% TeX code generated by batTeX. Don't edit this file; edit the  
% input file pdiff.tex instead.
\documentclass[12pt]{article}
\usepackage{mathptm}
\usepackage{battex}  
\usepackage{color}

\title{A positional derivative package for Maxima}
 \author{Barton Willis \\
   University of Nebraska at Kearney \\
   Kearney Nebraska}
\date{September 10, 2002\footnote{With minor updates November, 2006.}}

\begin{document}

\maketitle

\subsection*{Introduction}

\noindent  Working with derivatives of unknown functions\footnote{By
{\em unknown  function\/}, we mean a function that isn't bound to a formula and
that has a derivative that isn't known to Maxima.} can be  cumbersome in Maxima.
If we want, for example, the first order  Taylor polynomial of 
$f(x + x^2)$ about $x = 1$, we get


\begin{mcline}{c1}
taylor(f(x + x^2),x,1,1);
\end{mcline}



\begin{mdline}{d1}
   f\left(2\right)+\left(\left.{{d}\over{d\,x}}\,f\left(x^2+x\right)
 \right|_{x=1}\right)\,\left(x-1\right)+\cdots 
\end{mdline}

\noindent To ``simplify'' the Taylor polynomial, we must assign a gradient to $f$ 

\begin{mcline}{c2}
gradef(f(x),df(x))$
\end{mcline}

\begin{mcline}{c3}
taylor(f(x+x^2),x,1,1);
\end{mcline}

\begin{mdline}{d3}
   f\left(2\right)+3\,df\left(2\right)\,\left(x-1\right)+\cdots 
\end{mdline}

This method works well for simple problems, but it 
is tedious for functions of several variables or high order
derivatives. The positional derivative package {\tt pdiff} gives 
an alternative to using {\tt gradef} when working with derivatives 
of unknown functions.

\subsection*{Usage}

To use the positional derivative package, first load it from the Maxima input prompt. 

\begin{mcline}{c1}
     load(pdiff)$
\end{mcline}


\noindent Loading 
{\tt pdiff.lisp} sets the option variable {\tt use\_pdiff} to true; 
when {\tt use\_diff} is true, Maxima will 
indicate derivatives of  unknown functions positionally. To illustrate, 
the first three  derivatives of $f$ are


\begin{mcline}{c2}
[diff(f(x),x),
  diff(f(x),x,2),
  diff(f(x),x,3)];
\end{mcline}



\begin{mdline}{d2}
   \left[ f_{\left(1\right)}(x),f_{\left(2\right)}(x),f_{\left(3
 \right)}(x) \right] 
\end{mdline}

The subscript indicates the order of the derivative;  since $f$ is a function of 
one variable, the subscript has only one index.  When a function has more 
than one variable, the subscript has an index for each variable


\begin{mcline}{c3}
     [diff(f(x,y),x,0,y,1), diff(f(y,x),x,0,y,1)];
\end{mcline}

\begin{mdline}{d3}
   \left[ f_{\left(0,1\right)}(x,y),f_{\left(1,0\right)}(y,x)
  \right] 
\end{mdline}

\noindent Setting {\tt use\_pdiff} to false  (either locally or globally) inhibits 
derivatives from begin computed positionally


\begin{mcline}{c4}
      diff(f(x,x^2),x), use_pdiff : false;
\end{mcline}



\begin{mdline}{d4}
   {{d}\over{d\,x}}\,f\left(x,x^2\right)
\end{mdline}

\begin{mcline}{c5}
   diff(f(x,x^2),x), use_pdiff : true;
\end{mcline}



\begin{mdline}{d5}
   f_{\left(1,0\right)}(x,x^2)+2\,x\,f_{\left(0,1\right)}(x,x^2)
\end{mdline}


Taylor polynomials of unknown functions can be found without using
{\tt gradef}. An example


\begin{mcline}{c6}
     taylor(f(x+x^2),x,1,2);
\end{mcline}



\begin{mdline}{d6}
   f\left(2\right)+3\,f_{\left(1\right)}(2)\,\left(x-1\right)+{{
 \left(2\,f_{\left(1\right)}(2)+9\,f_{\left(2\right)}(2)\right)\,
 \left(x-1\right)^2}\over{2}}+\cdots 
\end{mdline}

\noindent Additionally, we can verify that $ y = f(x-c \, t) + g(x+c \,t)$ is
a solution to a wave equation without using {\tt gradef}


\begin{mcline}{c7}
     y : f(x-c*t) + g(x+c*t)$
\end{mcline}

\begin{mcline}{c8}
   ratsimp(diff(y,t,2) - c^2 * diff(y,x,2));
\end{mcline}



\begin{mdline}{d8}
   0
\end{mdline}

\begin{mcline}{c9}
   remvalue(y)$
\end{mcline}

\noindent Expressions involving  positional derivatives can be differentiated

\begin{mcline}{c10}
     e : diff(f(x,y),x);
\end{mcline}


\begin{mdline}{d10}
   f_{\left(1,0\right)}(x,y)
\end{mdline}

\begin{mcline}{c11}
   diff(e,y);
\end{mcline}



\begin{mdline}{d11}
   f_{\left(1,1\right)}(x,y)
\end{mdline}


\noindent The chain rule is applied when needed


\begin{mcline}{c12}
     [diff(f(x^2),x), diff(f(g(x)),x)];
\end{mcline}



\begin{mdline}{d12}
   \left[ 2\,x\,f_{\left(1\right)}(x^2),g_{\left(1\right)}(x)\,f_{
 \left(1\right)}(g\left(x\right)) \right] 
\end{mdline}


The positional derivative package doesn't alter the way known functions
are differentiated


\begin{mcline}{c13}
     diff(exp(-x^2),x);
\end{mcline}



\begin{mdline}{d13}
   -2\,x\,e^ {- x^2 }
\end{mdline}


\noindent To convert positional derivatives to standard Maxima derivatives,
use {\tt convert\_to\_diff}


\begin{mcline}{c14}
     e : [diff(f(x),x), diff(f(x,y),x,1,y,1)];
\end{mcline}



\begin{mdline}{d14}
   \left[ f_{\left(1\right)}(x),f_{\left(1,1\right)}(x,y) \right] 
\end{mdline}

\begin{mcline}{c15}
   e : convert_to_diff(e);
\end{mcline}



\begin{mdline}{d15}
   \left[ {{d}\over{d\,x}}\,f\left(x\right),{{d^2}\over{d\,y\,d\,x}}
 \,f\left(x,y\right) \right] 
\end{mdline}

\noindent To convert back to a positional derivative, use {\tt ev} with {\tt diff} 
as an argument


\begin{mcline}{c16}
     ev(e,diff);
\end{mcline}



\begin{mdline}{d16}
   \left[ f_{\left(1\right)}(x),f_{\left(1,1\right)}(x,y) \right] 
\end{mdline}

\noindent Conversion to standard derivatives  sometimes requires the 
introduction of  a dummy variable. Here's an example


\begin{mcline}{c17}
     e : diff(f(x,y),x,1,y,1);
\end{mcline}



\begin{mdline}{d17}
   f_{\left(1,1\right)}(x,y)
\end{mdline}

\begin{mcline}{c18}
   e : subst(p(s),y,e);
\end{mcline}



\begin{mdline}{d18}
   f_{\left(1,1\right)}(x,p\left(s\right))
\end{mdline}

\begin{mcline}{c19}
   e : convert_to_diff(e);
\end{mcline}



\begin{mdline}{d19}
   \left.{{d^2}\over{d\,\%x_0\,d\,x}}\,f\left(x,\%x_0\right)\right|_{
 \left[ \%x_0=p\left(s\right) \right] }
\end{mdline}

Dummy variables have the form ci, where i=0,1,2\dots and c is the 
value of the option variable {\tt dummy\_char}. The default value
for {\tt dummy\_char} is {\tt \%x}. If a user variable conflicts with a 
dummy variable,  the conversion process can give an 
incorrect value. To convert the previous expression back to a positional derivative,
use {\tt ev} with {\tt diff} and {\tt at} as arguments


\begin{mcline}{c20}
      ev(e,diff,at);
\end{mcline}



\begin{mdline}{d20}
   f_{\left(1,1\right)}(x,p\left(s\right))
\end{mdline}


Maxima correctly evaluates expressions involving positional derivatives 
if a formula is  later given to the function.  (Thus converting an unknown 
function into a known one.)  Here is an example;  let


\begin{mcline}{c21}
     e : diff(f(x^2),x);
\end{mcline}



\begin{mdline}{d21}
   2\,x\,f_{\left(1\right)}(x^2)
\end{mdline}

\noindent Now, give $f$ a formula


\begin{mcline}{c22}
     f(x) := x^5;
\end{mcline}



\begin{mdline}{d22}
   f\left(x\right):=x^5
\end{mdline}

\noindent and evaluate {\tt e}


\begin{mcline}{c23}
     ev(e);
\end{mcline}



\begin{mdline}{d23}
   10\,x^9
\end{mdline}


\noindent This result is the same as  



\begin{mcline}{c24}
     diff(f(x^2),x);
\end{mcline}



\begin{mdline}{d24}
   10\,x^9
\end{mdline}

\noindent In this calculation, Maxima first evaluates $f(x)$ to $x^{10}$ and then
does the derivative.  Additionally, we can substitute a value for $x$ 
before evaluating



\begin{mcline}{c25}
     ev(subst(2,x,e));
\end{mcline}



\begin{mdline}{d25}
   5120
\end{mdline}


\noindent We can duplicate this with



\begin{mcline}{c26}
     subst(2,x,diff(f(x^2),x));
\end{mcline}



\begin{mdline}{d26}
   5120
\end{mdline}

\begin{mcline}{c27}
   remfunction(f);
\end{mcline}



\begin{mdline}{d27}
   \left[ f \right] 
\end{mdline}


\noindent We can also evaluate a positional derivative using a local 
function definition


\begin{mcline}{c28}
     e : diff(g(x),x);
\end{mcline}



\begin{mdline}{d28}
   g_{\left(1\right)}(x)
\end{mdline}

\begin{mcline}{c29}
   e, g(x) := sqrt(x);
\end{mcline}



\begin{mdline}{d29}
   {{1}\over{2\,\sqrt{x}}}
\end{mdline}

\begin{mcline}{c30}
   e, g = sqrt;
\end{mcline}



\begin{mdline}{d30}
   {{1}\over{2\,\sqrt{x}}}
\end{mdline}

\begin{mcline}{c31}
   e, g = h;
\end{mcline}



\begin{mdline}{d31}
   h_{\left(1\right)}(x)
\end{mdline}

\begin{mcline}{c32}
   e, g = lambda([t],t^2);
\end{mcline}



\begin{mdline}{d32}
   2\,x
\end{mdline}


\subsection*{The {\tt pderivop} function}

If $F$ is an atom and $i_1, i_2, \dots i_n$ are nonnegative
integers, then $ \mbox{pderivop}(F,i_1,i_2, \dots i_n)$, is the 
function that has the formula 
\[
   \frac{\partial^{i_1 + i_2 + \cdots + i_n}}{\partial x_1^{i_1} 
     \partial x_2^{i_2}  \cdots \partial x_n^{i_n}} F(x_1,x_2, \dots x_n). 
\]
If any of the derivative arguments are not nonnegative integers,
we'll get an error


\begin{mcline}{c33}
     pderivop(f,2,-1);
\end{mcline}

Each derivative order must be a nonnegative integer


\noindent The {\tt pderivop} function can be composed with itself

\begin{mcline}{c34}
     pderivop(pderivop(f,3,4),1,2);
\end{mcline}



\begin{mdline}{d34}
   f_{\left(4,6\right)}
\end{mdline}
\noindent If the number of derivative arguments between two calls 
to {\tt pderivop} isn't the same, Maxima gives an error

\begin{mcline}{c35}
     pderivop(pderivop(f,3,4),1);
\end{mcline}

The function f expected 2 derivative argument(s), but it received 1


When {\tt pderivop} is applied to a known function, the result is
a lambda form\footnote{If you repeat theses calculations, you 
may get a different prefix for the {\tt gensym} variables.}

\begin{mcline}{c37}
      f(x) := x^2;
\end{mcline}



\begin{mdline}{d37}
   f\left(x\right):=x^2
\end{mdline}

\begin{mcline}{c38}
   df : pderivop(f,1);
\end{mcline}



\begin{mdline}{d38}
   \lambda\left(\left[ Q_{1253} \right] ,2\,Q_{1253}\right)
\end{mdline}

\begin{mcline}{c39}
   apply(df,[z]);
\end{mcline}



\begin{mdline}{d39}
   2\,z
\end{mdline}

\begin{mcline}{c40}
   ddf : pderivop(f,2);
\end{mcline}



\begin{mdline}{d40}
   \lambda\left(\left[ Q_{1254} \right] ,2\right)
\end{mdline}

\begin{mcline}{c41}
   apply(ddf,[10]);
\end{mcline}



\begin{mdline}{d41}
   2
\end{mdline}

\begin{mcline}{c42}
   remfunction(f);
\end{mcline}



\begin{mdline}{d42}
   \left[ f \right] 
\end{mdline}


If the first argument to {\tt pderivop} is a lambda form, the
result is another lambda form



\begin{mcline}{c43}
     f : pderivop(lambda([x],x^2),1);
\end{mcline}



\begin{mdline}{d43}
   \lambda\left(\left[ Q_{1255} \right] ,2\,Q_{1255}\right)
\end{mdline}

\begin{mcline}{c44}
   apply(f,[a]);
\end{mcline}



\begin{mdline}{d44}
   2\,a
\end{mdline}

\begin{mcline}{c45}
   f : pderivop(lambda([x],x^2),2);
\end{mcline}



\begin{mdline}{d45}
   \lambda\left(\left[ Q_{1256} \right] ,2\right)
\end{mdline}

\begin{mcline}{c46}
   apply(f,[a]);
\end{mcline}



\begin{mdline}{d46}
   2
\end{mdline}

\begin{mcline}{c47}
   f : pderivop(lambda([x],x^2),3);
\end{mcline}



\begin{mdline}{d47}
   \lambda\left(\left[ Q_{1257} \right] ,0\right)
\end{mdline}

\begin{mcline}{c48}
   apply(f,[a]);
\end{mcline}



\begin{mdline}{d48}
   0
\end{mdline}

\begin{mcline}{c49}
   remvalue(f)$
\end{mcline}


If the first argument to {\tt pderivop} isn't an atom or
a lambda form, Maxima will signal an error 



\begin{mcline}{c50}
     pderivop(f+g,1);
\end{mcline}

Non-atom g+f used as a function


You may use {\tt tellsimpafter} together with {\tt pderivop} to 
give a value to a derivative of a function at a point; an
example


\begin{mcline}{c51}
tellsimpafter(pderivop(f,1)(1),1)$
\end{mcline}

\begin{mcline}{c52}
tellsimpafter(pderivop(f,2)(1),2)$
\end{mcline}

\begin{mcline}{c53}
diff(f(x),x,2) + diff(f(x),x)$
\end{mcline}

\begin{mcline}{c54}
subst(1,x,%);
\end{mcline}

\begin{mdline}{d54}
   3
\end{mdline}

This technique works for functions of several variables as well

\begin{mcline}{c55}
kill(rules)$
\end{mcline}


\begin{mcline}{c56}
tellsimpafter(pderivop(f,1,0)(0,0),a)$
\end{mcline}

\begin{mcline}{c57}
tellsimpafter(pderivop(f,0,1)(0,0),b)$
\end{mcline}

\begin{mcline}{c58}
sublis([x = 0, y = 0], diff(f(x,y),x) + diff(f(x,y),y));
\end{mcline}



