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  15 <H2><A NAME="SEC1" HREF="dimension_toc.html#TOC1">Introduction to the Dimensional Analysis Package</A></H2>
  16
  17 <P>
  18 This document describes a new dimensional analysis package for Maxima.
  19 There is an older Maxima package for dimensional analysis.  The
  20 software described in this document differs from the older package.
  21 </P>
  22
  23
  24 <P>
  25 To use the dimensional analysis package, you must first load the file
  26 <B>dimension.mac</B>.  Provided this file has been installed in a directory
  27 that Maxima can  find, load it with the command
  28
  29 <PRE>
  30 (C1) load("dimension.mac")$
  31 </PRE>
  32
  33
  34 Use <B>qput</B> to define the dimension of a variable; thus
  35 to define <EM>x</EM> to be a length, <EM>t</EM> a time, and <EM>c</EM> a speed, use
  36
  37 <PRE>
  38 (C1) qput(x,"length",dimension)$
  39 (C2) qput(t,"time",dimension)$
  40 (C3) qput(c,"length"/"time",dimension)$
  41 </PRE>
  42
  43 <P>
  44 To find  the dimension of an expression, use the  <B>dimension</B> function.
  45 For example,
  46
  47 <PRE>
  48 (C4) dimension(c*t/5);
  49 (D4)                                length
  50 (C5) dimension(c-x/t);
  51                                     length
  52 (D5)                                ------
  53                                      time
  54 </PRE>
  55
  56 <P>
  57 When an expression is dimensionally inconsistent, <B>dimension</B>
  58 should signal an error
  59
  60 <PRE>
  61 (C6) dimension(c-x);
  62
  63 Expression is dimensionally inconsistent.
  64 </PRE>
  65
  66 <P>
  67 Any symbol may be used to represent a dimension; if you like,
  68 you can use "L" instead of "length" for the length dimension.  It
  69 isn't necessary to use strings for the dimensions; however, doing
  70 so reduces the chance of conflicting with other variables.
  71 Thus to use the dimensions "M","L", and "T", for mass, length, and time,
  72 respectively, use the commands
  73
  74 <PRE>
  75 (C1) qput(x,"L",dimension)$
  76 (C2) qput(t,"T",dimension)$
  77 (C3) qput(c,"L"/"T",dimension)$
  78 (C4) qput(m,"M",dimension)$
  79 </PRE>
  80
  81 <P>
  82 Then
  83
  84 <PRE>
  85 (C4) dimension(m * 'diff(x,t));
  86                                       L M
  87 (D4)                                  ---
  88                                        T
  89 </PRE>
  90
  91 <P>
  92 An <EM>equation</EM> is dimensionally consistent if either one side of
  93 the equality vanishes or if both sides have the same
  94 dimensions. Thus
  95
  96 <PRE>
  97 (C5) dimension(x=c*t);
  98
  99 (D5)                                   L
 100 (C6) dimension(x=0);
 101
 102 (D6)                                   L
 103 (C7) dimension(x=c);
 104
 105 Expression is dimensionally inconsistent.
 106 </PRE>
 107
 108 <P>
 109 The  dimension function supports most Maxima operators and it
 110 automatically maps over lists.  For example,
 111
 112 <PRE>
 113 (C1) dimension([x^3,x.t,x/t,x^^2]);
 114                                  3       L   2
 115 (D1)                           [L , L T, -, L ]
 116                                          T
 117 </PRE>
 118
 119 <P>
 120 The dimension of a subscripted symbol is the dimension of the
 121 non-subscripted variable.  For example,
 122
 123 <PRE>
 124 (C1) dimension(x[3]);
 125 (D1)                                   L
 126 </PRE>
 127
 128 <P>
 129 When the dimensions of a symbol are undefined, dimension returns
 130 an unevaluated expression
 131
 132 <PRE>
 133 (C2) dimension(z);
 134 (D2)                             dimension(z)
 135 </PRE>
 136
 137 <P>
 138 The remaining functions in the package, <B>dimensionless</B>,
 139 <B>dimension_as_list</B>, and <B>natural_unit</B> require that
 140 all expressions have dimensions that are members of the
 141 list <B>fundamental_dimensions</B>. The default value of this
 142 list is
 143
 144 <PRE>
 145 (C1) fundamental_dimensions;
 146
 147 (D1)                         [mass, length, time]
 148 </PRE>
 149
 150 <P>
 151 A user may redefine this list; to use "M", "L", and "T" for
 152 the fundamental dimensions, make the assignment
 153
 154 <PRE>
 155 (C2) fundamental_dimensions : ["M", "L", "T"]$
 156 </PRE>
 157
 158 <P>
 159 The list <B>fundamental_dimensions</B> must be nonempty, but it
 160 can have any finite cardinality.
