RegionMeasure
✖
RegionMeasure
更多信息和选项


- RegionMeasure 又被称为计数(零维),长度(一维),面积(二维),体积(三维)和勒贝格测度.
- 范例(行对应于嵌入维度和列对应于几何维度):
- 如果区域 reg 测度 d≥0,那么用 d-维测度.
- 零维测度对区域内的点进行计数.
- 在 RegionMeasure[x,{{t1,a1,b1},…,{tk,ak,bk}}] 中,如果 x 是一个标量,RegionMeasure 返回的是超曲面 {t1,…,tk,x} 在 k+1 维的测度.
- RegionMeasure 的第三个参数中的坐标图可按 CoordinateChartData 的第一个参数那样由三个参数 {coordsys,metric,dim} 来指定. 也可以使用把 dim 省略掉的简短形式.
- 可以指定下列选项:
-
AccuracyGoal Infinity 寻求绝对精度的数字 Assumptions $Assumptions 对参数所做的假设 GenerateConditions Automatic 是否产生参数条件 PerformanceGoal $PerformanceGoal 尝试优化的性能方面 PrecisionGoal Automatic 寻求精度的数字 WorkingPrecision Automatic 内部计算时使用的精度 - 假定积分的符号式极限为真并有序. 假定坐标图的符号式参数的范围由 CoordinateChartData 的属性 "ParameterRangeAssumptions" 给定.
- RegionMeasure 可用于 GeometricScene 中的符号区域.

范例
打开所有单元关闭所有单元基本范例 (6)常见实例总结
RegionMeasure 对应于零维区域的计数:

https://wolfram.com/xid/0enz1su6i-cwwt4l

https://wolfram.com/xid/0enz1su6i-bbjewl


https://wolfram.com/xid/0enz1su6i-e4clbm

RegionMeasure 对应于一维区域的曲线长度:

https://wolfram.com/xid/0enz1su6i-eos2w

https://wolfram.com/xid/0enz1su6i-f6a3zp


https://wolfram.com/xid/0enz1su6i-c7o0i0


https://wolfram.com/xid/0enz1su6i-bwacvn

RegionMeasure 对应于二维区域的表面积:

https://wolfram.com/xid/0enz1su6i-blm43a

https://wolfram.com/xid/0enz1su6i-godpmt


https://wolfram.com/xid/0enz1su6i-egqe2t


https://wolfram.com/xid/0enz1su6i-o87ms

RegionMeasure 对应于三维区域的体积:

https://wolfram.com/xid/0enz1su6i-bsqkvy

https://wolfram.com/xid/0enz1su6i-ggcgs3


https://wolfram.com/xid/0enz1su6i-e8xa1


https://wolfram.com/xid/0enz1su6i-4j1z3


https://wolfram.com/xid/0enz1su6i-i2xh4c


https://wolfram.com/xid/0enz1su6i-k8uqrq


https://wolfram.com/xid/0enz1su6i-n2s7qd

范围 (27)标准用法实例范围调查
特殊区域 (10)
Point 的测度对应于计数:

https://wolfram.com/xid/0enz1su6i-b1pwc1


https://wolfram.com/xid/0enz1su6i-jb580u


https://wolfram.com/xid/0enz1su6i-qdym8h

Line 的测度对应于弧长:

https://wolfram.com/xid/0enz1su6i-drodqz


https://wolfram.com/xid/0enz1su6i-b54rxb


https://wolfram.com/xid/0enz1su6i-hxbczw


https://wolfram.com/xid/0enz1su6i-o48rhs

Rectangle 可用在二维且测度对应于面积:

https://wolfram.com/xid/0enz1su6i-mwq77r


https://wolfram.com/xid/0enz1su6i-cchse5

Cuboid 可用在任意数目的维度:

https://wolfram.com/xid/0enz1su6i-ilz4pb


https://wolfram.com/xid/0enz1su6i-fv7lvj


https://wolfram.com/xid/0enz1su6i-e8og4x

Simplex 可以对应于点,线或二维三角形:

https://wolfram.com/xid/0enz1su6i-f9086r

https://wolfram.com/xid/0enz1su6i-dopwy


https://wolfram.com/xid/0enz1su6i-gtkeeh


https://wolfram.com/xid/0enz1su6i-bs7p7h


https://wolfram.com/xid/0enz1su6i-df2fxw


https://wolfram.com/xid/0enz1su6i-gr7zmu

Polygon 表示一个面积:

https://wolfram.com/xid/0enz1su6i-cfhpu7


https://wolfram.com/xid/0enz1su6i-fk8e8i


https://wolfram.com/xid/0enz1su6i-d27ft1


https://wolfram.com/xid/0enz1su6i-ku089w

Disk 可用在二维:

https://wolfram.com/xid/0enz1su6i-ec2e1a


https://wolfram.com/xid/0enz1su6i-vaaef

Ball 可用在任何维度并且测度是广义体积:

https://wolfram.com/xid/0enz1su6i-cg9yxi


https://wolfram.com/xid/0enz1su6i-fmfl3v


https://wolfram.com/xid/0enz1su6i-fc7zpk

Disk 作为一个椭圆可用在二维:

https://wolfram.com/xid/0enz1su6i-dpnaaq


https://wolfram.com/xid/0enz1su6i-cpdb8d

Ellipsoid 可用在任何维度:

https://wolfram.com/xid/0enz1su6i-godeh


https://wolfram.com/xid/0enz1su6i-krogt3


https://wolfram.com/xid/0enz1su6i-eham8g

Circle 可用在二维:

https://wolfram.com/xid/0enz1su6i-uanr3


https://wolfram.com/xid/0enz1su6i-mbaccm


https://wolfram.com/xid/0enz1su6i-gikvo


https://wolfram.com/xid/0enz1su6i-djl4gw

Cylinder 可用在三维:

https://wolfram.com/xid/0enz1su6i-bwzlj


https://wolfram.com/xid/0enz1su6i-g40lnz

Cone 可用在三维:

https://wolfram.com/xid/0enz1su6i-kon0a6


https://wolfram.com/xid/0enz1su6i-goq36h

公式区域 (2)
圆盘的测度表示为 ImplicitRegion:

https://wolfram.com/xid/0enz1su6i-len63u


https://wolfram.com/xid/0enz1su6i-kavh8q

圆盘的测度表示为 ParametricRegion:

https://wolfram.com/xid/0enz1su6i-8ufy


https://wolfram.com/xid/0enz1su6i-cl2ygx


https://wolfram.com/xid/0enz1su6i-bl8fm7

网格区域 (2)
在二维的 MeshRegion 的测度:

https://wolfram.com/xid/0enz1su6i-ib608p


https://wolfram.com/xid/0enz1su6i-esiyzj


https://wolfram.com/xid/0enz1su6i-mp72ll


https://wolfram.com/xid/0enz1su6i-cbepcr


https://wolfram.com/xid/0enz1su6i-dc3bst


https://wolfram.com/xid/0enz1su6i-ceqwoi

BoundaryMeshRegion 的测度:

https://wolfram.com/xid/0enz1su6i-ey15rx


https://wolfram.com/xid/0enz1su6i-b8mg48


https://wolfram.com/xid/0enz1su6i-dff1s9


https://wolfram.com/xid/0enz1su6i-o97lje

导出区域 (3)
RegionIntersection 的测度:

https://wolfram.com/xid/0enz1su6i-d92fo3

https://wolfram.com/xid/0enz1su6i-b45as9


https://wolfram.com/xid/0enz1su6i-pmbvb

TransformedRegion 的测度:

https://wolfram.com/xid/0enz1su6i-b6muk5

https://wolfram.com/xid/0enz1su6i-by9ied


https://wolfram.com/xid/0enz1su6i-cu2k8w

RegionBoundary 的测度:

https://wolfram.com/xid/0enz1su6i-bklovj

https://wolfram.com/xid/0enz1su6i-iob31i


https://wolfram.com/xid/0enz1su6i-ca4mg4

地理区域 (2)

https://wolfram.com/xid/0enz1su6i-sq1ypd

https://wolfram.com/xid/0enz1su6i-uy30p5

Polygon 与 GeoPosition 一起使用所得的区域:

https://wolfram.com/xid/0enz1su6i-1lk1dy

https://wolfram.com/xid/0enz1su6i-8vjhh8

Polygon 与 GeoGridPosition 一起使用所得区域的度量:

https://wolfram.com/xid/0enz1su6i-ipsiqu

https://wolfram.com/xid/0enz1su6i-yjciq7

参数化公式 (8)

