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1、四川大学数学学院徐小湛http:/June2011视觉微积分：视觉微积分：Mamikon方法方法2011.6.10四川大学数学学院徐小湛http:/June20111959年年Mamikon发现了一种求面积的发现了一种求面积的方法，很巧妙的方法。不需要积分。有方法，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。确定的时，这种方法特别方便。MamikonMnatsakanian四川大学数学学院徐小湛http:/June2011根据根据Mamikon的方
2、法，图中环形区域的面积等于右上方的方法，图中环形区域的面积等于右上方圆的面积。圆的面积。理由：它们都是红色的直线段旋转理由：它们都是红色的直线段旋转360度扫出的图形，所度扫出的图形，所以面积相等。以面积相等。看了这个看了这个动画动画，你对这种方法就有直观的了解，你对这种方法就有直观的了解：http:/www.cco.caltech.edu/mamikon/CircIkon.html四川大学数学学院徐小湛http:/June2011用用Mamikon的方法，曳物线（的方法，曳物线（Tractix）（见下图）（见下图）（动画点（动画点这里这里）与其渐近线之间的面积可以轻易得出。）与其渐近线之间的
3、面积可以轻易得出。解释如下：解释如下：假设切线（拉绳）的长度是假设切线（拉绳）的长度是LL，则小孩在曳物的整个过程中，切线转，则小孩在曳物的整个过程中，切线转动了动了90度，所以根据度，所以根据Mamikon的结论，曳物线与其渐近线之间的面积等的结论，曳物线与其渐近线之间的面积等于半径为于半径为L 的的1/4圆的面积，即圆的面积，即（见下图）。如果用定积分来求这（见下图）。如果用定积分来求这个面积，计算是比较复杂的。个面积，计算是比较复杂的。四川大学数学学院徐小湛http:/June2011去看看动画去看看动画：http:/www.cco.caltech.edu/mamikon/TraxIko
4、n.html四川大学数学学院徐小湛http:/June2011这个方法在以下主页可以看到：视觉微积分主页：视觉微积分主页：http:/www.cco.caltech.edu/mamikon/calculus.html四川大学数学学院徐小湛http:/June2011微积分教材的作者Apostol对这种方法的介绍：AVISUALApproachtoCALCULUSproblemsAtalkbyTOMM.APOSTOLhttp:/www.its.caltech.edu/mamikon/VisualCalc.html四川大学数学学院徐小湛http:/June2011下面，下面，我们用数学软件我们用数
5、学软件Maple绘制一些图形。绘制一些图形。四川大学数学学院徐小湛http:/June2011环形域:L:切线长切线长四川大学数学学院徐小湛http:/June2011with(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):quxian2:=spacecurve(x(t)-L*D(x)(t)/R(t),y(t)-L*D(y)(t)/R(t),0,t=0.2*
6、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=-2.3,v=0.0.01,thickness=3):xyz:=display(x_axis,y_axis,z_axis,thickn
7、ess=3):display(qumian,quxian1,quxian2,orientation=-90,0,tickmarks=4,4,4,axes=none,lightmodel=light2,scaling=constrained);切线长切线长四川大学数学学院徐小湛http:/June2011with(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:=spacecurve(x(t),y(t),0,t=0.2*Pi,thickness=3,color=red):
8、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=30,2):quxian3:=spacecurve(-L*D(x)(t)/R(t),-L*D(y)(t)
9、/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):display(qumian,quxian1,quxian2,orientation=-90,0,tickmarks=4,4
10、,4,axes=none,lightmodel=light2,scaling=constrained);display(qumian2,quxian3,orientation=-90,0,tickmarks=4,4,4,axes=none,lightmodel=light2,scaling=constrained);四川大学数学学院徐小湛http:/June2011with(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:=spacecurve(x(t),y(t),0,
11、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):y_axis:=plot3d(0,u,0,u=-3.3,v=0.0.01,thi
12、ckness=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);切线长切线长四川大学数学学院徐小湛http:/June2011with(plots):a:=2:b:=1:L:=3:x:=t-a*cos(t):y:=t-b*
13、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=30,2):qumian2:=plot3d(-s*D(x
14、)(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=0.0.01,thickness=3):xyz:=disp
15、lay(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);四川大学数学学院徐小湛http:/June2011with(plots):a:=2
16、: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=0.2*Pi,s=0.L,g
17、rid=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:=plot3d(0,0,u,u=-2.3
18、,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=constrained);四川大学数学学院徐
19、小湛http:/June2011with(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(t)-s
20、*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:forifrom1toKdoti:=i*2*Pi/K:linei:=s
21、pacecurve(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);四川大学数学学院徐小湛http:/June2011两个图形面积相等两个图形面积相等四川大学数学学院徐小湛http:/J
22、une2011with(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(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)/
23、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:forifrom1toKdoti:=i*2*Pi/K:linei:=spacecurve
