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1、三重积分的计算柱面球面三重积分的计算柱面球面柱面坐标Cylindrical Coordinates柱面坐标 极坐标竖坐标柱面坐标第1页/共60页规定：第2页/共60页第3页/共60页柱面坐标的三组坐标面常数圆柱面：常数柱面坐标因此得名第4页/共60页常数半平面第5页/共60页常数平面with(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
2、):for i from 1 to 40 dopingmiani:=plot3d(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6页/共60页柱面坐标的坐标面第7页/共60页柱面坐标的坐标面动画run in Maplewith(plots):x_axis:=plot3d(u,0,0,u=0.2,v=0.0.0
3、1,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):pingmiani:=plot3d(x,y,2*i/40,x=-2.2,y=-2.2,color=blue,style=patchnogr
4、id):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:=display(seq(zhumiani,i=1.40),insequence=true):display(pingmian,banp
5、ingmian,zhumian,x_axis,y_axis,z_axis,orientation=24,67);第8页/共60页用以上三组坐标面划分区域两张相邻的水平面两个相邻的圆柱面两张相邻的半平面第9页/共60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
6、=0.3,z=0.2,coords=cylindrical,style=patch,color=green):banpingmian2:=plot3d(r,Pi/3,z,r=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=
7、patch,color=red):pingmian1:=plot3d(r,theta,1,r=0.3,theta=0.2*Pi,coords=cylindrical,style=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,orientat
8、ion=5,60);第10页/共60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=1.2,z=1.1.5,coords=cylindrical,style=patchnogrid,color=green):banpingmian2:=plot3d(r,Pi/
9、3,z,r=1.2,z=1.1.5,coords=cylindrical,style=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,
10、theta=Pi/6.Pi/3,coords=cylindrical,style=patchnogrid,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11页/共60页第12页/共60页常见曲面
11、的柱面坐标方程直角坐标方程柱面坐标方程半球面第13页/共60页直角坐标方程柱面坐标方程圆锥面旋转抛物面with(plots):qumian:=implicitplot3d(z=x2+y2,x=-2.2,y=-2.2,z=0.2,color=yellow,grid=15,15,15):x_axis:=plot3d(u,0,0,u=-2.2,v=0.0.01,thickness=2):y_axis:=plot3d(0,u,0,u=-2.2,v=0.0.01,thickness=2):z_axis:=plot3d(0,0,u,u=0.3,v=0.0.01,thickness=2):display(q
12、umian,x_axis,y_axis,z_axis,orientation=23,66,scaling=constrained);with(plots):x_axis:=plot3d(u,0,0,u=0.1.2,v=0.0.01,thickness=2):y_axis:=plot3d(0,u,0,u=0.1.2,v=0.0.01,thickness=2):z_axis:=plot3d(0,0,u,u=0.1.5,v=0.0.01,thickness=2):zuobiaoxi:=display(x_axis,y_axis,z_axis):qumian:=plot3d(u*cos(theta),
13、u*sin(theta),u,u=0.1,theta=0.2*Pi,numpoints=800,color=green):display(qumian,zuobiaoxi,orientation=35,60);第14页/共60页直角坐标方程柱面坐标方程圆柱面圆柱面第15页/共60页常见曲面的柱面坐标方程直角坐标方程柱面坐标方程半球面圆锥面旋转抛物面圆柱面圆柱面第16页/共60页例 螺旋面 柱面坐标方程直角坐标方程第17页/共60页参数方程：第18页/共60页当 在 xOy 面上的投影区域 D 是圆域时，用柱面坐标计算三重积分比较方便在“先一后二”的二重积分中需要用极坐标积分时，我们实际上就在使
14、用柱面坐标计算三重积分第19页/共60页例4.4是一个圆锥体它的投影区域是一个圆域宜用柱面坐标计算第20页/共60页第21页/共60页例4.5交线的投影柱面投影区域：第22页/共60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):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
15、.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,orientation=23,66,scaling=constrained);第23页/共60页例4.57.4 三重积分（柱、球）第24页/共60页作 的剖面图：上边界曲面：下边界曲面：第25页/共60页题1(3)在 xOy 面上的投影区域
16、：第26页/共60页下边界曲面：上边界曲面：锥面球面第27页/共60页第28页/共60页球面坐标Spherical coordinates球面坐标第29页/共60页第30页/共60页球面坐标的三组坐标面常数球面：常数球面坐标因此得名第31页/共60页常数半平面第32页/共60页常数锥面：第33页/共60页球面坐标的坐标面第34页/共60页球面坐标的坐标面动画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
17、):z_axis:=plot3d(0,0,u,u=0.3.3,v=0.0.01,thickness=2):for i from 0 to 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.
18、3,color=brown,style=patchnogrid,coords=cylindrical):od:banpingmian:=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,
19、67,scaling=constrained);第35页/共60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
20、(r,Pi/2.5,z,r=0.3,z=0.2,coords=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,c
21、oords=spherical,style=patch,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36页/共60页with(plots):x_axis:=plot3d(u,0,0,u=0.1.5,v=0.0.01,thickn
22、ess=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=patch,color=green)
23、: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,color=red):qiumia
24、n2:=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37页/共60页with(plots):x_axis:=plot3d(u,0,0,u=0.1.5,v=0.0.01,thickness=2):y_axis:=plot3d(0,u,0,u=0
25、.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=spherical,style=wireframe,color=green):banpingmian2:=plot3d(r,P
26、i/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=spherical,style=patch,color=yellow):zuimian3:=plot3d(r,theta,Pi/5,
27、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=spherical,style=wireframe,color=blue):qiumian1:=plot3d(1,theta,phi,ph
28、i=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,coords=spherical,style=patch,color=red):qiumian4:=plot3d(1.5,the
29、ta,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,orientation=5,74);第38页/共60页用以上三组坐标面划分区域两张相邻的球面面两个相邻的圆锥面两张相邻的半平
30、面一个典型的小区域如图其体积约为第39页/共60页球面坐标下的体积元素：第40页/共60页三重积分从直角坐标变换为球面坐标第41页/共60页常见曲面的球面坐标方程直角坐标方程球面坐标方程球面with(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,v=0.0.01,thickness=2):zuobiaoxi:=display(x_axis,y_axis,z_axis):qumia
31、n:=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42页/共60页直角坐标方程球面坐标方程球面例4.6第43页/共60页直角坐标方程球面坐标方程正圆锥面圆锥面第44页/共60页直角坐标球面坐标：球形区域上、下限全是常数球面坐标系中的“长方体”第45页/共60页直角坐标球面坐标：球形区域第46页/共60页球面坐标球顶锥体上、下限全是常数第47页/共60页with(plots):qiumian:=plot3d
32、(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=-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(qiu
33、mian,zuimian,x_axis,y_axis,z_axis,orientation=23,66,scaling=constrained);第48页/共60页球面坐标球顶锥体第49页/共60页例4.7解：原式第50页/共60页with(plots):zuimian:=plot3d(r*cos(t),r*sin(t),r,t=0.2*Pi,r=0.1,color=yellow):qiumian:=plot3d(sin(s)*cos(t),sin(s)*sin(t),cos(s)+1,s=0.Pi/2,t=0.2*Pi,color=green):x_axis:=plot3d(u,0,0,u=
34、-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=20,70,scaling=constrained);第51页/共60页Example第52页/共60页第53页/共60页第54页/共60页Example平顶锥体解在 xOy 面上的投影区域为圆域：第55页/共60页直角坐标第56页/共60页柱面坐标第57页/共60页球面坐标第58页/共60页似乎柱面坐标最简单比较：第59页/共60页感谢您的观看。第60页/共60页