https://wolfram.com/xid/0enz1su6i-eue79v


https://wolfram.com/xid/0enz1su6i-n92qmn


https://wolfram.com/xid/0enz1su6i-j9ke5e


https://wolfram.com/xid/0enz1su6i-zbc7h9


https://wolfram.com/xid/0enz1su6i-1pitot


https://wolfram.com/xid/0enz1su6i-5sxeem


https://wolfram.com/xid/0enz1su6i-qoz8ox


https://wolfram.com/xid/0enz1su6i-ur3sdt


https://wolfram.com/xid/0enz1su6i-b0p8cn


https://wolfram.com/xid/0enz1su6i-30kii9


https://wolfram.com/xid/0enz1su6i-ja7l9k


https://wolfram.com/xid/0enz1su6i-vskcwn


https://wolfram.com/xid/0enz1su6i-41lep6


https://wolfram.com/xid/0enz1su6i-i1igt0

选项 (4)各选项的常用值和功能
Assumptions (2)

https://wolfram.com/xid/0enz1su6i-0ta4jo


https://wolfram.com/xid/0enz1su6i-23dx17


https://wolfram.com/xid/0enz1su6i-z34ih3


https://wolfram.com/xid/0enz1su6i-7q3tjf

应用 (13)用该函数可以解决的问题范例
点 (2)

https://wolfram.com/xid/0enz1su6i-cif1gc


https://wolfram.com/xid/0enz1su6i-v0a3c


https://wolfram.com/xid/0enz1su6i-i88zx

对于变化的点质量函数 ,用 Integrate:

https://wolfram.com/xid/0enz1su6i-hg78yp

https://wolfram.com/xid/0enz1su6i-n8ksx

曲线 (4)

https://wolfram.com/xid/0enz1su6i-b6dqkv


https://wolfram.com/xid/0enz1su6i-p0agot


https://wolfram.com/xid/0enz1su6i-tcs1i


https://wolfram.com/xid/0enz1su6i-e0gaow

https://wolfram.com/xid/0enz1su6i-kliyuh


https://wolfram.com/xid/0enz1su6i-ooen5m


https://wolfram.com/xid/0enz1su6i-zt1i8

对于变化的密度 ,用 Integrate:

https://wolfram.com/xid/0enz1su6i-bl3d

https://wolfram.com/xid/0enz1su6i-lu4yjg

曲面 (2)

https://wolfram.com/xid/0enz1su6i-gacd2u


https://wolfram.com/xid/0enz1su6i-hpp179

https://wolfram.com/xid/0enz1su6i-p5kud2

有由 给出的变化质量密度,用 Integrate:

https://wolfram.com/xid/0enz1su6i-zp4h1z

实体 (3)
有恒定密度 的 Ball 的总面积:

https://wolfram.com/xid/0enz1su6i-mdblx

对于一个变化密度函数 ,用 Integrate:

https://wolfram.com/xid/0enz1su6i-b38w1m

https://wolfram.com/xid/0enz1su6i-bihrxh

找到一个 Cone 的乙醇的质量:

https://wolfram.com/xid/0enz1su6i-iy06ki

https://wolfram.com/xid/0enz1su6i-6bgb0m


https://wolfram.com/xid/0enz1su6i-6fqwu0


https://wolfram.com/xid/0enz1su6i-o6nunu

找到由 定义的不均匀质量密度 Cylinder 的质量:

https://wolfram.com/xid/0enz1su6i-fdm2kc

https://wolfram.com/xid/0enz1su6i-7dnlfn


https://wolfram.com/xid/0enz1su6i-bkchve


https://wolfram.com/xid/0enz1su6i-ygw8ad


https://wolfram.com/xid/0enz1su6i-baocp9

属性和关系 (10)函数的属性及与其他函数的关联
区域 ℛ 的 RegionMeasure 由积分 给出:

https://wolfram.com/xid/0enz1su6i-iskrma

https://wolfram.com/xid/0enz1su6i-ervl1h


https://wolfram.com/xid/0enz1su6i-bfhpvr

https://wolfram.com/xid/0enz1su6i-bg88sb

ArcLength 是一维区域的 RegionMeasure 的一种特殊情况:

https://wolfram.com/xid/0enz1su6i-e1ug52

Area 是二维区域的 RegionMeasure 的一种特殊情况:

https://wolfram.com/xid/0enz1su6i-c2642e

Volume 是三维区域的 RegionMeasure 的一种特殊情况:

https://wolfram.com/xid/0enz1su6i-vuxyk

测度由 RegionDimension 决定,包括维度0的计数:

https://wolfram.com/xid/0enz1su6i-y27re

https://wolfram.com/xid/0enz1su6i-cvbynk


https://wolfram.com/xid/0enz1su6i-zmhte

https://wolfram.com/xid/0enz1su6i-de4vxu


https://wolfram.com/xid/0enz1su6i-ek8t75

https://wolfram.com/xid/0enz1su6i-bidx3x


https://wolfram.com/xid/0enz1su6i-i7dwxi

https://wolfram.com/xid/0enz1su6i-ecxf11

对包含混合维度的区域,RegionDimension 给出最大维度:

https://wolfram.com/xid/0enz1su6i-fohsee

https://wolfram.com/xid/0enz1su6i-egnzy0

RegionMeasure[x,{t},c] 等价于 ArcLength[x,t,c]:

https://wolfram.com/xid/0enz1su6i-hmf8as


https://wolfram.com/xid/0enz1su6i-uuk9as

RegionMeasure[x,{s,t},c] 等价于 Area[x,s,t,c]:

https://wolfram.com/xid/0enz1su6i-ka9rno


https://wolfram.com/xid/0enz1su6i-fenb50

RegionMeasure[x,{s,t,u},c] 等价于 Volume[x,s,t,u,c]:

https://wolfram.com/xid/0enz1su6i-mvk8pn


https://wolfram.com/xid/0enz1su6i-oasr4j

RegionCentroid 等同于 Integrate[p,p∈ℛ]/m 且 m=RegionMeasure[ℛ]:

https://wolfram.com/xid/0enz1su6i-dfqxjw

https://wolfram.com/xid/0enz1su6i-qvg69

可能存在的问题 (3)常见隐患和异常行为
RegionMeasure 对离散点用计数测度:

https://wolfram.com/xid/0enz1su6i-b5b3gy

https://wolfram.com/xid/0enz1su6i-cunlnx


https://wolfram.com/xid/0enz1su6i-pdar1

参数化形式以参数化为基础,并计入多重覆盖(count multiple coverings):

https://wolfram.com/xid/0enz1su6i-5lhswu


https://wolfram.com/xid/0enz1su6i-s567zl

当无法算出精确答案时,RegionMeasure 使用机器算术:

https://wolfram.com/xid/0enz1su6i-x4tql3

巧妙范例 (1)奇妙或有趣的实例

https://wolfram.com/xid/0enz1su6i-kg6a48

https://wolfram.com/xid/0enz1su6i-smj


https://wolfram.com/xid/0enz1su6i-e1zjrd


https://wolfram.com/xid/0enz1su6i-fgce23


https://wolfram.com/xid/0enz1su6i-b5uasf

Wolfram Research (2014),RegionMeasure,Wolfram 语言函数,https://reference.wolfram.com/language/ref/RegionMeasure.html (更新于 2019 年).
文本
Wolfram Research (2014),RegionMeasure,Wolfram 语言函数,https://reference.wolfram.com/language/ref/RegionMeasure.html (更新于 2019 年).
Wolfram Research (2014),RegionMeasure,Wolfram 语言函数,https://reference.wolfram.com/language/ref/RegionMeasure.html (更新于 2019 年).
CMS
Wolfram 语言. 2014. "RegionMeasure." Wolfram 语言与系统参考资料中心. Wolfram Research. 最新版本 2019. https://reference.wolfram.com/language/ref/RegionMeasure.html.
Wolfram 语言. 2014. "RegionMeasure." Wolfram 语言与系统参考资料中心. Wolfram Research. 最新版本 2019. https://reference.wolfram.com/language/ref/RegionMeasure.html.
APA
Wolfram 语言. (2014). RegionMeasure. Wolfram 语言与系统参考资料中心. 追溯自 https://reference.wolfram.com/language/ref/RegionMeasure.html 年
Wolfram 语言. (2014). RegionMeasure. Wolfram 语言与系统参考资料中心. 追溯自 https://reference.wolfram.com/language/ref/RegionMeasure.html 年
BibTeX
@misc{reference.wolfram_2025_regionmeasure, author="Wolfram Research", title="{RegionMeasure}", year="2019", howpublished="\url{https://reference.wolfram.com/language/ref/RegionMeasure.html}", note=[Accessed: 06-April-2025
]}
BibLaTeX
@online{reference.wolfram_2025_regionmeasure, organization={Wolfram Research}, title={RegionMeasure}, year={2019}, url={https://reference.wolfram.com/language/ref/RegionMeasure.html}, note=[Accessed: 06-April-2025
]}