Arnold Trivium 2 1993

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Uspekhi Mat. Nauk 48:1 (1993), 211-2 22 Russian Math. Surveys 48:1 (1993), 217-232

A mathematical trivium II

V.I.Arnol d

It seems almost a miracle thatcontemporary methods ofteaching have not yetcompletely stifled holycuriosity. A. Einstein

My appeal in A math ematical trivium [1] that written examinationsshould be used in mathematics provoked many replies with criticism of oraland written examinations both from Russia and from other countries inEu rope and America. The autho rs of man y letters from Russia reckoned thatinstructors could solve, on average, one-third of the problems of the trivium[1]. On this basis they thoug ht tha t the problems of the trivium were toodifficult. This should pro bab ly be set against the fact that in the US SR abo ut40% of heads of mathematics departments have had no mathematicaleducation, and the position will probably deteriorate even further.

Written examinations in various mathematical subjects have been tried outin several Russian higher education institutions, so some conclusions can bedrawn . The majority of the instructors taking part considered tha t it waseasier to conduct w ritten examinations th an oral ones. Supprising as it mayseem, copying is less of a danger than might have been expected, since it iseasily detected (incorrect so lutions are copied on the whole). The increase inthe number of unsatisfactory marks (because students who have learnt nothingare shown up) can hardly be regarded as a shortcoming of a writtenexamination.The selection of examination problems aroused the most frequent criticism.Here strange things sometimes happened (the appearance of which, however, isalso useful: the perso nality of the setter and local traditio ns clearly appear inthe character and the very formulation of the questions).

For example, the official national American written examination (known bythree initials, which I have forgotten) contained the following test-problem in1992:


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athematical Trivium II 219

A written examination for the course in ordinary differential equations inthe Mechanics and Mathematics Faculty of Moscow State University was firstheld in the spring of 1991. Two hours were allotted to the problems givenbelow. Correction (10 15 pieces of work per instructor) took an hour, afterwhich the results were announced to the students. For another hour thestudents looked through their work and analysed their mistakes (with the aidof the instructors). This part of the examination was voluntary, but almost allthe students wanted to discuss their work with the instructors.

The criteria for marking were made precise after the correction of thework, but in general were roughly as follows satisfactorymore than onecorrect solution, goodmore than two, excellentmore than three.When another group takes the same examination in a few days time, allthe problems are completely changed. So that the reader can see the degreeof similarity between the versions used on the same day, (1 ) and the degree ofdifference in subsequent days, versions for all the days are given below. Theyare compiled taking into account the fact that the students examined laterknow the problems from previous days (which in itself is not bad for thestudents, but requires additional effort from the setter). Setting the questionswould be easier if the whole class could be examined simultaneously, but thiscould not be organized for technical reasons.

Examination in differential equationsMechanics and Mathematics Faculty Moscow State University 1991irstday (one of 6 versions).1. F ind the image of the vector (1,0) based at the point (, 0) under theaction of the transformation at time t = 1 of the phase flow of the system = y,y = sin x.

1. In what coordinate system do the variables separate in the equat ion

3. Does the Cauchy problemdu 3 du

have a solution in the neighbourhood of the point x0 = 0, y = yo), and is itunique?4. Is the solution of the system

= yz, y = xz, i = 0

with initial condition x0, yo, z 0) stable in the sense of Lyapunov? 1 )I n Russia it is customary for the questions for a single examination to be set in severaldifferent versions, which are then allocated to the students at random (Translator s note).


