三重积分的计算柱面球面.pptx

1、College of mathematics27 November 20247.4.2 三重积分计算 II.利用柱面坐标和球面坐标计算三重积分Triple Integrals in Cylindrical and Spherical Coordinates第1页College of mathematics27 November 2024上一页|首页|下一页柱面坐标柱面坐标Cylindrical Coordinates(,)P x y z柱面坐标柱面坐标 极坐标极坐标 竖坐标竖坐标zr(,)P rzO柱面坐标柱面坐标(,)x ycosxrsinyrzz第2页College of mathemat

2、ics27 November 2024上一页|首页|下一页0r 02z 要求：要求：(,)P x y zzr(,)P rzO第3页College of mathematics27 November 2024上一页|首页|下一页(,)P x y zzr(,)P rzOcosxrsinyrzz222xyr22rxyarctanyxzz:T1:T第4页College of mathematics27 November 2024上一页|首页|下一页柱面坐标三组坐标面柱面坐标三组坐标面r 常数常数圆柱面：圆柱面：22xy常数常数rR柱面坐标所以得名柱面坐标所以得名第5页College of mathem

3、atics27 November 2024上一页|首页|下一页常数常数半平面半平面第6页College of mathematics27 November 2024上一页|首页|下一页z 常数常数平面平面zhhwith(plots):x_axis:=plot3d(u,0,0,u=0.2,v=0.0.01,thickness=2):y_axis:=plot3d(0,u,0,u=0.2,v=0.0.01,thickness=2):z_axis:=plot3d(0,0,u,u=0.11,v=0.0.01,thickness=2):for i from 1 to 40 dopingmiani:=plo

4、t3d(t,s,i/4,t=0.2,s=0.2,color=blue,style=patchnogrid):od:pingmian:=display(seq(pingmiani,i=1.40),insequence=true):display(pingmian,x_axis,y_axis,z_axis,orientation=24,67);第7页College of mathematics27 November 2024上一页|首页|下一页柱面坐标坐标面柱面坐标坐标面zhrR第8页College of mathematics27 November 2024上一页|首页|下一页柱面坐标坐标面动画

5、柱面坐标坐标面动画run in Maplewith(plots):x_axis:=plot3d(u,0,0,u=0.2,v=0.0.01,thickness=2):y_axis:=plot3d(0,u,0,u=0.2,v=0.0.01,thickness=2):z_axis:=plot3d(0,0,u,u=0.2.3,v=0.0.01,thickness=2):for i from 1 to 40 dozhumiani:=plot3d(2*i/40,theta,z,theta=0.2*Pi,z=0.2,color=green,style=patch,coords=cylindrical):pi

6、ngmiani:=plot3d(x,y,2*i/40,x=-2.2,y=-2.2,color=blue,style=patchnogrid):banpingmiani:=plot3d(r,2*Pi*i/40,z,r=0.3,z=0.2,color=brown,style=patchnogrid,coords=cylindrical):od:banpingmian:=display(seq(banpingmiani,i=1.40),insequence=true):pingmian:=display(seq(pingmiani,i=1.40),insequence=true):zhumian:=

7、display(seq(zhumiani,i=1.40),insequence=true):display(pingmian,banpingmian,zhumian,x_axis,y_axis,z_axis,orientation=24,67);第9页College of mathematics27 November 2024上一页|首页|下一页用以上三组坐标面划分区域用以上三组坐标面划分区域 drxyzodzdr rdddVdVzd dzrdrd d两张两张相邻水平面相邻水平面两个相邻圆柱面两个相邻圆柱面两张相邻半平面两张相邻半平面第10页College of mathematics27 N

8、ovember 2024上一页|首页|下一页with(plots):x_axis:=plot3d(u,0,0,u=0.3,v=0.0.01,thickness=2):y_axis:=plot3d(0,u,0,u=0.3,v=0.0.01,thickness=2):z_axis:=plot3d(0,0,u,u=0.3.3,v=0.0.01,thickness=2):banpingmian1:=plot3d(r,Pi/6,z,r=0.3,z=0.2,coords=cylindrical,style=patch,color=green):banpingmian2:=plot3d(r,Pi/3,z,r