24、(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);四川大学数学学院徐小湛http:/June2011with(plots):a:=2:b:=3:L:=5:x:=t-a*cos(t):y:
25、=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(-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:=plot3
26、d(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:forifrom1toKdoti:=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
27、=2)od:line:=display(seq(linei,i=1.K):display(qumian,quxian2,line,orientation=0,0,tickmarks=4,4,4,axes=none,lightmodel=light2,scaling=constrained);四川大学数学学院徐小湛http:/June2011两个图形面积相等两个图形面积相等四川大学数学学院徐小湛http:/June2011with(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(
28、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:=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,th
29、ickness=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:forifrom1toKdoti:=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:=dis
30、play(seq(linei,i=1.K):display(qumian,quxian1,quxian2,line,xyz,orientation=-90,0,tickmarks=4,4,4,axes=none,lightmodel=light2,scaling=constrained);四川大学数学学院徐小湛http:/June2011with(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=
31、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,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=
32、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:forifrom1toKdoti:=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=-9
33、0,0,tickmarks=4,4,4,axes=none,lightmodel=light2,scaling=constrained);四川大学数学学院徐小湛http:/June2011两个图形面积相等两个图形面积相等四川大学数学学院徐小湛http:/June2011with(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:
34、=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):z_axis:=plot3d(0
35、,0,u,u=-2.3,v=0.0.01,thickness=1):xyz:=display(x_axis,y_axis,z_axis,thickness=3):K:=20:forifrom1toKdoti:=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
36、,0,tickmarks=4,4,4,axes=none,lightmodel=light2,scaling=constrained);四川大学数学学院徐小湛http:/June2011with(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(-L*D(x)(t)/R(t),-L*D(y)(t)/R(t
37、),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:=display(x_axis,y_ax
38、is,z_axis,thickness=3):K:=20:forifrom1toKdoti:=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,orientation=-90,0,tickmarks=4,4,4,axes=none,lightmodel=light2,scaling=constrained);四川大学数学学院徐小湛h
39、ttp:/June2011两个图形面积相等两个图形面积相等四川大学数学学院徐小湛http:/June2011with(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)(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:=
40、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):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:forif
41、rom1toKdoti:=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,lightmodel=light2,scaling=constrained);四川大学数学学院徐小湛http:/June2
42、011with(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)(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,style=patc
43、hnogrid,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:forifrom1toKdoti:=A+i*(B-A)/K:linei:=spacecurve(s*D(x)(ti)/R(ti),s*D(y)(ti)/
44、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);四川大学数学学院徐小湛http:/June2011两个图形面积相等两个图形面积相等四川大学数学学院徐小湛http:/June2011下面下面用数学软件用数学软件Mathematica验证验证Mamikon的结论的结论四川大学数学学院徐
45、小湛http:/June2011a=3;b=1;L=2;xt_:=aCost;yt_:=bSint;Rt_:=Sqrtxt2+yt2;r0t_:=xt,yt;rt_:=xt-L*xt/Rt,yt-L*yt/Rt;ParametricPlotr0t,rt,t,0,2Pi四川大学数学学院徐小湛http:/June2011a=3;b=1;L=2;xt_:=aCost;yt_:=bSint;Rt_:=Sqrtxt2+yt2;Xt_:=xt+L*xt/Rt;yt_:=yt+L*yt/Rt;Integratext*yt-xt*yt,t,0,2*Pi四川大学数学学院徐小湛http:/June2011xt_:=aCost;yt_:=bSint;Rt_:=Sqrtxt2+yt2;Xt_:=L*xt/Rt;Yt_:=L*yt/Rt;IntegrateXt*Yt-Xt*Yt,t,0,2*Pi/2利用格林公式计算环形利用格林公式计算环形区域面积：区域面积：Mathematica的计算结果：环形区域面积为
