视觉微积分+Mamikon方法.ppt

1、视觉微积分：Mamikon方法2011.6.10,1959年Mamikon发现了一种求面积的方法，很巧妙的方法。不需要积分。有些曲线所围成的面积用这种方法很方便计算，尤其是当曲线的切线的长度容易确定的时，这种方法特别方便。,Mamikon Mnatsakanian,根据Mamikon的方法，图中环形区域的面积等于右上方圆的面积。 理由：它们都是红色的直线段旋转360度扫出的图形，所以面积相等。 看了这个动画，你对这种方法就有直观的了解： /mamikon/CircIkon.html,用Mamikon的方法，曳物线（Tractix）（见下图）（动

2、画点 这里 ）与其渐近线之间的面积可以轻易得出。解释如下：,假设切线（拉绳）的长度是LL，则小孩在曳物的整个过程中，切线转动了90度，所以根据Mamikon的结论，曳物线与其渐近线之间的面积等于半径为 L 的1/4圆的面积，即 （见下图）。如果用定积分来求这个面积，计算是比较复杂的。,去看看动画：/mamikon/TraxIkon.html,这个方法在以下主页可以看到： 视觉微积分主页： /mamikon/calculus.html,微积分教材的作者Apostol对这种方法的介绍： AVISUA

3、L Approach to CALCULUS problems A talk by TOM M. APOSTOL /mamikon/VisualCalc.html,下面， 我们用数学软件Maple绘制一些图形。,环形域:,L: 切线长,with(plots): a:=2:b:=2:L:=3: x:=t-a*cos(t):y:=t-b*sin(t): R:=t-sqrt(D(x)(t)2+D(y)(t)2): quxian1:=spacecurve(x(t),y(t),0,t=0.2*Pi,thickness=5,color=red): quxi

4、an2:=spacecurve(x(t)-L*D(x)(t)/R(t),y(t)-L*D(y)(t)/R(t),0,t=0.2*Pi,thickness=5,color=blue): qumian:=plot3d(x(t)-s*D(x)(t)/R(t),y(t)-s*D(y)(t)/R(t),0,t=0.2*Pi,s=0.L,grid=30,2): x_axis:=plot3d(u,0,0,u=-3.3,v=0.0.01,thickness=3): y_axis:=plot3d(0,u,0,u=-3.3,v=0.0.01,thickness=3): z_axis:=plot3d(0,0,u,u

5、=-2.3,v=0.0.01,thickness=3): xyz:=display(x_axis,y_axis,z_axis,thickness=3): display(qumian,quxian1,quxian2,orientation=-90,0,tickmarks=4,4,4,axes=none,lightmodel=light2,scaling=constrained);,切线长,with(plots): a:=2:b:=1:L:=3: x:=t-a*cos(t):y:=t-b*sin(t): R:=t-sqrt(D(x)(t)2+D(y)(t)2): quxian1:=spacecu

6、rve(x(t),y(t),0,t=0.2*Pi,thickness=3,color=red): quxian2:=spacecurve(x(t)-L*D(x)(t)/R(t),y(t)-L*D(y)(t)/R(t),0,t=0.2*Pi,thickness=3,color=blue): qumian:=plot3d(x(t)-s*D(x)(t)/R(t),y(t)-s*D(y)(t)/R(t),0,t=0.2*Pi,s=0.L,grid=30,2): qumian2:=plot3d(-s*D(x)(t)/R(t),-s*D(y)(t)/R(t),0,t=0.2*Pi,s=0.L,grid=3

7、0,2): quxian3:=spacecurve(-L*D(x)(t)/R(t),-L*D(y)(t)/R(t),0,t=0.2*Pi,thickness=3,color=blue): x_axis:=plot3d(u,0,0,u=-3.3,v=0.0.01,thickness=3): y_axis:=plot3d(0,u,0,u=-3.3,v=0.0.01,thickness=3): z_axis:=plot3d(0,0,u,u=-2.3,v=0.0.01,thickness=3): xyz:=display(x_axis,y_axis,z_axis,thickness=3): displ