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220 v.I . Arnol'd

Secondday. Consider the system (6 versions):( i = y, i = y3, ( i = y, i = y2, x = y4,\ y = x2 . | JI= x . I 3 = x4 . (_ jr = x 2 . | y . \ = x 2

1) Find the positions of equilibrium and investigate their stability.2) Can all the solutions of the system be extended indefinitely?3) How many non zero solutions for which y(0) = x(l) = 0 does the

system have?4) Find the derivative with respect to #at # = 0 of the solution with

initial condition JC(O) = y(0) a.Third day. Consider the system (6 versions):

= x'y, $ i V , $ = x V, i = xyz, I x = xy0, I i = xy4,y= xy . [y = xy . [ y = iy' . [ j/ = I^J/ . [ j, = x'yz. [ y =

1) and 2) as for the previous day.3) F ind a diffeomorphism of the plane rectifying the direction field of the

phase curves in the neighbourhood of the point (1, 1).4) Find all the first integrals continuous on the whole plane that coincide

with y on the y axis.Fourthday. Given th e system (6 versions):

2 2 2 2 2 2Z = I Z . 2 = 2 IZ . = ZZ . = XZZ . IZ .

1)as for the previous days.2) Find all initial conditions for which the solutions can be extended

forwards indefinitely.3) Find the image of the vector (0, 1) based at the point 0 under the action

of the transformation of the phase flow at time t = 1.4) Find all first integrals continuous in the neighbourhood of the point

= 1 and equal to 1 on the real axis.Fifth day. Consider the problem (one of 6 versions):

1) Does the problem have an unbounded solution defined on the wholeplane?

2) Is the quantity u bounded on the characteristics?3) D o all th e characteristics intersect the surface y = x + u2l4) Does the equation of characteristics have a first integral whose derivativewith respect to u is equal to 1 at the origin? Find the derivative of this

derivative with respect to u along a characteristic vector.


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Mathematical Trivium I I 221

Sixth day. Consider the equation 6versions):r + i = s i n h 3 z . + = sinh 3x/ 2.

i + s ii i = z . & + sin ' = ' / 2.

1) Can the solution with initial condition x(0) = 1,x(0)= 0 beextendedto the whole i axis?

2) Does the solution with initial condition x(0) = a,x(0)=0haveabounded third derivative with respect to a t = 0?

3) Compute the value ofthis derivative fort = 2.4) Compute the tenth derivative with respect toaa ta =0 ofthe solution

with initial condition x(0) = a,x(0)=0.Seventh day. G iven the equation 6versions):

2 ^x = si n t. = cos f.x si n " . =cos"t .

') 0 0 'Xx = sinh ' sin " t. =sinh" cos t.1) Find the third derivative a tzero ofthe solution with initial condition

x(0) = 0.2) Does this equation extend tothe whole ofthe i axis?3) Does the equation have unbounded solutions?4) Find the numberofasymptotically stable periodic solutions ofthe

equation.The examination problems forfirst and third year students ofvarious

European universities given below were published in [2].Universityof Warwick England) 1 )

During the first year the students study from 10 to 15subjects ("modules")of which 8are compulsory. Inthe third year the numberofsubjectsisbetween 8and 15(but, beginning with the second year, there are also easierprogrammes with fewer subjects, which qualify foranother degree). Eachsubject corresponds to 30hours oflectures, bu t teachingismostly givenbypostgraduate student instructors ("tu to rs"). In thefirst year oneofthecompulsory subjects issupervised work, and in the third year the onlycompulsory subject ofspecialization, "applied mathematics", consists ofcoursework corresponding totwo subjects.( ) T h e author quotes from [2], so theo riginal questions have been translated from Englishto French toRussian toEnglish. Thefina l versiongivenhereisunlikelyto bewordforword thesameas that originally set (Translator's note).


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V.I. Arnold

The examinations are short (1lA to 2 hours) and consist of three to fiveindependent questions. Almost all the questions contain problems from thecourse. If questions are given, the mark is derived from the best n 1answers. To illustrate the variety of subjects the following questions havebeen selected:

F or first year studen ts: four questions in the subjects "Analysis I I(compulsory), Lin ear algebra (compulsory), G ro up theory B (compulsoryfor specialists in pure mathematics), and "Life in three dimensions(compulsory for specialists in applied mathematics).

F or third year students: three questions in the subjects Complex analysis( the only compulsory subject for specialists in pure mathematics), "Algebraictopology , and Catastrophe theory .AnalysisII .Define accurately what is meant by the uniform continuity of a functionf: I * R, and formulate the negation of this definition, that is, what is meantby a function not being uniformly continuous.