9、=0.3,z=0.2,coords=cylindrical,style=patch,color=green):zhumian1:=plot3d(1,theta,z,theta=0.2*Pi,z=0.2,coords=cylindrical,style=patch,color=red):zhumian2:=plot3d(2,theta,z,theta=0.2*Pi,z=0.2,coords=cylindrical,style=patch,color=red):pingmian1:=plot3d(r,theta,1,r=0.3,theta=0.2*Pi,coords=cylindrical,sty

10、le=patch,color=blue):pingmian2:=plot3d(r,theta,1.5,r=0.3,theta=0.2*Pi,coords=cylindrical,style=patch,color=grey):display(pingmian1,pingmian2,banpingmian1,banpingmian2,zhumian1,zhumian2,x_axis,y_axis,z_axis,orientation=5,60);第11页College of mathematics27 November 2024上一页|首页|下一页with(plots):x_axis:=plot

11、3d(u,0,0,u=0.3,v=0.0.01,thickness=2):y_axis:=plot3d(0,u,0,u=0.3,v=0.0.01,thickness=2):z_axis:=plot3d(0,0,u,u=0.3.3,v=0.0.01,thickness=2):banpingmian1:=plot3d(r,Pi/6,z,r=1.2,z=1.1.5,coords=cylindrical,style=patchnogrid,color=green):banpingmian2:=plot3d(r,Pi/3,z,r=1.2,z=1.1.5,coords=cylindrical,style=

12、patchnogrid,color=green):zhumian1:=plot3d(1,theta,z,theta=Pi/6.Pi/3,z=1.1.5,coords=cylindrical,style=patchnogrid,color=red):zhumian2:=plot3d(2,theta,z,theta=Pi/6.Pi/3,z=1.1.5,coords=cylindrical,style=patchnogrid,color=red):pingmian1:=plot3d(r,theta,1,r=1.2,theta=Pi/6.Pi/3,coords=cylindrical,style=pa

13、tchnogrid,color=blue):pingmian2:=plot3d(r,theta,1.5,r=1.2,theta=Pi/6.Pi/3,coords=cylindrical,style=patchnogrid,color=blue):display(banpingmian1,banpingmian2,zhumian1,zhumian2,pingmian1,pingmian2,x_axis,y_axis,z_axis,orientation=10,70);第12页College of mathematics27 November 2024上一页|首页|下一页rdrdVzd drddx

14、drydzzd d(,)f x y z dxdydz(cos,sin,)f rrzddzr rd drxyzodzdr rdd第13页College of mathematics27 November 2024上一页|首页|下一页常见曲面柱面坐标方程常见曲面柱面坐标方程直角坐标方程直角坐标方程柱面坐标方程柱面坐标方程半球面半球面222zaxy22zar22zara第14页College of mathematics27 November 2024上一页|首页|下一页直角坐标方程直角坐标方程柱面坐标方程柱面坐标方程圆锥面圆锥面22zxyzr旋转抛物面旋转抛物面22zxy2zrzr2zr第15页C

15、ollege of mathematics27 November 2024上一页|首页|下一页直角坐标方程直角坐标方程柱面坐标方程柱面坐标方程圆柱面圆柱面222xyara圆柱面圆柱面222xyax2 cosraraa2a2 cosra第16页College of mathematics27 November 2024上一页|首页|下一页常见曲面柱面坐标方程常见曲面柱面坐标方程直角坐标方程直角坐标方程柱面坐标方程柱面坐标方程半球面半球面222zaxy22zar圆锥面圆锥面22zxyzr旋转抛物面旋转抛物面22zxy2zr圆柱面圆柱面222xyara圆柱面圆柱面222xyax2 cosra第17页

16、College of mathematics27 November 2024上一页|首页|下一页例例 螺旋面螺旋面 zarctanyzx柱面坐标方程柱面坐标方程直角坐标方程直角坐标方程第18页College of mathematics27 November 2024上一页|首页|下一页参数方程：参数方程：cossinxstystzvtarctanvyvzx第19页College of mathematics27 November 2024上一页|首页|下一页当当 在在 xOy 面上投影区域面上投影区域 D 是圆域时，用是圆域时，用柱面坐标计算三重积分比较方便柱面坐标计算三重积分比较方便在在“

