rlm@2: #+TITLE:An Unambiguous Notation for Derivatives
rlm@2: #+author: Dylan Holmes
rlm@2: #+EMAIL: rlm@mit.edu
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rlm@2: 
rlm@2: * Calculus of Infinitesimals
rlm@2: ** Differential Objects
rlm@2: 
rlm@2: A *differential object* is a pair $[x,\,dx]$ consisting of a variable
rlm@2: and an infinitely small increment of it. We want differential objects
rlm@2: to enable us to compute derivatives of functions. 
rlm@2: 
rlm@2: Differential objects are for
rlm@2: calculating derivatives of functions: the derivative of $f$ with
rlm@2: respect to $x$ 
rlm@2: 
rlm@2: You can \ldquo{}apply\rdquo{}
rlm@2: functions to differential objects; the result is:
rlm@2: 
rlm@2: \([x,dx]\xrightarrow{\quad f \quad}[f(x), Df(x)\cdot dx].\)
rlm@2: 
rlm@2: Loosely speaking, the interaction of $f$ and a differential object
rlm@2: of $x$ is a differential object of $f$.
rlm@2: 
rlm@2: #As a linguistic convention, we'll call this interaction /applying f
rlm@2: #to the differential object/. This is not to be confused with the
rlm@2: #=apply= function in Clojure.
rlm@2: 
rlm@2: ** Interactions obey the chain rule
rlm@2: 
rlm@2: The interaction of $f$ and the differential object $[x, dx]$ is
rlm@2: a differential object $[f(x), Df(x)\cdot dx]$. Because of the rule for
rlm@2: interactions, if you apply another function $g$, you get the
rlm@2: chain-rule answer you expect:
rlm@2: 
rlm@2: \([f(x), Df(x)\cdot dx]\xrightarrow{\quad g\quad}\left[g(f(x)),\,
rlm@2: Dg(f(x))\cdot Df(x)\cdot dx\right]\)
rlm@2: 
rlm@2: 
rlm@2: #+begin_src clojure :tangle deriv.clj
rlm@2: 
rlm@2: #+end_src
rlm@2: 
rlm@2: #+results:
rlm@2: : nil
rlm@2: 
rlm@2: 
rlm@2: