三重积分的计算柱面球面省公共课一等奖全国赛课获奖课件.pptx

College of mathematics04 August 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 mathematics04 August 2024上一页|首页|下一页柱面坐标柱面坐标Cylindrical Coordinates柱面坐标柱面坐标 极坐标极坐标 竖坐标竖坐标柱面坐标柱面坐标第2页College of mathematics04 August 2024上一页|首页|下一页要求：要求：第3页College of mathematics04 August 2024上一页|首页|下一页第4页College of mathematics04 August 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常数常数圆柱面：圆柱面：常数常数柱面坐标所以得名柱面坐标所以得名第5页College of mathematics04 August 2024上一页|首页|下一页常数常数半平面半平面第6页College of mathematics04 August 2024上一页|首页|下一页常数常数平面平面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):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7页College of mathematics04 August 2024上一页|首页|下一页柱面坐标坐标面柱面坐标坐标面第8页College of mathematics04 August 2024上一页|首页|下一页柱面坐标坐标面动画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):pingmiani:=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:=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 mathematics04 August 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两张相邻水平面两张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 mathematics04 August 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=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,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,orientation=5,60);第11页College of mathematics04 August 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=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=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=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12页College of mathematics04 August 2024上一页|首页|下一页第13页College of mathematics04 August 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第14页College of mathematics04 August 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旋转抛物面旋转抛物面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(qumian,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),u*sin(theta),u,u=0.1,theta=0.2*Pi,numpoints=800,color=green):display(qumian,zuobiaoxi,orientation=35,60);第15页College of mathematics04 August 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圆柱面圆柱面第16页College of mathematics04 August 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圆锥面圆锥面旋转抛物面旋转抛物面圆柱面圆柱面圆柱面圆柱面第17页College of mathematics04 August 2024上一页|首页|下一页例例 螺旋面螺旋面 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 mathematics04 August 2024上一页|首页|下一页参数方程：参数方程：第19页College of mathematics04 August 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“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 mathematics04 August 2024上一页|首页|下一页例例4.4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第21页College of mathematics04 August 2024上一页|首页|下一页第22页College of mathematics04 August 2024上一页|首页|下一页例例4.5交线投影柱面交线投影柱面投影区域：投影区域：第23页College of mathematics04 August 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):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,orientation=23,66,scaling=constrained);第24页College of mathematics04 August 2024上一页|首页|下一页例4.57.4 三重积分（柱、球）第25页College of mathematics04 August 2024上一页|首页|下一页作作 剖面图：剖面图：上边界曲面：上边界曲面：下边界曲面：下边界曲面：第26页College of mathematics04 August 2024上一页|首页|下一页题题1(3)在在 xOy 面上投影区域：面上投影区域：第27页College of mathematics04 August 2024上一页|首页|下一页下边界曲面：下边界曲面：上边界曲面：上边界曲面：锥面锥面球面球面第28页College of mathematics04 August 2024上一页|首页|下一页第29页College of mathematics04 August 2024上一页|首页|下一页球面坐标球面坐标Spherical coordinates球面坐标球面坐标第30页College of mathematics04 August 2024上一页|首页|下一页第31页College of mathematics04 August 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常数常数球面：球面：常数常数球面坐标所以得名球面坐标所以得名第32页College of mathematics04 August 2024上一页|首页|下一页常数常数半平面半平面第33页College of mathematics04 August 2024上一页|首页|下一页常数常数锥面：锥面：第34页College of mathematics04 August 2024上一页|首页|下一页球面坐标坐标面球面坐标坐标面第35页College of mathematics04 August 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 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:=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 mathematics04 August 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/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,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,coords=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37页College of mathematics04 August 2024上一页|首页|下一页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=0.1.5,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):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):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 mathematics04 August 2024上一页|首页|下一页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=0.1.5,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,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=spherical,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=spherical,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,coords=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,orientation=5,74);第39页College of mathematics04 August 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两张相邻球面面两张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一个经典小区域如图一个经典小区域如图其体积约为其体积约为第40页College of mathematics04 August 2024上一页|首页|下一页球面坐标下体积元素：球面坐标下体积元素：第41页College of mathematics04 August 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第42页College of mathematics04 August 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):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 of mathematics04 August 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例例4.6第44页College of mathematics04 August 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圆锥面圆锥面第45页College of mathematics04 August 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：球形区域球形区域上、下限全是常数上、下限全是常数球面坐标系中球面坐标系中“长方体长方体”第46页College of mathematics04 August 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：球形区域球形区域第47页College of mathematics04 August 2024上一页|首页|下一页球面坐标球面坐标球顶锥体球顶锥体上、下限全是常数上、下限全是常数第48页College of mathematics04 August 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=-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 mathematics04 August 2024上一页|首页|下一页球面坐标球面坐标球顶锥体球顶锥体第50页College of mathematics04 August 2024上一页|首页|下一页例例4.7解：解：原式原式第51页College of mathematics04 August 2024上一页|首页|下一页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=-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52页College of mathematics04 August 2024上一页|首页|下一页Example第53页College of mathematics04 August 2024上一页|首页|下一页第54页College of mathematics04 August 2024上一页|首页|下一页第55页College of mathematics04 August 2024上一页|首页|下一页Example平顶锥体平顶锥体解解在在 xOy 面上投影区域面上投影区域为圆域：为圆域：第56页College of mathematics04 August 2024上一页|首页|下一页直角坐标直角坐标第57页College of mathematics04 August 2024上一页|首页|下一页柱面坐标柱面坐标第58页College of mathematics04 August 2024上一页|首页|下一页球面坐标球面坐标第59页College of mathematics04 August 2024上一页|首页|下一页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