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1、The Feminine Intuition of Princess DidoOn OptimumLife on MathematicsA story spread among the Phoenicians People always want to do their best.Middle school students who study hard always dream ofgetting full marks in the final exams.People who hurry onalways choose shortcuts to get to the meeting pla
2、ce as quicklyas possible.A young girl lives an abnormally frugal life only expecting tobuy a Gucci scarf before the Spring Festival.A Phoenician StoryInheritance of the throne causes the murder.Across the sea to escape from murder.Is the Area Largest?The land Dido chose was sea-basedLaid the hide ou
3、t in a semi-circle.Maximum problemwhen the perimeter of a rectangular piece of grassland isconstant,what are the length and the width making the largestarea?Constrained optimizationConditional extremummax.22SxystxyL=+=constraintyxmaximum valueQuestion：what are the length x and the width y maximizing
4、 thelargest area of the rectangle?Solution max.22SxystxyL=+=12yLx=22211()()22416LLSxLxxLxx=+=+4L4LWhen,the area of the rectangle at thistime takes the maximum value.4Lxy=2max16LS=The Land Area ProblemNo idea of the lands shape!To find a function to maximize the area:()yy x=The function of the curve:
5、()yy x=2max()s.t.1()babaSy x dxLydxC=+=Maximize Subject to Perimeter of the hideCan the functions we usually mention act as independent variablesWhat is the microelement method?Can the functions we usually mention act as independent variables?What kind of mathematical thinking lies in the design of
6、the slide?What is the calculus of variations?The length of the curve?iyixiSThe length of the curve is2222111()()nnniiiiiiiiiiixyLsxyttt=+=+Microelement,DifferentialDivide parameter t into n small segments,and record the length of every segment as(1,2,)=it inWhen is very small,the lengths ,and are sm
7、all too.By the formula of arc differential,there isisixiy22+=iiisxyThe thinking of microelement methodiyixiS2222111()()nnniiiiiiiiiiixyLsxyttt=+=+0it 2122()()ttttLxydt=+The length of the curvetxtydtMicroelement,DifferentialTwo Kinds of Optimization Problemsxymaxs.t.22SxyxyL=+=()yy x=2max()s.t.1()bab
8、aSy x dxLydxC=+=To find y(x)make S be maximum To find x,y make S be maximum A Class of New“Functions”Independent variable:y(x)21()baLydx=+You can calculate a value when you give an integrable function.Two ExamplesConsider following straight line ,=yxaxb We can get the area enclosed by the axis x:22(
9、)2=bbaabay x dxxdx OxyabThe equation of semicircle：22,=yrxrxr The area enclosed by axis x is as following:22212=rrrx dxr xyOrrFunctionalFunction spacemappingReal number domainHow to solve the extremum of functionals?21()baLydxC=+=滑梯的故事（镜头15.1，主讲台沈)关于泛函的极值问题早在关于泛函的极值问题早在18世纪就有人注意到世纪就有人注意到，一个著名的一个著名的问
10、题就是最速降线问题问题就是最速降线问题。幼儿园往往少不了滑梯幼儿园往往少不了滑梯，滑梯的乐趣更多来自于从顶端到底滑梯的乐趣更多来自于从顶端到底端直接滑下来的快感端直接滑下来的快感。所以所以，滑梯的良好设计应该包含一个优化滑梯的良好设计应该包含一个优化问题：如何设计滑梯滑道的形状问题：如何设计滑梯滑道的形状，使得乘坐滑梯者以最短的时间使得乘坐滑梯者以最短的时间滑完全程滑完全程？牛顿牛顿(1642-1727)：英国著名物理学家和数学家，经典力学奠基：英国著名物理学家和数学家，经典力学奠基人。他与莱布尼兹分享创立微积分的荣誉，著有自然哲学之数人。他与莱布尼兹分享创立微积分的荣誉，著有自然哲学之数学原
11、理，对近代数学做出卓越的贡献。学原理，对近代数学做出卓越的贡献。约翰约翰 伯努利伯努利(1667-1748)：瑞士数学家，其数学成果丰富，如求：瑞士数学家，其数学成果丰富，如求积分的变量替换法等。求极限的洛必达法则也是他最先给出。他积分的变量替换法等。求极限的洛必达法则也是他最先给出。他提出的提出的“最速降线问题最速降线问题”征解，催发了变分法的诞生。征解，催发了变分法的诞生。名人介紹（镜头15.2，坐姿娟)雅可比雅可比(1804-1851)：德国数学家，在函数论、数学分析、数论、：德国数学家，在函数论、数学分析、数论、几何学、力学方面作出重要贡献，是椭圆函数理论的奠基者之一，几何学、力学方面
12、作出重要贡献，是椭圆函数理论的奠基者之一，雅可比矩阵与雅可比行列式用途广泛。雅可比矩阵与雅可比行列式用途广泛。洛必达洛必达(1661-1704)：法国数学家，相继求解出：法国数学家，相继求解出“摆线难题摆线难题”及及“最最速降线问题速降线问题”。17世纪末撰写无限小分析一书，书中介绍了洛世纪末撰写无限小分析一书，书中介绍了洛必达法则。必达法则。名人介紹（镜头15.3，坐姿桃)变分法（镜头16，主讲台立刚)然而，这个问题并没有因为这些数学巨匠的完美工作而戛然而止。然而，这个问题并没有因为这些数学巨匠的完美工作而戛然而止。30年后，约翰年后，约翰 伯努利的学生欧拉将最速降线问题进一步延伸，给伯努利
13、的学生欧拉将最速降线问题进一步延伸，给出了泛函的极值问题，并最终创立了解决泛函极值问题的变分法。出了泛函的极值问题，并最终创立了解决泛函极值问题的变分法。Calculus of VariationsAssume the extreme curve is()=yy x,give a set of functions:()()+J y xx The above formula can be regarded as a function of the parameter:()()()=+J y xx xyO)(xy)()(xxy+()=yy x is the extremal curve of()(
14、)+J y xx 0=is the extremum of the function().Princess Didos Area ProblemFor a closed curve with a constantperimeter,thecircleenclosesthelargest area.Guess:Princess Didos sense of beauty Princess Dido stood on the coastline.The beautifulscenerymade her temporarilyput asidethedepression and misery in
