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<HTML>
<HEAD>
<!-- This HTML file has been created by texi2html 1.52
from dimension.texi on 7 April 2002 -->
<TITLE>Untitled Document</TITLE>
</HEAD>
<BODY>
<H1>Untitled Document</H1>
<P>
<P><HR><P>
<H2><A NAME="SEC1" HREF="dimension_toc.html#TOC1">Introduction to the Dimensional Analysis Package</A></H2>
<P>
This document describes a new dimensional analysis package for Maxima.
There is an older Maxima package for dimensional analysis that
is similar to the one that was shipped with the commercial Macsyma system.
The software described in this document differs greatly from the
older package.
</P>
<P>
The new dimensional analysis code was written by Barton Willis of the
University of Nebraska at Kearney. The complete distribution contains
source code, texinfo documentation, a file for testing (in standard
Maxima "rtest" form), and a TeX file that provides additional documentation
and examples of how to use the package. You may download
the distribution from www.unk.edu/acad/math/people/willisb
and you may contact the author at willisb@unk.edu.
</P>
<P>
To use the dimensional analysis package, you must first load the file
<B>dimension.mac</B>. Provided this file has been installed in a directory
that Maxima can find, load it with the command
<PRE>
(C1) load("dimension.mac")$
</PRE>
<P>
Alternatively, you can compile <B>dimension.mac</B> with the command
<PRE>
(C1) compile_file("dimension.mac")$
</PRE>
<P>
After it has been compiled, load it with
<PRE>
(C2) load("dimension.o")$
</PRE>
<P>
The compiled code runs about thirty percent faster than the interpreted
code.
</P>
<P>
Use <B>qput</B> to define the dimension of a variable; thus
to define <EM>x</EM> to be a length, <EM>t</EM> a time, and <EM>c</EM> a speed, use
<PRE>
(C1) qput(x,"length",dimension)$
(C2) qput(t,"time",dimension)$
(C3) qput(c,"length"/"time",dimension)$
</PRE>
<P>
To find the dimension of an expression, use the <B>dimension</B> function.
For example,
<PRE>
(C4) dimension(c*t/5);
(D4) length
(C5) dimension(c-x/t);
length
(D5) ------
time
</PRE>
<P>
When an expression is dimensionally inconsistent, <B>dimension</B>
should signal an error
<PRE>
(C6) dimension(c-x);
Expression is dimensionally inconsistent.
</PRE>
<P>
Any symbol may be used to represent a dimension; if you like,
you can use "L" instead of "length" for the length dimension. It
isn't necessary to use strings for the dimensions; however, doing
so reduces the chance of conflicting with other variables.
Thus to use the dimensions "M","L", and "T", for mass, length, and time,
respectively, use the commands
<PRE>
(C1) qput(x,"L",dimension)$
(C2) qput(t,"T",dimension)$
(C3) qput(c,"L"/"T",dimension)$
(C4) qput(m,"M",dimension)$
</PRE>
<P>
Then
<PRE>
(C4) dimension(m * 'diff(x,t));
L M
(D4) ---
T
</PRE>
<P>
An <EM>equation</EM> is dimensionally consistent if either one side of
the equality vanishes or if both sides have the same
dimensions. Thus
<PRE>
(C5) dimension(x=c*t);
(D5) L
(C6) dimension(x=0);
(D6) L
(C7) dimension(x=c);
Expression is dimensionally inconsistent.
</PRE>
<P>
The dimension function supports most Maxima operators and it
automatically maps over lists. For example,
<PRE>
(C1) dimension([x^3,x.t,x/t,x^^2]);
3 L 2
(D1) [L , L T, -, L ]
T
</PRE>
<P>
The dimension of a subscripted symbol is the dimension of the
non-subscripted variable. For example,
<PRE>
(C1) dimension(x[3]);
(D1) L
</PRE>
<P>
When the dimensions of a symbol are undefined, dimension returns
an unevaluated expression
<PRE>
(C2) dimension(z);
(D2) dimension(z)
</PRE>
<P>
The remaining functions in the package, <B>dimensionless</B>,
<B>dimension_as_list</B>, and <B>natural_unit</B> require that
all expressions have dimensions that are members of the
list <B>fundamental_dimensions</B>. The default value of this
list is
<PRE>
(C1) fundamental_dimensions;
(D1) [mass, length, time]
</PRE>
<P>
A user may redefine this list; to use "M", "L", and "T" for
the fundamental dimensions, make the assignment
<PRE>
(C2) fundamental_dimensions : ["M", "L", "T"]$
</PRE>
<P>
The list <B>fundamental_dimensions</B> must be nonempty, but it
can have any finite cardinality.