Prove in detail, verifying that your definition is satisfied, thatf(x) = isuniformly continuous on R, but g(x) = x2 is not. Let / : [a, b] * R bebounded and integrable on [a, b],and let F : [a, b] * R be defined byF{x) = f(t)dt.

Assuming as known all the properties of the integral that you need (whichshould be clearly stated), prove that F is uniformly continuous on [a, b]; if inaddition / is cont inuous, prove that F is differentiable and th at F = f.Linearalgebra.

1. % : V * V is a linear transformation from a vector space V to itself.Define the linear transformations T 2 : V > V and T 3 : V > V.

2. V is a three dimensional vector space over R and % : V * V is a lineartransformation such that %2 & 0, but T 3 = 0. Let e\ be a vector in V sucht ha t T 2e\ & 0. Let e2 = Te\ and e3 = T 2e\. Prove that e\, e2, e^ form abasis of V.

3. Find the matrix of the transformation % : V > V in the basis el s e2, e3.4. Prove that any 3 x3 matrix A over R for which '2 ( 0, but A3 = 0,is similar to the matrix

Grouptheory.Define homomorphism and kernel. Prove that the kernel of a homomorphismis a normal subgroup.State the first homomorphism theorem.Prove that the dihedral group D^ has no surjective homomorphisms on to

th e cyclic group Cn of order if > 2.Produce an injective homomorphism of Cn into Dn.


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Mathematical Trivium II 223

Life in threedimensions.1. Let S = {((2+ c os JC)cos y, (2+ c os x) sin y, sin x)\0


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224 V I Arnold

Sketch the curve y2 = x ^ l x2) and (without calculation) try toun derstand what i ts caustic looks like. (Th e Se rre t F ren et formulae may beused without proof.)

2. D efine th e curve of the centres of m ass of displaced water a nd th emetacentric curve of a (two dimensional) vessel. Show tha t the curve of th ecentres of mass of displaced water of an elliptical vessel is also an ellipse anddescribe its m etac en tric curve. D escribe briefly how the equilibrium of theship varies when its centre of gravity moves along an axis of the ellipse(distinguishing the major and minor axes).

(You need not prove that l inear transformations map the centre of gravityof a plane domain into the centre of gravity of its image.)

University of CopenhagenThere are fewer subjects here (4 for first year stude n ts specializing inmathematics an d ph ysics). T here is a specific set of courses for eac hspeciality. E ach subject is studied by lectures, acco m pan ied by exercises. T h eexaminations are writ ten, as a rule, and con sist of 3 or 4 indepen dentquestions, which m ay include problem s from the course. But these are n otalways exam inatio ns in the classical sense of the word: two or three of th eexaminations for third year students, given below, consist of exercises forwhich the students are given two and a half days and are on their honour towork independently.There follow:for first year studentsone question in analysis and one in linear algebra;

for third year stu den ts on e question in grou p theo ry, one in differentialgeometry, and two in systems of differential equations.Question 1. Let ( = {(, y) e R 2 | x ( 0} andf{x, y) =^ ^ for (x, y) e (.

a) Calculate df/ dx = Dx~f(x, y) an d df /dy = D2~f{x,y) for all (x, y) ,(.b) Find a power series = 0 ant such that

2^antn for all t eR \ {0},n 0

and hence deduce that g : R \ {0} R can be extended to a C func tiong : R > R. F indg(0) an d g'(0).c) P ro ve t h a t / : ( > R can be extended t o a C f u n c t i o n / : R 2 >R .

F i n d / ( 0 , >),D i / ( O , y) an d >2/ (0 , j) for all y e R .sin xyH int. f(x,y) =y ygyxy) for & 0,y ( 0.