\begin{mdline}{d58}
   b+a
\end{mdline}


\subsection*{\TeX{}-ing positional derivatives}

Several option variables control how positional derivatives
are converted to \TeX{}. When the option variable
{\tt tex\_uses\_prime\_for\_derivatives} is true (default false), 
makes functions of one variable \TeX{} as superscripted primes


\begin{mcline}{c59}
tex_uses_prime_for_derivatives : true$
\end{mcline}

\begin{mcline}{c60}
tex(makelist(diff(f(x),x,i),i,1,3))$
\end{mcline}

\begin{mdline}{d60}
\left[ f^{\prime}(x),f^{\prime\prime}(x),f^{\prime\prime\prime}(x)
  \right] 
\end{mdline}

\begin{mcline}{c61}
   tex(makelist(pderivop(f,i),i,1,3))$
\end{mcline}

\[
\left[ f^{\prime},f^{\prime\prime},f^{\prime\prime\prime} \right] 
\]

When the derivative order exceeds the value of the option
variable {\tt tex\_prime\_limit}, (default 3) 
derivatives are indicated with parenthesis delimited superscripts


\begin{mcline}{c62}
   tex(makelist(pderivop(f,i),i,1,5)), tex_prime_limit : 0$
\end{mcline}

\[
\left[ f^{(1)},f^{(2)},f^{(3)},f^{(4)},f^{(5)} \right] 
\]

\begin{mcline}{c63}
   tex(makelist(pderivop(f,i),i,1,5)), tex_prime_limit : 5$
\end{mcline}


\[
\left[ f^{\prime},f^{\prime\prime},f^{\prime\prime\prime},f^{\prime
 \prime\prime\prime},f^{\prime\prime\prime\prime\prime} \right] 
\]

The value of {\tt tex\_uses\_prime\_for\_derivatives} doesn't change the way 
functions of two or more variables are converted to \TeX{}.


\begin{mcline}{c64}
     tex(pderivop(f,2,1));
\end{mcline}

\[
f_{\left(2,1\right)}
\]



When the option variable {\tt tex\_uses\_named\_subscripts\_for\_derivatives}
(default false) is true, a derivative with respect to the i-th argument is 
indicated by a subscript that is the i-th element of the option variable
{\tt tex\_diff\_var\_names}.  An example is the clearest way to 
describe this.


\begin{mcline}{c65}
     tex_uses_named_subscripts_for_derivatives : true$
\end{mcline}

\begin{mcline}{c66}
   tex_diff_var_names;
\end{mcline}



\begin{mdline}{d66}
   \left[ x,y,z \right] 
\end{mdline}

\begin{mcline}{c67}
  
  tex([pderivop(f,1,0), pderivop(f,0,1), pderivop(f,1,1), pderivop(f,2,0)]);
\end{mcline}

$$\left[ f_{x},f_{y},f_{xy},f_{xx} \right] $$


\begin{mcline}{c68}
   tex_diff_var_names : [a,b];
\end{mcline}



\begin{mdline}{d68}
   \left[ a,b \right] 
\end{mdline}

\begin{mcline}{c69}
  
  tex([pderivop(f,1,0), pderivop(f,0,1), pderivop(f,1,1), pderivop(f,2,0)]);
\end{mcline}

$$\left[ f_{a},f_{b},f_{ab},f_{aa} \right] $$



\begin{mcline}{c70}
   tex_diff_var_names : [x,y,z];
\end{mcline}



\begin{mdline}{d70}
   \left[ x,y,z \right] 
\end{mdline}

\begin{mcline}{c71}
   tex([diff(f(x,y),x), diff(f(y,x),y)]);
\end{mcline}

$$\left[ f_{x}(x,y),f_{x}(y,x) \right] $$



When the derivative order exceeds {tt tex\_prime\_limit}, revert to the default
method for converting to \TeX{}