17、先一后二先一后二”二重积分中需要用极坐标积二重积分中需要用极坐标积分时，我们实际上就在使用柱面坐标计算三分时，我们实际上就在使用柱面坐标计算三重积分重积分第20页College of mathematics27 November 2024上一页|首页|下一页例例4.422()xydV22:2xyz是一个圆锥体是一个圆锥体它投影区域是一个圆域它投影区域是一个圆域宜用柱面坐标计算宜用柱面坐标计算2z 222zxyz22:4D xyD22zxy第21页College of mathematics27 November 2024上一页|首页|下一页2z 2zr22zxyzr22()xydV22rrrd

18、z20dr20d2rdz230r dr2230(2)rr dr245 20112 25rr165第22页College of mathematics27 November 2024上一页|首页|下一页例例4.52222zxyzxy221xy交线投影柱面交线投影柱面投影区域：投影区域：22:1D xy第23页College of mathematics27 November 2024上一页|首页|下一页 with(plots):zuimian:=implicitplot3d(z=sqrt(x2+y2),x=-2.2,y=-2.2,z=0.1.6,color=black,grid=20,20,20

19、):paowumian:=implicitplot3d(z=x2+y2,x=-2.2,y=-2.2,z=0.1.8,color=red,grid=20,20,20):x_axis:=plot3d(u,0,0,u=-1.2.1.2,v=0.0.01,thickness=2):y_axis:=plot3d(0,u,0,u=-1.2.1.2,v=0.0.01,thickness=2):z_axis:=plot3d(0,0,u,u=-0.2.2,v=0.0.01,thickness=2):display(zuimian,paowumian,x_axis,y_axis,z_axis,orientatio

20、n=23,66,scaling=constrained);22zxy22zxy22:1D xy第24页College of mathematics27 November 2024上一页|首页|下一页例4.522zxy22zxy7.4 三重积分（柱、球）第25页College of mathematics27 November 2024上一页|首页|下一页11作作 剖面图：剖面图：上上边界曲面：边界曲面：22zxyr下边界曲面：下边界曲面：22zxy2rVdV2rrrdz10dr20d2120()r rrdr6zr2zr第26页College of mathematics27 November

21、2024上一页|首页|下一页题题1(3)2222242008 xyxxyzddydxz:02,x204,yx22228xyzxy在在 xOy 面上投影区域：面上投影区域：22:4,0,0D xyxy:0,022rD20第27页College of mathematics27 November 2024上一页|首页|下一页下边界曲面：下边界曲面：22zxyr上边界曲面：上边界曲面：22288zxyr锥面锥面球面球面22222220408 xxyxyzyddxdz220082rrdrzrdzd28rzr第28页College of mathematics27 November 2024上一页|首页

22、|下一页22222242008 xyxxyzddydxz220082rrdrzrdzd28220122rrrzdr2220(8)4rrrdr220第29页College of mathematics27 November 2024上一页|首页|下一页球面坐标球面坐标Spherical coordinates(,)P x y zzr(,)P r O球面坐标球面坐标(,)x y0r 020第30页College of mathematics27 November 2024上一页|首页|下一页cosxOQcosinsrsinyOQsisinnrcoszr(,)P x y zzr(,)P r OQxy

23、ysincosxrsinsinyrcoszr2222xyzr第31页College of mathematics27 November 2024上一页|首页|下一页球面坐标三组坐标面球面坐标三组坐标面r 常数常数球面：球面：222xyz常数常数rR球面坐标所以得名球面坐标所以得名rR第32页College of mathematics27 November 2024上一页|首页|下一页常数常数半平面半平面第33页College of mathematics27 November 2024上一页|首页|下一页常数常数锥面：锥面：22cotzxy第34页College of mathematics

24、27 November 2024上一页|首页|下一页rR球面坐标坐标面球面坐标坐标面第35页College of mathematics27 November 2024上一页|首页|下一页球面坐标坐标面动画球面坐标坐标面动画run in Maplewith(plots):x_axis:=plot3d(u,0,0,u=0.3,v=0.0.01,thickness=2):y_axis:=plot3d(0,u,0,u=0.3,v=0.0.01,thickness=2):z_axis:=plot3d(0,0,u,u=0.3.3,v=0.0.01,thickness=2):for i from 0 to

