在非笛卡尔坐标系下计算面积和体积
通过围绕 轴旋转曲线
定义出的这个“坚果”很容易在柱坐标中参数化.
In[1]:=
![Click for copyable input](assets.zh/compute-areas-and-volumes-in-non-cartesian-coordin/In_151.png)
ParametricPlot3D[
CoordinateTransform[ "Cylindrical" -> "Cartesian", {Sin[z], t, z}] //
Evaluate, {z, 0, Pi}, {t, 0, 2 Pi}, PlotTheme -> "Business"]
Out[1]=
![](assets.zh/compute-areas-and-volumes-in-non-cartesian-coordin/O_85.png)
用 Area 直接在柱坐标里求表面积.
In[2]:=
![Click for copyable input](assets.zh/compute-areas-and-volumes-in-non-cartesian-coordin/In_152.png)
Area[{Sin[z], t, z}, {z, 0, Pi}, {t, 0, 2 Pi}, "Cylindrical"]
Out[2]=
![](assets.zh/compute-areas-and-volumes-in-non-cartesian-coordin/O_86.png)
Volume(体积)也可以在非笛卡尔坐标系里计算.
In[3]:=
![Click for copyable input](assets.zh/compute-areas-and-volumes-in-non-cartesian-coordin/In_153.png)
Volume[{r Sin[z], t, z}, {z, 0, Pi}, {t, 0, 2 Pi}, {r, 0,
1}, "Cylindrical"]
Out[3]=
![](assets.zh/compute-areas-and-volumes-in-non-cartesian-coordin/O_87.png)
通过使用 RegionMeasure,可以在任何维数下进行这些计算.
In[4]:=
![Click for copyable input](assets.zh/compute-areas-and-volumes-in-non-cartesian-coordin/In_154.png)
RegionMeasure[{Sin[z], t,
z}, {{z, 0, Pi}, {t, 0, 2 Pi}}, "Cylindrical"]
Out[4]=
![](assets.zh/compute-areas-and-volumes-in-non-cartesian-coordin/O_88.png)
In[5]:=
![Click for copyable input](assets.zh/compute-areas-and-volumes-in-non-cartesian-coordin/In_155.png)
RegionMeasure[{r Sin[z], t,
z}, {{z, 0, Pi}, {t, 0, 2 Pi}, {r, 0, 1}}, "Cylindrical"]
Out[5]=
![](assets.zh/compute-areas-and-volumes-in-non-cartesian-coordin/O_89.png)