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Mathematical TriviumII 225

Question 2. Consider the matrix B(c), c e R ,(c 0 0\

B(c)= 2 c + 1 2 .V V

a) F or each c e R, define a polynomial qc of degree 2 such thatPB(c)(t) = (ct)qc(t) for all c and ( e l , wherep is the characteristicpolynomial of B(c).b) Prove that qc(c) = 0 for c = 1 . Show that for c # 1,B(c) has threeeigenvalues.c) Find the set D = {c, R |5 ( c ) is R diagonalizable}. Prove that the set

4 = {c e R|i?(c) is diagonalizable in an orthogonal basis} is empty.Question3. Let S be a regular surface, U an open subset of R2, and let , U) ,)) be a parametrization of 5, defined in U. Throughout thisproblem it is assumed that the coefficients E, F, G of the first fundamentalform of S, calculated by means of the parametrization (X, U), have thefollowing properties: 5 depends only on u, G only on),and F is identicallyzero.

a) Let w(t) be a vector field along a parametrized curve X(u(t), v(t)).Prove that the angle between Xu(u(t), o(i)) and w(t) does not depend on t.b) Show by means of a) (or independently) that the Gaussian curvature.is identically zero.c) Prove that X(u) is locally isometric to R2, using a parametrization of R 2of the form (u, o) * (


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I I1) The plane is referred to an orthonormal basis (Ox, Oy). A function & ofclass C1 is given on an interval / of the axis R. For each t in /, let M tdenote the point with coordinates (&(), t2+t), let P, be the point (t2, 0), and%& the curve described by the point M, as t varies in /.Show that a necessary and sufficient condition for the tangent to %& at M,to pass through the point P, for each t except t = 1/2 is that the function

& satisfies a certain differential equation , and find this equation .2) Let (2s) be the differential equation

I ( I + l)y (2i + l)y + x2{2x + 1) = 0 .a) For which points of the plane may it be stated a priori that locally one

and only one integral curve passes through each of them? For which pointsof the plane may it be stated a priori that no integral curve passes througheach of them?

b) Determine the set of points where the tangents to all the integral curvesof equation (E) are horizontal.

c) Find the solutions of the differential equation (E) on the intervals o o , 1 an d ]0,+ oo[.

d) Can these solutions be extended to 1? To 0? Is there a solution on R?3) We consider the curves ,, defined in 1) for the functions & that satisfyth e equation (E) on R \ { 1}, extended by continuity to 1. A sketch of the

curves is attached.a) The sketch suggests that the curves %& have two common points. Provethis and determine the tangents to the curves ;& at these points.b) The sketch suggests that each curve ;& has a cusp Rv and that thepoints .& lie on a straight line. Prove this and find the equation of this line.

Examination in the University of Paris VII (third year)Parts I, II and III are fairly independent.

ILet g be a bounded Borel function on the interval [0, 1].1) Show that for any t e [0, T] andfixed

2) Assume henceforth that there exists a constant


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228 V.I. Arnoldfor each integer n. Show that for each t e [0, T] the left hand side ofequation 1) tends to 0 as 2 tends to infinity. Deduce that for all f < 7*

/ g(s)ds = 0.JT t

3) Show that g is zero almost everywhere. If g is continuous, then it isidentically zero.

4) Deduce from the foregoing that i f/ i s a continuous function on [1,)]such that s < < oonf x)dxl\for all integers n, then / is identically zero.

5) I f/ is a continuous function on [0, T] such that ^t f{t)dt = 0 forall n, then / is identically zero. (This can be deduced from 4), but can also beproved directly using the Stone Weierstrass theorem and results from thepresent course.)I I

L e t / b e a locally boun ded Borel function that vanishes on ] oo, 0]. F orfixed% > 0 put

/ = [Te

ns(f*f)(s)ds.Jo

1) By making the change of variables w = %), s = 2To w, show that/ = / dv [ en{2Tvw)f{T v)f{T w)dw.Jo JT V

2)Henceforth we assume that / * / is identically zero. Show that(f e nvf(T v)dv)2= if e n{v+w)f(T v)f(T xo)dvdw,Jo JJA

where is the region {(), w)\0 =3), w


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Mathematical TriviumII 2292) Deduce from this that for any$ and t

I f(t u)xing(u)du = O.