25、 40 dozuimiani:=plot3d(r,theta,Pi*i/40,r=0.3,theta=0.2*Pi,color=green,style=patch,coords=spherical):qiumiani:=plot3d(2*i/40,theta,phi,theta=0.2*Pi,phi=0.Pi,style=patch,coords=spherical):banpingmiani:=plot3d(r,2*Pi*i/40,z,r=0.3,z=-3.3,color=brown,style=patchnogrid,coords=cylindrical):od:banpingmian:=

26、display(seq(banpingmiani,i=1.40),insequence=true):qiumian:=display(seq(qiumiani,i=1.40),insequence=true):zuimian:=display(seq(zuimiani,i=1.40),insequence=true):display(qiumian,banpingmian,zuimian,x_axis,y_axis,z_axis,orientation=24,67,scaling=constrained);第36页College of mathematics27 November 2024上一

27、页|首页|下一页with(plots):x_axis:=plot3d(u,0,0,u=0.3,v=0.0.01,thickness=2):y_axis:=plot3d(0,u,0,u=0.3,v=0.0.01,thickness=2):z_axis:=plot3d(0,0,u,u=0.3.3,v=0.0.01,thickness=2):banpingmian1:=plot3d(r,Pi/4,z,r=0.3,z=0.2,coords=cylindrical,style=patch,color=green):banpingmian2:=plot3d(r,Pi/2.5,z,r=0.3,z=0.2,c

28、oords=cylindrical,style=patch,color=grey):zuimian1:=plot3d(r,theta,Pi/4,theta=0.2*Pi,r=0.2,coords=spherical,style=patch,color=red):zuimian2:=plot3d(r,theta,Pi/3,theta=0.2*Pi,r=0.3,coords=spherical,style=patch,color=yellow):qiumian1:=plot3d(1,theta,phi,phi=0.Pi,theta=0.2*Pi,coords=spherical,style=pat

29、ch,color=brown):qiumian2:=plot3d(1.5,theta,phi,phi=0.Pi,theta=0.2*Pi,coords=spherical,style=patch,color=grey):display(banpingmian1,banpingmian2,zuimian1,zuimian2,qiumian2,x_axis,y_axis,z_axis,orientation=5,60);第37页College of mathematics27 November 2024上一页|首页|下一页with(plots):x_axis:=plot3d(u,0,0,u=0.1

30、.5,v=0.0.01,thickness=2):y_axis:=plot3d(0,u,0,u=0.1.5,v=0.0.01,thickness=2):z_axis:=plot3d(0,0,u,u=0.1.5,v=0.0.01,thickness=2):banpingmian1:=plot3d(r,Pi/5,phi,r=1.1.5,phi=Pi/5.Pi/3,coords=spherical,style=patch,color=green):banpingmian2:=plot3d(r,Pi/3,phi,r=1.1.5,phi=Pi/5.Pi/3,coords=spherical,style=

31、patch,color=green):zuimian1:=plot3d(r,theta,Pi/5,theta=Pi/5.Pi/3,r=1.1.5,coords=spherical,style=patch,color=yellow):zuimian2:=plot3d(r,theta,Pi/3,theta=Pi/5.Pi/3,r=1.1.5,coords=spherical,style=patch,color=yellow):qiumian1:=plot3d(1,theta,phi,phi=Pi/5.Pi/3,theta=Pi/5.Pi/3,coords=spherical,style=patch

32、,color=red):qiumian2:=plot3d(1.5,theta,phi,phi=Pi/5.Pi/3,theta=Pi/5.Pi/3,coords=spherical,style=patch,color=red):display(banpingmian1,banpingmian2,zuimian1,zuimian2,qiumian1,qiumian2,x_axis,y_axis,z_axis,orientation=5,74);第38页College of mathematics27 November 2024上一页|首页|下一页with(plots):x_axis:=plot3d

33、(u,0,0,u=0.1.5,v=0.0.01,thickness=2):y_axis:=plot3d(0,u,0,u=0.1.5,v=0.0.01,thickness=2):z_axis:=plot3d(0,0,u,u=0.1.5,v=0.0.01,thickness=2):banpingmian1:=plot3d(r,Pi/5,phi,r=1.1.5,phi=Pi/5.Pi/3,coords=spherical,style=patch,color=green):banpingmian3:=plot3d(r,Pi/5,phi,r=0.1.6,phi=Pi/7.Pi/2.5,coords=sp