8、ay(qumian,quxian1,quxian2,orientation=-90,0,tickmarks=4,4,4,axes=none,lightmodel=light2,scaling=constrained); display(qumian2,quxian3,orientation=-90,0,tickmarks=4,4,4,axes=none,lightmodel=light2,scaling=constrained);,with(plots): a:=2:b:=1:L:=3: x:=t-a*cos(t):y:=t-b*sin(t): R:=t-sqrt(D(x)(t)2+D(y)(

9、t)2): quxian1:=spacecurve(x(t),y(t),0,t=0.2*Pi,thickness=3,color=red): quxian2:=spacecurve(x(t)-L*D(x)(t)/R(t),y(t)-L*D(y)(t)/R(t),0,t=0.2*Pi,thickness=3,color=blue): qumian:=plot3d(x(t)-s*D(x)(t)/R(t),y(t)-s*D(y)(t)/R(t),0,t=0.2*Pi,s=0.L,grid=30,2): x_axis:=plot3d(u,0,0,u=-3.3,v=0.0.01,thickness=3)

10、: y_axis:=plot3d(0,u,0,u=-3.3,v=0.0.01,thickness=3): z_axis:=plot3d(0,0,u,u=-2.3,v=0.0.01,thickness=3): xyz:=display(x_axis,y_axis,z_axis,thickness=3): display(qumian,quxian1,quxian2,orientation=-90,0,tickmarks=4,4,4,axes=none,lightmodel=light2,scaling=constrained);,切线长,with(plots): a:=2:b:=1:L:=3:

11、x:=t-a*cos(t):y:=t-b*sin(t): R:=t-sqrt(D(x)(t)2+D(y)(t)2): quxian1:=spacecurve(x(t),y(t),0,t=0.2*Pi,thickness=3,color=red): quxian2:=spacecurve(x(t)-L*D(x)(t)/R(t),y(t)-L*D(y)(t)/R(t),0,t=0.2*Pi,thickness=3,color=blue): qumian:=plot3d(x(t)-s*D(x)(t)/R(t),y(t)-s*D(y)(t)/R(t),0,t=0.2*Pi,s=0.L,grid=30,

12、2): qumian2:=plot3d(-s*D(x)(t)/R(t),-s*D(y)(t)/R(t),0,t=0.2*Pi,s=0.L,grid=30,2): quxian3:=spacecurve(-L*D(x)(t)/R(t),-L*D(y)(t)/R(t),0,t=0.2*Pi,thickness=3,color=blue): x_axis:=plot3d(u,0,0,u=-3.3,v=0.0.01,thickness=3): y_axis:=plot3d(0,u,0,u=-3.3,v=0.0.01,thickness=3): z_axis:=plot3d(0,0,u,u=-2.3,v

13、=0.0.01,thickness=3): xyz:=display(x_axis,y_axis,z_axis,thickness=3): display(qumian,quxian1,quxian2,orientation=-90,0,tickmarks=4,4,4,axes=none,lightmodel=light2,scaling=constrained); display(qumian2,quxian3,orientation=-90,0,tickmarks=4,4,4,axes=none,lightmodel=light2,scaling=constrained);,with(pl

14、ots): a:=2:b:=1:L:=3: x:=t-a*cos(t):y:=t-b*sin(t): R:=t-sqrt(D(x)(t)2+D(y)(t)2): quxian1:=spacecurve(x(t),y(t),0,t=0.2*Pi,thickness=3,color=red): quxian2:=spacecurve(x(t)-L*D(x)(t)/R(t),y(t)-L*D(y)(t)/R(t),0,t=0.2*Pi,thickness=3,color=blue): qumian:=plot3d(x(t)-s*D(x)(t)/R(t),y(t)-s*D(y)(t)/R(t),0,t

15、=0.2*Pi,s=0.L,grid=30,2): qumian2:=plot3d(-s*D(x)(t)/R(t),-s*D(y)(t)/R(t),0,t=0.2*Pi,s=0.L,grid=30,2): quxian3:=spacecurve(-L*D(x)(t)/R(t),-L*D(y)(t)/R(t),0,t=0.2*Pi,thickness=3,color=blue): x_axis:=plot3d(u,0,0,u=-3.3,v=0.0.01,thickness=3): y_axis:=plot3d(0,u,0,u=-3.3,v=0.0.01,thickness=3): z_axis:

16、=plot3d(0,0,u,u=-2.3,v=0.0.01,thickness=3): xyz:=display(x_axis,y_axis,z_axis,thickness=3): display(qumian,quxian1,quxian2,orientation=-90,0,tickmarks=4,4,4,axes=none,lightmodel=light2,scaling=constrained); display(qumian2,quxian3,orientation=-90,0,tickmarks=4,4,4,axes=none,lightmodel=light2,scaling

17、=constrained);,with(plots): a:=2:b:=2:L:=5: x:=t-a*cos(t):y:=t-b*sin(t): R:=t-sqrt(D(x)(t)2+D(y)(t)2): quxian1:=spacecurve(x(t),y(t),0,t=0.2*Pi,thickness=4,color=red): quxian2:=spacecurve(x(t)-L*D(x)(t)/R(t),y(t)-L*D(y)(t)/R(t),0,t=0.2*Pi,thickness=3,color=blue): qumian:=plot3d(x(t)-s*D(x)(t)/R(t),y

18、(t)-s*D(y)(t)/R(t),0,t=0.2*Pi,s=0.L,style=patchnogrid,color=green): x_axis:=plot3d(u,0,0,u=-3.3,v=0.0.01,thickness=3): y_axis:=plot3d(0,u,0,u=-3.3,v=0.0.01,thickness=3): z_axis:=plot3d(0,0,u,u=-2.3,v=0.0.01,thickness=3): xyz:=display(x_axis,y_axis,z_axis,thickness=3): K:=20: for i from 1 to K do ti:

19、=i*2*Pi/K: linei:=spacecurve(x(ti)-s*D(x)(ti)/R(ti),y(ti)-s*D(y)(ti)/R(ti),0,s=0.L,color=brown,thickness=2) od: line:=display(seq(linei,i=1.K): display(qumian,quxian1,quxian2,line,orientation=-90,0,tickmarks=4,4,4,axes=none,lightmodel=light2,scaling=constrained);,两个图形面积相等,with(plots): a:=2:b:=3:L:=5

20、: x:=t-a*cos(t):y:=t-b*sin(t): R:=t-sqrt(D(x)(t)2+D(y)(t)2): quxian1:=spacecurve(x(t),y(t),0,t=0.2*Pi,thickness=4,color=red): quxian2:=spacecurve(x(t)-L*D(x)(t)/R(t),y(t)-L*D(y)(t)/R(t),0,t=0.2*Pi,thickness=3,color=blue): qumian:=plot3d(x(t)-s*D(x)(t)/R(t),y(t)-s*D(y)(t)/R(t),0,t=0.2*Pi,s=0.L,style=

21、patchnogrid,color=green): x_axis:=plot3d(u,0,0,u=-3.3,v=0.0.01,thickness=3): y_axis:=plot3d(0,u,0,u=-3.3,v=0.0.01,thickness=3): z_axis:=plot3d(0,0,u,u=-2.3,v=0.0.01,thickness=3): xyz:=display(x_axis,y_axis,z_axis,thickness=3): K:=20: for i from 1 to K do ti:=i*2*Pi/K: linei:=spacecurve(x(ti)-s*D(x)(

22、ti)/R(ti),y(ti)-s*D(y)(ti)/R(ti),0,s=0.L,color=brown,thickness=2) od: line:=display(seq(linei,i=1.K): display(qumian,quxian1,quxian2,line,orientation=-90,0,tickmarks=4,4,4,axes=none,lightmodel=light2,scaling=constrained);,with(plots): a:=2:b:=3:L:=5: x:=t-a*cos(t):y:=t-b*sin(t): R:=t-sqrt(D(x)(t)2+D

23、(y)(t)2): quxian1:=spacecurve(x(t),y(t),0,t=0.2*Pi,thickness=4,color=red): quxian2:=spacecurve(-L*D(x)(t)/R(t),-L*D(y)(t)/R(t),0,t=0.2*Pi,thickness=3,color=blue): qumian:=plot3d(-s*D(x)(t)/R(t),-s*D(y)(t)/R(t),0,t=0.2*Pi,s=0.L,style=patchnogrid,color=green): x_axis:=plot3d(u,0,0,u=-3.3,v=0.0.01,thic

