RegionMeasure

RegionMeasure[reg]

gives the measure of the region reg.

RegionMeasure[reg,d]

gives the d-dimensional measure of the region reg.

RegionMeasure[{x1,,xn},{{t1,a1,b1},,{tk,ak,bk}}]

gives the k-measure of the parametric formula whose Cartesian coordinates xi are functions of tj.

RegionMeasure[{x1,,xn},{{t1,a1,b1},,{tk,ak,bk}},chart]

interprets the xi as coordinates in the specified coordinate chart.

Details and Options

  • RegionMeasure is also known as count (0D), length (1D), area (2D), volume (3D), and Lebesgue measure.
  • Example cases where rows correspond to embedding dimension and columns to geometric dimension:
  • If the region reg is of dimension d0, then the d-dimensional measure is used.
  • The zero-dimensional measure counts the number of points in the region.
  • In RegionMeasure[x,{{t1,a1,b1},,{tk,ak,bk}}], if x is a scalar, RegionMeasure returns the measure of the hypersurface {t1,,tk,x} in k+1 dimensions.
  • Coordinate charts in the third argument of RegionMeasure can be specified as triples {coordsys,metric,dim} in the same way as in the first argument of CoordinateChartData. The short form in which dim is omitted may be used.
  • The following options can be given:
  • AccuracyGoalInfinitydigits of absolute accuracy sought
    Assumptions$Assumptionsassumptions to make about parameters
    GenerateConditionsAutomaticwhether to generate conditions on parameters
    PerformanceGoal$PerformanceGoalaspects of performance to try to optimize
    PrecisionGoalAutomaticdigits of precision sought
    WorkingPrecisionInfinitythe precision used in internal computations
  • Symbolic limits of integration are assumed to be real and ordered. Symbolic coordinate chart parameters are assumed to be in range given by the "ParameterRangeAssumptions" property of CoordinateChartData.
  • RegionMeasure can be used with symbolic regions in GeometricScene.

Examples

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Basic Examples  (6)

RegionMeasure corresponds to count for zero-dimensional regions:

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RegionMeasure corresponds to curve length for one-dimensional regions:

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RegionMeasure corresponds to surface area for two-dimensional regions:

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RegionMeasure corresponds to volume for three-dimensional regions:

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Area of a bow-tie figure:

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Volume of a cylinder expressed in cylindrical coordinates:

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Scope  (25)

Options  (6)

Applications  (13)

Properties & Relations  (10)

Possible Issues  (2)

Neat Examples  (1)

Introduced in 2014
(10.0)
|
Updated in 2019
(12.0)