 +functions to differential objects; the result is:
    1.27 +
    1.28 +\([x,dx]\xrightarrow{\quad f \quad}[f(x), Df(x)\cdot dx].\)
    1.29 +
    1.30 +Loosely speaking, the interaction of $f$ and a differential object
    1.31 +of $x$ is a differential object of $f$.
    1.32 +
    1.33 +#As a linguistic convention, we'll call this interaction /applying f
    1.34 +#to the differential object/. This is not to be confused with the
    1.35 +#=apply= function in Clojure.
    1.36 +
    1.37 +** Interactions obey the chain rule
    1.38 +
    1.39 +The interaction of $f$ and the differential object $[x, dx]$ is
    1.40 +a differential object $[f(x), Df(x)\cdot dx]$. Because of the rule for
    1.41 +interactions, if you apply another function $g$, you get the
    1.42 +chain-rule answer you expect:
    1.43 +
    1.44 +\([f(x), Df(x)\cdot dx]\xrightarrow{\quad g\quad}\left[g(f(x)),\,
    1.45 +Dg(f(x))\cdot Df(x)\cdot dx\right]\)
    1.46 +
    1.47 +
    1.48 +#+begin_src clojure :tangle deriv.clj
    1.49 +
    1.50 +#+end_src
    1.51 +
    1.52 +#+results:
    1.53 +: nil
    1.54 +
    1.55 +
    1.56 +