24、kness=3): y_axis:=plot3d(0,u,0,u=-3.3,v=0.0.01,thickness=3): z_axis:=plot3d(0,0,u,u=-2.3,v=0.0.01,thickness=3): xyz:=display(x_axis,y_axis,z_axis,thickness=3): K:=20: for i from 1 to K do ti:=i*2*Pi/K: linei:=spacecurve(-s*D(x)(ti)/R(ti),-s*D(y)(ti)/R(ti),0,s=0.L,color=brown,thickness=2) od: line:=d

25、isplay(seq(linei,i=1.K): display(qumian,quxian2,line,orientation=0,0,tickmarks=4,4,4,axes=none,lightmodel=light2,scaling=constrained);,两个图形面积相等,with(plots): a:=3:b:=3:L:=5:A:=0:B:=3.5: x:=t-a*cos(t):y:=t-b*sin(t): R:=t-sqrt(D(x)(t)2+D(y)(t)2): quxian1:=spacecurve(x(t),y(t),0,t=A.B,thickness=4,color=

26、red): quxian2:=spacecurve(x(t)-L*D(x)(t)/R(t),y(t)-L*D(y)(t)/R(t),0,t=A.B,thickness=3,color=blue): qumian:=plot3d(x(t)-s*D(x)(t)/R(t),y(t)-s*D(y)(t)/R(t),0,t=A.B,s=0.L,style=patchnogrid,color=green): x_axis:=plot3d(u,0,0,u=-3.3,v=0.0.01,thickness=1): y_axis:=plot3d(0,u,0,u=-3.3,v=0.0.01,thickness=1)

27、: z_axis:=plot3d(0,0,u,u=-2.3,v=0.0.01,thickness=1): xyz:=display(x_axis,y_axis,z_axis,thickness=3): K:=20: for i from 1 to K do ti:=i*(B-A)/K: linei:=spacecurve(x(ti)-s*D(x)(ti)/R(ti),y(ti)-s*D(y)(ti)/R(ti),0,s=0.L,color=brown,thickness=2) od: line:=display(seq(linei,i=1.K): display(qumian,quxian1,

28、quxian2,line,xyz,orientation=-90,0,tickmarks=4,4,4,axes=none,lightmodel=light2,scaling=constrained);,with(plots): a:=3:b:=3:L:=5:A:=0:B:=3.5: x:=t-a*cos(t):y:=t-b*sin(t): R:=t-sqrt(D(x)(t)2+D(y)(t)2): quxian1:=spacecurve(x(t),y(t),0,t=A.B,thickness=4,color=red): quxian2:=spacecurve(-L*D(x)(t)/R(t),-

29、L*D(y)(t)/R(t),0,t=A.B,thickness=3,color=blue): qumian:=plot3d(-s*D(x)(t)/R(t),-s*D(y)(t)/R(t),0,t=A.B,s=0.L,style=patchnogrid,color=green): x_axis:=plot3d(u,0,0,u=-3.3,v=0.0.01,thickness=1): y_axis:=plot3d(0,u,0,u=-3.3,v=0.0.01,thickness=1): z_axis:=plot3d(0,0,u,u=-2.3,v=0.0.01,thickness=1): xyz:=d

30、isplay(x_axis,y_axis,z_axis,thickness=3): K:=20: for i from 1 to K do ti:=i*(B-A)/K: linei:=spacecurve(-s*D(x)(ti)/R(ti),-s*D(y)(ti)/R(ti),0,s=0.L,color=brown,thickness=2) od: line:=display(seq(linei,i=1.K): display(qumian,quxian2,line,xyz,orientation=-90,0,tickmarks=4,4,4,axes=none,lightmodel=light

31、2,scaling=constrained);,两个图形面积相等,with(plots): a:=3:b:=3:L:=5:A:=0:B:=5.5: x:=t-a*cos(t):y:=t-b*sin(t): R:=t-sqrt(D(x)(t)2+D(y)(t)2): quxian1:=spacecurve(x(t),y(t),0,t=A.B,thickness=4,color=red): quxian2:=spacecurve(x(t)-L*D(x)(t)/R(t),y(t)-L*D(y)(t)/R(t),0,t=A.B,thickness=3,color=blue): qumian:=plot