15、her heart.Her relaxedspirit made her draw a full circle on the vast land.海土地 Pythagoras and the poet Dante believed that thecircle was the most beautiful figure the sun isround,and the full moon is round.The circle iscrisp,concise,uniform and symmetrical.Many praises have been given for circles.Dant
16、e AlighieriCircles express beautiful wishesOlympic ringsLogo of Harbin Engineering UniversityArchitectural beautyThe ancient Roman ArenaThe Temple of Heaven in BeijingThe Sydney Opera HouseWhy Do People Love Circular Patterns?The extremum of functionalsWhy is the circle full of optimized genes?Whent
17、he curved roofs of ancient Chinese building glowwithbeauty,howdoesthedesignadvancepractical functions in an unexpected way?Ancient Chinese ArchitectureCurved shape not only for beauty but also for practicality of evacuating rainwater?Brachistochrone problem Among all the curves connecting two points
18、 ABthat are not on the same vertical line,we need tofind a curve so that for a particle with an initialvelocity equal to zero,it takes the shortest timeto reach the point B from A descending along thecurve under the influence of gravity.ABiSCalculate the timeABiS2iiiiisstvgy=11222212(1)1()2(2)oBxAxy
19、yTdxdxgygy+=0isisisis11222111()22nniiiiiinniiiiiiiisTtvyxyxxgygy=+=Time travelingSpeed travelingy-axis coordinate of泛函极值（镜头27:上课讲台立刚)求一条曲线()yy x=使得时间1221()2oxxyTdxgy+=最小。下降曲线的参数方程为 121(sin)2(1cos)2cxccy=+=摆线摆线：是圆沿着一条直线滚动时，圆上某一点的轨迹。12rc=The cycloid is the curve generated by a point on the circumferen
20、ce of a circle that rolls along a straight line.12rc=Extremum problem of functionals The parameter equation of the decreasing curve is()()121sin21cos2cxccy=+=Find a curve()yy x=makes the time 01221()2xxyTdxgy+=is minimized.Decreasing curveMotion Characteristics of The CircleThe raging tornado Moving
21、 bicyclesThe stars in the universeWhat the circle reminds people of is more dynamic than stationary.Is Beauty an Answer to Scientific Questions?Princess Didos area problem reminds us of beauty.The mathematical principle in the construction of Chinese charactersWhat kind of truth does the ever-changi
22、ng nature make people seek in optimization?What is the mathematical principle in the construction of Chinese characters?A BDaughters QuestionWhy is the line between the two points the shortest distance?Thinking:to draw an auxiliary line?Or use the Pythagorean theorem?The“economic essence”of natureIn
23、 40 AD,Greek engineer Hiro emphasized the economic nature of natural phenomena in his explanation of light,and proposed the principle of the shortest path of light.He believes that light travels between two points in space along the shortest path.Later,Fermat corrected this statement,thinking that l
24、ight traveled along the shortest path between two points in space.Robert Grosseteste(11751253),Chancellor of Oxford University,said that nature always moves and changes in the smallest and most optimal way mathematically.As early as the 6th century,the Greek philosopher Olympiodorus the Younger advo
25、cated the“economic essence”of nature.He believed that nature did not do anything unnecessary.How to Walk?Goose step“Cloud step”Step length?Consideration:how to consume the least energy per step?Degenerated tailsChimpanzees without tailApe with tailThe joy of talking nonsenseThe writer Su Qin discove
26、red the fact from a psychologist:If more than 90%of a persons words are nonsense,he will be happy;if less than 50%of nonsense,he will not be happy.Writing and TalkingOral expression:lengthy and wordy.Written words：concise.To use a knife to scrape the wrong character.Essences of Chinese charactersChi
27、nese characters:use the fewest strokes to express accurate meaning.Chinese vocabulary:use the least words to express the most information.To save the resources and energies in writing.The character 男(nan)男男(Male)：Labors in the field.The character fu婦(The traditional Chinese character of fu):sweeping the floor.妇(Concise strokes of the character fu):raising kids,cooking,laundry,and so on.Shede“舍得”She:to give something.De:to get something”.Shede:acquisition needs a sacrifice first.