Thats the singular solution of the given clairaut differential equation. For the love of physics walter lewin may 16, 2011 duration. In mathematics he is also credited with clairauts equation and clairauts relation. Alexis claude clairaut and the true figure of the earth. Clairauts equation is a firstorder differential equation of the form. The above expression for g has been supplanted by the somigliana equation. He was the first to discuss the equation of the second degree in three variables, and to classify the surfaces represented by it. Now, the term general solution is also unfortunate since clearly the general solutions and singular solutions are distinct. Clairaut 1 who was the first to point out the difference between the general and the singular solutions of an equation of this form. Plotting of solutions is more prominently displayed in the section on damped secondorder equations. We say that an orthogonal patch xu,v is a clairaut parametrization in u if e v 0 and g v 0. Clairaut, alexis 171765 french mathematician whose precocity had been expressed early in childhood. Ordinary differential equationsold contents wikibooks. We have shown that the sphere is clairaut in vand the torus is clairaut in u.
Alexis clairaut 17 1765 mactutor history of mathematics. Thats the equation of the parabola given here by the red lines, red curve. Alexis claude clairaut mactutor history of mathematics. Pdf constrained systems and the clairaut equation researchgate. Dan has written several books on mathematics on the topics of differential equations, integration, statistics, and general mathematics. Clairauts difference equation is a special case of lagranges equation sokolnikoff and redheffer 1958 defined by. By criteria analogous to those used in the classification of conics he obtained. Clairaut who was the first to point out the difference between the general and the singular solutions of an equation of this form. Use the result of b to find another solutions of the clairaut equation of part a.
Rouse ball a short account of the history of mathematics. We demonstrate the origin of the dirac primary constraints, along with their explicit form, and this is. Depending upon the domain of the functions involved we have ordinary di. Clairaut s theorem is a general mathematical law giving the surface gravity on a viscous rotating ellipsoid in equilibrium under the action of its gravitational field and centrifugal force. This gives the parametric form of the characteristic curves as dx dt a, dy dt b, du dt c. Clairauts differential equation is a first order ordinary differential equation which can be put into the form. Alexis claude clairaut was a french mathematician, astronomer, and geophysicist. The equation is named for the 18thcentury french mathematician and physicist alexisclaude clairaut, who devised it. Differentiate both sides with respect to and obtain. Such an equation has an easily obtained general solution. The topics covered are plane and solid analytic geometry, vector algebra, analysis, derivatives, integrals, series, differential equations etc. This part is an equation of the envelope, y is equal to negative x plus four and squared over four.
The couple had 20 children, however only a few of them survived childbirth. This work was still incomplete at the time of his death. A book on algebra was published in 1749, in which he took the subject up to the solution of equations of degree four. A history of mathematicsmodern europeeuler, lagrange. However, being that the highest order derivatives in these equation are of second order, these are second order partial differential. Summaries the history of differential forms is examined from its origins in the work of clairaut on the theory of differential equations through the end of the 19th century. The legendreclairaut transformation presented is involutive. They are solved by differentiation and, in addition to the general integral, also allow for a singular solution. Katz also coedited a collection of historical articles taken from maa journals of the past 90 years, sherlock holmes in babylon and other tales of mathematical history. Examples of some of the partial differential equation treated in this book are shown in table 2. Which paper book did dalembert publish his essay on the theory of winds in.
Clairauts theorem is a general mathematical law giving the surface gravity on a viscous rotating ellipsoid in equilibrium under the action of its gravitational field and centrifugal force. Clairaut saw this trend, and wrote the book in an attempt to make the subject. Then, according to clairaut s theorem alexis claude clairaut, 171765, mixed partial derivatives are the same. It was published in 1743 by alexis claude clairaut in a treatise which synthesized physical and geodetic evidence that the earth is an oblate rotational ellipsoid. The singular solution is obtained by eliminating the parameter from the equations and. The following year clairaut studied the differential equations now known as clairaut s differential equations and gave a singular solution in addition to the general integral of the equations. History of science and mathematics stack exchange is a question and answer site for people interested in the history and origins of science and mathematics. Clairauts equation, in mathematics, a differential equation of the form y x. The history of differential forms from clairaut to. Clairauts difference equation from wolfram mathworld. We will not need the general chain rule or any of its consequences during the course of the proof, but we. Solve the following differential equations by converting to clairaut s form through suitable substitutions.