3) Show finally that / and g are identically zero.Below there follow problems from the examinations on differential

equations in 1992 in the Mechanics and Mathematics Faculty of MoscowState University.

Examination in linear theory N.Kh. Rozov, January 1992; 3 hours)1)Consider thelinear equat ion

(1) ( i + ) jT

with the additional conditions(2) y(a) = l, y'(a) = A, y (a)=a;

here a, a, A are real parameters.a) (2 marks) Find all the values of the parameters a, a, A for which the

existence and uniqueness theorem guarantees that there is a unique solution ofthe Cauchy problem for the third order differential equat ion (1) with initialconditions (2).

b) (1 mark) What is the domain of definition of the non extendablesolution of the initial problem (l) (2) in the case a. = \ ,a = 2,A=V.

c) (1 mark) Represent the initial problem (l) (2) in the form of a Cauchyproblem for a normal linear system of differential equations.

2. Let (3) denote the linear homogeneous equation corresponding to thenon homogeneous equation (1) with # = 1. For this equation (3) considerthe three Cauchy problems with the respective initial conditions

(4) 3/(1) = 2, j/ '(l) = 0, i"( l) = 0;(5) 3,(1) = 1 , y'(l)=l j / " ( l ) = l ;(6) 3/(1) = 0. / '( ) = 0, j/ "(l) = 2 ;

let q>\(t),q>2(t), q>3(t), t e / , be non extendable solutions of these problems.a) (1 mark) F ind the interval / explicitly and prove that {&0(),&2(),>is a fundamental system of solutions of equation (3) on / .b) (1 mark) Obtain an expression for the Wronskian determinant of the

solutions 4'()> 6&= valid on the interval / .c) (2 marks) Write down the solution of the Cauchy problem for equation(3) with initial conditions

in terms of the functions>\ (), >0{),


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230 V.I.Arnold

3. (3m arks) D oesthe set of all real solutions of theequationy + jj = aco s i, a = const> 0,

contain a periodic function?4. (5m arks) C alculate the fun dam ental m atrix e'A for the linearhomogeneous system

with constant matr ix( 2 1 2 \

.4 = 1 2 2 \ 3 3 5 /

5. (4m arks) LetK(t,5) be theC auchy m atrix (fundam ental m atrixnormalized at thepo in t5) for thelinear ho m ogeneo us systemx =A(t)x, i 6 S ,

where them atr ix A(t) is con t inuous on R. Express thederivative dK(t,5)/ 95in terms of them atrices A andK.Examination innon linear d ifferential equations N.Kh.Rozov,June1992

1. (3m arks) It is known tha t th efun ction u(t) e C ([0, oo])satisfiesthedouble inequality0


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a) (1 mark) Asymptotically stable?b) (2 marks) Stable in the sense of Lyapunov, but not asymptotically

stable?c) (1 mark) Unstable?4. Consider the phase portrait of the equation

6 + 26x 0on the phase plane (x, y = x).

a) (1 mark) Determine the type of the phase portrait of this equation foreach (real)value of the parameter?.

b) (2 marks) Sketch the phase portraits of thegiven equation for


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232 V.I. ArnoVd

4. F ind the derivative with respect to the param eter at = 0 of thesolution of the problem

5. a) State the theorem on the existence of a solution of the Cauchyproblem for a quasilinear partial differential equation,

b) Solve the Cauchy problemdz dz . 2xy 1xz = yz, curveL : ' = 1,+1 +y .dx dy

References[1] V.I.Amol'd, Amathematical trivium, UspekhiM at.Nauk46:1(1991), 225 232.= Russian M ath . Surveys 46:1 (1991), 271 278.[2] M.Artigue, Sujets d'examens d'ici et d'ailleurs, GazettedesMathematiciens1992,

N o. 53,77 83.Translated by C.J.Shaddock

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Arnold Trivium 2 1993