</P>
<P>
To find the dimensions of an expression as a list of the exponents
of the fundamental dimensions, use the <B>dimension_as_list</B> function
<PRE>
(C3) dimension_as_list('diff(x,t,3));
(D3) [0, 1, - 3]
</PRE>
<P>
Thus the dimension of the third derivative of x
with respect to t is L / T^3.
</P>
<P>
To find the dimensionless expressions that can be formed as a
product of elements of a list of atoms, use the
<B>dimensionless</B> function. For example,
<PRE>
(C1) dimensionless([c,t,x]);
Dependent equations eliminated: (1)
x
(D1) [---, 1]
c t
</PRE>
<P>
Finally, to find quantities that have a given dimension, use
the <B>natural_unit</B> function. To find an expression with
dimension time, use the command
<PRE>
(C1) natural_unit("T",[x,c]);
Dependent equations eliminated: (1)
x
(D1) [-]
c
</PRE>
<H2><A NAME="SEC2" HREF="dimension_toc.html#TOC2">Definitions for Dimensional Analysis</A></H2>
<a name="FUNDAMENTAL_DIMENSIONS"></a><P>
<DL>
<DT><U>Variable:</U> <B>FUNDAMENTAL_DIMENSIONS</B>
<DD><A NAME="IDX1"></A>
FUNDAMENTAL_DIMENSIONS is a list of the fundamental dimensions. The
default value is ["mass", "length", "time"]; however, a user may
redefine it to be any nonempty list of atoms. The function
<B>dimension</B> doesn't use this list, but the other functions
in the dimensional analysis package do use it.
</P>
</DL>
<a name="DIMENSION"></a><P>
<DL>
<DT><U>Function:</U> <B>DIMENSION</B> <I>(e)</I>
<DD><A NAME="IDX2"></A>
Return the dimension of the expression e. If
e is dimensionally inconsistent, signal an error.
The dimensions of the symbols in the expression e
should be first defined using <B>qput</B>.
</P>
</DL>
<a name="DIMENSION_AS_LIST"></a><P>
<DL>
<DT><U>Function:</U> <B>DIMENSION_AS_LIST</B> <I>(e)</I>
<DD><A NAME="IDX3"></A>
Return the dimension of the expression e as a list of
the exponents of the dimensions in the list
<B>fundamental_dimensions</B>. If e is dimensionally
inconsistent, signal an error.
</P>
</DL>
<a name="DIMENSIONLESS"></a><P>
<DL>
<DT><U>Function:</U> <B>DIMENSIONLESS</B> <I>([e1,e2,...,en])</I>
<DD><A NAME="IDX4"></A>
</P>
<P>
Return a basis for the dimensionless quantities that can be formed as
a product of powers of the expressions e1, e2, ..., en.
</P>
</DL>
<a name="NATURAL_UNIT"></a><P>
<DL>
<DT><U>Function:</U> <B>NATURAL_UNIT</B> <I>(q, [e1,e2,...,en])</I>
<DD><A NAME="IDX5"></A>
</P>
<P>
Return a basis for the quantities that can be formed as a product of
powers of the expressions e1, e2, ..., en that have the
same dimension as does q.
</P>
</DL>
<P>
</P>
<P>
<P><HR><P>
This document was generated on 7 April 2002 using the
<A HREF="http://wwwcn.cern.ch/dci/texi2html/">texi2html</A>
translator version 1.51a.</P>
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