\begin{mcline}{c72}
     tex(diff(f(x,y,z),x,1,y,1,z,1)), tex_prime_limit : 4$
\end{mcline}

\[
f_{xyz}(x,y,z)
\]

\begin{mcline}{c73}
tex(diff(f(x,y,z),x,1,y,1,z,1)), tex_prime_limit : 1$
\end{mcline}

$$f_{\left(1,1,1\right)}(x,y,z)$$
\subsection*{A longer example}

We'll use the positional derivative package to change the independent
variable of the differential equation


\begin{mcline}{c74}
      de :  4*x^2*'diff(y,x,2) + 4*x*'diff(y,x,1) + (x-1)*y = 0;
\end{mcline}



\begin{mdline}{d74}
   4\,x^2\,\left({{d^2}\over{d\,x^2}}\,y\right)+4\,x\,\left({{d
 }\over{d\,x}}\,y\right)+\left(x-1\right)\,y=0
\end{mdline}

With malice aforethought, we'll assume a solution of the form
$ y = g(x^n)$, where $n$ is a number.  Substituting $y \rightarrow g(x^n)$
in the  differential equation  gives


\begin{mcline}{c75}
     de : subst(g(x^n),y,de);
\end{mcline}



\begin{mdline}{d75}
   4\,x^2\,\left({{d^2}\over{d\,x^2}}\,g\left(x^{n}\right)\right)+4\,
 x\,\left({{d}\over{d\,x}}\,g\left(x^{n}\right)\right)+\left(x-1
 \right)\,g\left(x^{n}\right)=0
\end{mdline}

\begin{mcline}{c76}
   de : ev(de, diff);
\end{mcline}



\begin{mdline}{d76}
   4\,x^2\,\left(n^2\,x^{2\,n-2}\,g^{\prime\prime}(x^{n})+\left(n-1
 \right)\,n\,x^{n-2}\,g^{\prime}(x^{n})\right)+4\,n\,x^{n}\,g^{\prime
 }(x^{n})+\left(x-1\right)\,g\left(x^{n}\right)=0
\end{mdline}

Now let $x \rightarrow t^{1/n}$


\begin{mcline}{c77}
     de : radcan(subst(x^(1/n),x, de));
\end{mcline}



\begin{mdline}{d77}
   4\,n^2\,x^2\,g^{\prime\prime}(x)+4\,n^2\,x\,g^{\prime}(x)+\left(x
 ^{{{1}\over{n}}}-1\right)\,g\left(x\right)=0
\end{mdline}

Setting $n \rightarrow 1/2$, we recognize that $g$ is the order 1 Bessel
equation


\begin{mcline}{c78}
     subst(1/2,n, de);
\end{mcline}



\begin{mdline}{d78}
   x^2\,g^{\prime\prime}(x)+x\,g^{\prime}(x)+\left(x^2-1\right)\,g
 \left(x\right)=0
\end{mdline}


\subsection*{Limitations}

\begin{itemize}
  \item Positional derivatives of subscripted functions are not allowed.

  \item Derivatives of unknown functions with symbolic orders are not 
   computed positionally.

  \item The {\tt pdiff.lisp} code alters the Maxima functions {\tt mqapply}
   and {\tt sdiffgrad} Although I'm unaware  of any problems associated 
   with these altered functions, there may be some.  Setting {\tt use\_pdiff} to
   false should restore {\tt mqapply} and {\tt sdiffgrad} to their
   original functioning.

\end{itemize}

\subsection*{Conclusion}

The {\tt pdiff} package provides a simple way of working with 
derivatives of unknown functions.  If you find a bug in the package,
or if you have a comment or a question, please send it to
{\tt willisb@unk.edu}. 

The {\tt pdiff} package could serve as a basis for a Maxima package
differential and integral operators.

\end{document}