32、3d(x(t)-s*D(x)(t)/R(t),y(t)-s*D(y)(t)/R(t),0,t=A.B,s=0.L,style=patchnogrid,color=green): x_axis:=plot3d(u,0,0,u=-3.3,v=0.0.01,thickness=1): y_axis:=plot3d(0,u,0,u=-3.3,v=0.0.01,thickness=1): z_axis:=plot3d(0,0,u,u=-2.3,v=0.0.01,thickness=1): xyz:=display(x_axis,y_axis,z_axis,thickness=3): K:=20: for

33、 i from 1 to K do ti:=i*(B-A)/K: linei:=spacecurve(x(ti)-s*D(x)(ti)/R(ti),y(ti)-s*D(y)(ti)/R(ti),0,s=0.L,color=brown,thickness=2) od: line:=display(seq(linei,i=1.K): display(qumian,quxian1,quxian2,line,xyz,orientation=-90,0,tickmarks=4,4,4,axes=none,lightmodel=light2,scaling=constrained);,with(plots

34、): a:=3:b:=3:L:=5:A:=0:B:=5.5: x:=t-a*cos(t):y:=t-b*sin(t): R:=t-sqrt(D(x)(t)2+D(y)(t)2): quxian1:=spacecurve(x(t),y(t),0,t=A.B,thickness=4,color=red): quxian2:=spacecurve(-L*D(x)(t)/R(t),-L*D(y)(t)/R(t),0,t=A.B,thickness=3,color=blue): qumian:=plot3d(-s*D(x)(t)/R(t),-s*D(y)(t)/R(t),0,t=A.B,s=0.L,st

35、yle=patchnogrid,color=green): x_axis:=plot3d(u,0,0,u=-3.3,v=0.0.01,thickness=1): y_axis:=plot3d(0,u,0,u=-3.3,v=0.0.01,thickness=1): z_axis:=plot3d(0,0,u,u=-2.3,v=0.0.01,thickness=1): xyz:=display(x_axis,y_axis,z_axis,thickness=3): K:=20: for i from 1 to K do ti:=i*(B-A)/K: linei:=spacecurve(-s*D(x)(

36、ti)/R(ti),-s*D(y)(ti)/R(ti),0,s=0.L,color=brown,thickness=2) od: line:=display(seq(linei,i=1.K): display(qumian,quxian2,line,orientation=-90,0,tickmarks=4,4,4,axes=none,lightmodel=light2,scaling=constrained);,两个图形面积相等,with(plots): L:=5:A:=-1.732:B:=1.732: x:=t-t2:y:=t-t3-3*t: R:=t-sqrt(D(x)(t)2+D(y)

37、(t)2): quxian1:=spacecurve(x(t),y(t),0,t=A.B,thickness=4,color=red): quxian2:=spacecurve(x(t)+L*D(x)(t)/R(t),y(t)+L*D(y)(t)/R(t),0,t=A.B,thickness=3,color=blue): qumian:=plot3d(x(t)+s*D(x)(t)/R(t),y(t)+s*D(y)(t)/R(t),0,t=A.B,s=0.L,style=patchnogrid,color=green): x_axis:=plot3d(u,0,0,u=-3.3,v=0.0.01,

38、thickness=1): y_axis:=plot3d(0,u,0,u=-3.3,v=0.0.01,thickness=1): z_axis:=plot3d(0,0,u,u=-2.3,v=0.0.01,thickness=1): xyz:=display(x_axis,y_axis,z_axis,thickness=3): K:=20: for i from 1 to K do ti:=A+i*(B-A)/K: linei:=spacecurve(x(ti)+s*D(x)(ti)/R(ti),y(ti)+s*D(y)(ti)/R(ti),0,s=0.L,color=brown, thickness=2) od: line:=display(seq(linei,i=1.K): display(qumian,quxian1,quxian2,line,xyz,orientation=-90,0,tickmarks=4,4,4,axes=none,