The Volume command computes the volume of a polyhedral set in the generalized sense of a Lebesgue measure, defining a $d$-volume of a set in higher dimensions.  For sets in 3, 2, and 1 dimensional spaces, this corresponds to the standard definitions of volume, area, and length, respectively.

The optional parameter d specifies which $d$-volume to compute for the set, where d is less than or equal to the number of polyset's coordinates.  For sets of dimension greater than d, the sum of the volumes for the $d$-faces is computed.

The Area and Length commands are used to compute the 2-volume and 1-volume of polyset, respectively, and are equivalent to Volume(polyset, 2) and Volume(polyset, 1).

$\mathrm{with}\left(\mathrm{PolyhedralSets}\right)\:$

$c≔\mathrm{ExampleSets}:-\mathrm{Cube}\left(\right)$

$c≔\{\begin{array}{lll}\mathrm{Coordinates}& \:& \left[x_1\,x_2\,x_3\right]\\ \mathrm{Relations}& \:& \left[-x_3\le 1\,x_3\le 1\,-x_2\le 1\,x_2\le 1\,-x_1\le 1\,x_1\le 1\right]\end{array}$

$\mathrm{c\_edges}≔\mathrm{Edges}\left(c\right)\:$$\mathrm{map}\left(\mathrm{Length}\,\mathrm{c\_edges}\right)$

$\left[2\,2\,2\,2\,2\,2\,2\,2\,2\,2\,2\,2\right]$

$\mathrm{Volume}\left(c\right)$

$8$

$\mathrm{Area}\left(c\right)$

$24$

$\mathrm{ps}≔\mathrm{PolyhedralSet}\left(\left[\left[1\,0\,0\,0\right]\,\left[0\,1\,0\,0\right]\,\left[0\,0\,1\,0\right]\,\left[0\,0\,0\,1\right]\right]\,\left[x\,y\,z\,u\right]\right)$

$\mathrm{ps}≔\{\begin{array}{lll}\mathrm{Coordinates}& \:& \left[x\,y\,z\,u\right]\\ \mathrm{Relations}& \:& \left[-z\le 0\,-y\le 0\,-x\le 0\,x+y+z\le 1\,x+y+z+u=1\right]\end{array}$

$\mathrm{Volume}\left(\mathrm{ps}\right)$

$0$

$\mathrm{Volume}\left(\mathrm{ps}\,3\right)$

$\frac{1}{3}$

The PolyhedralSets[Volume], PolyhedralSets[Area] and PolyhedralSets[Length] commands were introduced in Maple 2015.

For more information on Maple 2015 changes, see Updates in Maple 2015.