They are solved by differentiation and, in addition to the general integral, also allow for a. Solve the following differential equations by converting. It contains many worked examples that illustrate the theoretical material and serve as models for solving problems. Graph the envelope of part b and explain how it is related to the family of lines in part a. The plot shows that here the singular solution plotted in red is an envelope of the oneparameter family of solutions making. The history of differential forms from clairaut to poincare. The clairaut equation is a particular case of the lagrange equation. Alexis claude clairaut was born at paris on may, 17, and died there on may 17, 1765. He is noted for his work on differential equations and on curves and for formulating clairaut s theorem dealing with geodesic lines on the surface of an ellipsoid. Introduction to nonlinear differential and integral equations dover books on mathematics by harold t. It is an equation for an unknown function yx that expresses a relationship between the unknown function and.
Lagrange equation is a more general setup that includes clairaut s equation as a special case in terms of method, albeit a special case that is qualitatively somewhat different. It was published in 1743 by alexis claude clairaut in a treatise 1 which synthesized physical and geodetic evidence that the earth is an oblate rotational ellipsoid. The history of differential forms is examined from its origins in the work of clairaut on the theory of differential equations through the end of the 19th century. Piskunov this text is designed as a course of mathematics for higher technical schools. A course in calculus is a gateway to other, more advanced. History of calculus university of california, davis. The history of differential forms from clairaut to poincari by victor j.
It was written, as were all his similar works, in the style of the bernoulli brothers, by likening the infinitesimal arc of the curve of three elementary rectilinear segments. The solution family for the general solution is, with. He has edited or coedited two recent books dealing with this subject, learn from the masters 1994 and using history to teach mathematics 2000. For solving the equation we use an auxiliary variable p. Schwarzs proof of clairauts theorem history of science. Clairaut s theorem the next theorem shows that the order of differentiation does not matter, provided that the considered function is sufficiently differentiable. A clairaut s equation is a differential equation of the form y p. Concise introduction for undergraduates includes, among other topics, a survey of first order equations, discussions of complexvalued solutions, linear differential operators, inverse operators and variation of parameters method, the laplace transform, picards existence theorem, and an. Plotting solutions with mathematica is not prominent, for example, in the chapter on firstorder equations the text only demonstrates plotting a family of solutions of a bernoulli equation, and plotting the solutions of the clairaut equation. While solving this problem, he was the first mathematician to derive the formula. Calculusthe chain rule and clairauts theorem wikibooks. This is a highly nonlinear equation so its behavior is quite different from the linear des we tend to focus upon. Clairaut equation this is a classical example of a differential equation possessing besides its general solution a socalled singular solution. There exist bullen, 1975 three famous solutions of this.
He is editorinchief of the chemical rubber companys crcs standard mathematical tables and formulae, and is on the editorial board for crcs handbook of chemistry and physics. Clairaut worked on a wide range of problems within mathematics. To learn more, see our tips on writing great answers. Clairaut wrote some important memoirs on the topic, studying the theory as well as conducting optical experiments. Its general solution is a oneparameter family of straight lines. Clairaut was one of the key figures in the expedition to lapland that helped to confirm newtons theory for the. He belongs to the small group of children who, though of exceptional precocity, survive and maintain their powers when grown up. Pdf simplest solutions of clairauts equation and the.
In clairauts equation 18thcentury french mathematician and physicist alexisclaude clairaut, who devised it. A history of mathematicsmodern europeeuler, lagrange, and laplace. Katz, history of mathematics, a classic version, 3rd. Alexis clairaut was born in paris, france, to jeanbabtiste clairaut, who taught mathematics in paris and was elected to the berlin academy, and his wife catherine petit. Differential equations has its origins rooted in a small number of mathematical. Clairauts differential equation wolfram demonstrations. In mathematical analysis, clairauts equation or the clairaut equation is a differential equation. He was a prominent newtonian whose work helped to establish the validity of the principles and results that sir isaac newton had outlined in the principia of 1687. In mathematical analysis, clairauts equation or the clairaut equation is a differential equation of the form. A history of the subject, and more detailed equations for g can be found in khan. Clairauts differential equation from wolfram mathworld. The list of topics covered is quite exhaustive and the book has over 2500 problems and solutions.
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