34、herical,style=wireframe,color=green):banpingmian2:=plot3d(r,Pi/3,phi,r=1.1.5,phi=Pi/5.Pi/3,coords=spherical,style=patch,color=grey):banpingmian4:=plot3d(r,Pi/3,phi,r=0.1.6,phi=Pi/7.Pi/2.5,coords=spherical,style=wireframe,color=green):zuimian1:=plot3d(r,theta,Pi/5,theta=Pi/5.Pi/3,r=1.1.5,coords=spher

35、ical,style=patch,color=yellow):zuimian3:=plot3d(r,theta,Pi/5,theta=Pi/7.Pi/2.5,r=0.1.8,coords=spherical,style=wireframe,color=blue):zuimian2:=plot3d(r,theta,Pi/3,theta=Pi/5.Pi/3,r=1.1.5,coords=spherical,style=patch,color=yellow):zuimian4:=plot3d(r,theta,Pi/3,theta=Pi/7.Pi/2.5,r=0.1.8,coords=spherica

36、l,style=wireframe,color=blue):qiumian1:=plot3d(1,theta,phi,phi=Pi/5.Pi/3,theta=Pi/5.Pi/3,coords=spherical,style=patch,color=red):qiumian3:=plot3d(1,theta,phi,phi=Pi/7.Pi/2.5,theta=Pi/7.Pi/2.5,coords=spherical,style=wireframe,color=red):qiumian2:=plot3d(1.5,theta,phi,phi=Pi/5.Pi/3,theta=Pi/5.Pi/3,coo

37、rds=spherical,style=patch,color=red):qiumian4:=plot3d(1.5,theta,phi,phi=Pi/7.Pi/2.5,theta=Pi/7.Pi/2.5,coords=spherical,style=wireframe,color=red):display(banpingmian1,banpingmian2,zuimian1,zuimian2,qiumian1,qiumian2,banpingmian3,banpingmian4,zuimian3,zuimian4,qiumian3,qiumian4,x_axis,y_axis,z_axis,o

38、rientation=5,74);第39页College of mathematics27 November 2024上一页|首页|下一页rxyzor sinr用以上三组坐标面划分区域用以上三组坐标面划分区域两张两张相邻球面面相邻球面面两个相邻圆锥面两个相邻圆锥面两张相邻半平面两张相邻半平面sinr r一个经典小区域如图一个经典小区域如图其其体积约为体积约为sinVrrrsinr第40页College of mathematics27 November 2024上一页|首页|下一页sinVrrr球面坐标下体积元素：球面坐标下体积元素：2sin drdddrV2sindxdydrdddrz第41

39、页College of mathematics27 November 2024上一页|首页|下一页三重积分从直角坐标变换为球面坐标三重积分从直角坐标变换为球面坐标2sindxdydrdddrz(,)f x y z dxdydz2(sin cos,sin sisn,cos)in f rrrdrdrd 第42页College of mathematics27 November 2024上一页|首页|下一页常见曲面球面坐标方程常见曲面球面坐标方程直角坐标方程直角坐标方程球面坐标方程球面坐标方程球面球面2222xyzRrRRrRwith(plots):x_axis:=plot3d(u,0,0,u=0.

40、2,v=0.0.01,thickness=2):y_axis:=plot3d(0,u,0,u=0.2,v=0.0.01,thickness=2):z_axis:=plot3d(0,0,u,u=0.2,v=0.0.01,thickness=2):zuobiaoxi:=display(x_axis,y_axis,z_axis):qumian:=plot3d(sin(u)*cos(v),sin(u)*sin(v),cos(u),u=0.Pi,v=0.2*Pi,numpoints=800):display(qumian,zuobiaoxi,orientation=35,60);第43页College