 161
 162 </P>
 163 <P>
 164 To find the dimensions of an expression as a list of the exponents
 165 of the fundamental dimensions, use the <B>dimension_as_list</B> function
 166
 167 <PRE>
 168 (C3) dimension_as_list('diff(x,t,3));
 169
 170 (D3)                              [0, 1, - 3]
 171 </PRE>
 172
 173 <P>
 174 Thus the dimension of the third derivative of x
 175 with respect to t  is L / T^3.
 176
 177 </P>
 178 <P>
 179 To find the dimensionless expressions that can be formed as a
 180 product of elements of a list of atoms, use the
 181 <B>dimensionless</B> function. For example,
 182
 183 <PRE>
 184 (C1) dimensionless([c,t,x]);
 185
 186 Dependent equations eliminated:  (1)
 187                                      x
 188 (D1)                               [---, 1]
 189                                     c t
 190 </PRE>
 191
 192 <P>
 193 Finally, to find quantities that have a given dimension, use
 194 the <B>natural_unit</B> function.  To find an expression with
 195 dimension time, use the command
 196
 197 <PRE>
 198 (C1) natural_unit("T",[x,c]);
 199
 200 Dependent equations eliminated:  (1)
 201                                        x
 202 (D1)                                  [-]
 203                                        c
 204 </PRE>
 205
 206
 207
 208 <H2><A NAME="SEC2" HREF="dimension_toc.html#TOC2">Definitions for Dimensional Analysis</A></H2>
 209
 210 <a name="FUNDAMENTAL_DIMENSIONS"></a><P>
 211 <DL>
 212 <DT><U>Variable:</U> <B>FUNDAMENTAL_DIMENSIONS</B>
 213 <DD><A NAME="IDX1"></A>
 214 FUNDAMENTAL_DIMENSIONS is a list of the fundamental dimensions. The
 215 default value is ["mass", "length", "time"]; however, a user may
 216 redefine it to be any nonempty list of atoms. The function
 217 <B>dimension</B> doesn't use this list, but the other functions
 218 in the dimensional analysis package do use it.
 219
 220 </P>
 221 </DL>
 222
 223 <a name="DIMENSION"></a><P>
 224 <DL>
 225 <DT><U>Function:</U> <B>DIMENSION</B> <I>(e)</I>
 226 <DD><A NAME="IDX2"></A>
 227 Return the dimension of the expression e. If
 228 e is dimensionally inconsistent, signal an error.
 229 The dimensions of the symbols in the expression e
 230 should be first defined using <B>qput</B>.
 231
 232 </P>
 233 </DL>
 234
 235 <a name="DIMENSION_AS_LIST"></a><P>
 236 <DL>
 237 <DT><U>Function:</U> <B>DIMENSION_AS_LIST</B> <I>(e)</I>
 238 <DD><A NAME="IDX3"></A>
 239 Return the dimension of the expression e as a list of
 240 the exponents of the dimensions in the list
 241 <B>fundamental_dimensions</B>. If e is dimensionally
 242 inconsistent, signal an error.
 243
 244 </P>
 245 </DL>
 246
 247 <a name="DIMENSIONLESS"></a><P>
 248 <DL>
 249 <DT><U>Function:</U> <B>DIMENSIONLESS</B> <I>([e1,e2,...,en])</I>
 250 <DD><A NAME="IDX4"></A>
 251
 252 </P>
 253 <P>
 254 Return a basis for the dimensionless quantities that can be formed as
 255 a product of powers of the expressions e1, e2, ..., en.
 256
 257 </P>
 258 </DL>
 259
 260 <a name="NATURAL_UNIT"></a><P>
 261 <DL>
 262 <DT><U>Function:</U> <B>NATURAL_UNIT</B> <I>(q, [e1,e2,...,en])</I>
 263 <DD><A NAME="IDX5"></A>
 264
 265 </P>
 266 <P>
 267 Return a basis for the quantities that can be formed as a product of
 268 powers of the expressions e1, e2, ..., en that have the
 269 same dimension as does q.
 270
 271 </P>
 272 </DL>
 273
 274 <P>
 275
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