41、of mathematics27 November 2024上一页|首页|下一页直角坐标方程直角坐标方程球面坐标方程球面坐标方程球面球面2222xyzazcos2ar2ra2 cosra2 cosra例例4.6第44页College of mathematics27 November 2024上一页|首页|下一页直角坐标方程直角坐标方程球面坐标方程球面坐标方程正圆锥面正圆锥面22zxy4圆锥面圆锥面22cotzxy44第45页College of mathematics27 November 2024上一页|首页|下一页直角坐标直角坐标球面坐标：球面坐标：球形区域球形区域2222:xyzR:0

42、2,0,0rR上、下限全是常数上、下限全是常数RrR(,)f x y z dxdydz00220(,)sinRrddF rdr球面坐标系中球面坐标系中“长方体长方体”第46页College of mathematics27 November 2024上一页|首页|下一页直角坐标直角坐标球面坐标：球面坐标：球形区域球形区域222:2xyzaz:02,0,202 cosra2a2 cosra(,)f x y z dxdydz2 co220s020(,sin)addF rrrd 第47页College of mathematics27 November 2024上一页|首页|下一页R球面坐标球面坐标

43、球顶球顶锥体锥体22222:xyzRxy0,40rR上、下限全是常数上、下限全是常数:02,22zxy222zRxy第48页College of mathematics27 November 2024上一页|首页|下一页with(plots):qiumian:=plot3d(r*cos(t),r*sin(t),r,t=0.2*Pi,r=0.1/sqrt(2),color=yellow):zuimian:=plot3d(sin(s)*cos(t),sin(s)*sin(t),cos(s),s=0.Pi/4,t=0.2*Pi,color=green):x_axis:=plot3d(u,0,0,u=

44、-1.1,v=0.0.01,thickness=2):y_axis:=plot3d(0,u,0,u=-1.1,v=0.0.01,thickness=2):z_axis:=plot3d(0,0,u,u=0.1,v=0.0.01,thickness=2):display(qiumian,zuimian,x_axis,y_axis,z_axis,orientation=23,66,scaling=constrained);第49页College of mathematics27 November 2024上一页|首页|下一页球面坐标球面坐标球顶球顶锥体锥体22222:xyzRxy0,40rR:02,

45、22zxy222zRxy(,)f x y z dxdydz002240(,)sinRrddF rdr 第50页College of mathematics27 November 2024上一页|首页|下一页例例4.732222()xyzedxdydz222:1xyz:02,0,01r解：解：3222()00201sinrddedrr原式原式3220010sinrdde r dr 310012 cos 3re 4(1)3e11r 第51页College of mathematics27 November 2024上一页|首页|下一页222xyzz22zxy第52页College of mathe

46、matics27 November 2024上一页|首页|下一页Example222xyzz22zxy22zxy222xyzz第53页College of mathematics27 November 2024上一页|首页|下一页:02,0,40cosr1cosr1cosr4Vdxdydz2cos0020sindrrdd第54页College of mathematics27 November 2024上一页|首页|下一页Vdxdydzcos020240sinrdddr24002cos0sindddrr340cos1si2n3d44021cos348第55页College of mathema

47、tics27 November 2024上一页|首页|下一页Example222()f xyzdxdydz22:xyzR平顶锥体平顶锥体22zxyzR解解在在 xOy 面上投影区域面上投影区域为为圆域：圆域：222:D xyRD第56页College of mathematics27 November 2024上一页|首页|下一页:D22yRx22yRx RRD22zxyzR,RxR2222,RxyRx22xyzR222()f xyzdxdydz22222()Rxyf xdzyzRRdx2222RxRxdy直角坐标直角坐标第57页College of mathematics27 Novembe

48、r 2024上一页|首页|下一页:DrRRDzrzR02,0,rRrzR222()f xyzdxdydz22()Rrdrzzf20d0Rdr柱面坐标柱面坐标第58页College of mathematics27 November 2024上一页|首页|下一页:02,04,0secrR222()f xyzdxdydz2ec2s0sin()Rrfdrr20d40d球面坐标球面坐标zRcosrRsecrRD4seczR4第59页College of mathematics27 November 2024上一页|首页|下一页222()f xyzdxdydz22222()Rxyf xdzyzRRdx2222RxRxdy22()Rrdrzzf20d0Rdr2ec2s0sin()Rrfdrr20d40d似乎柱面坐标最简单似乎柱面坐标最简单比较：比较：第60页