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EFFICIENT METHODS OF MATHEMATICAL ANALYSIS Gennadi Malaschonok
(Tambov State University, 392622 Tambov, Russia e-mail: malaschonok(a^math-iu.tambov.su)
The special course «Efficient methods of mathematical analysis» for fourth year students of mathematical department was introduced at Tambov State University. It is a four month course with 2 hours a week. The aim of this course is to teach students to use the Mathematica 3.0 for solving problems in mathematical analysis.
The course was constructed taking into account the fact that the students know the mathematical aspect of the problem and can not use the computer system. The basic support for teaching was the reasonable character of ”Mathematica” language, its closeness to the natural mathematical language, easiness of self-education by means of’’help”- promptings.
The lectures were divided into three parts:
1. Introduction - the review of Computer Algebra systems and their possibilities.
2. Main part - study of Mathematica, solving of simple exercises, study of using of the ’’help”-promptings.
3. Special part - solving of difficult problems of mathematical analysis.
Lectures made up the third part of the course, the original students’ work and consultations of the lecturer - the rest. As the result every student consequently, had to solve a series of problems from some part of mathematical analysis and pass a test in this work.
The main themes of mathematical analysis that were developed for teaching by means of Mathematica were:
1. Introduction to analysis: functions of single real variable, limits.
2. Differential calculus of functions of single real variable.
3. Integral calculus of functions of single real variable.
4. Series.
5. Differential calculus of functions of several real variables.
6. Integral calculus of functions of several real variables.
7. Ordinary differential equations.
There were found some problems, not easy for the system. They needed special methods, that had to be devised (for example, the calculation of improper integrals, the simplification of solutions towards the needed variant, etc.).
The aim of such work is evident, because the great amount of modelling problems are reduced to such analytic elements.
It turned out that the best guidance for teaching is the book of S.Wolfram ’’Mathematica”. It is characterized by suitable organization of the material (from easy to difficult), its completeness (the most complete), the easiness and clearness of examples in each chapter.
In our presentation we will show some problems of mathematical analysis, solved by means of’’Mathematica”.
Example 1
Problem
Find minimum of the function u = ax2 + by2 + cz2, where a, b, c - are positive constants, and x, y, z satisfy the condition x + y + z = 1.
Solution
Clear[x, y, z, a, b, c, X, f, A]; u = axA2+byA2 + czA2;
Compose the Lagrange's function:
$ = a x2 + by2+cz2+A(x + y + z- l);
»x = D[«, x] ; 2y = D[«, y] ; *z = D[*, z] ;
Solve the Lagrange system of equations:
Solve [ {Sx == 0, *y == 0, *2 == 0, x + y + z - 1 == 0}, {x, y, z, A}]
ff„ ) be______ ( a c ______ab______ 2abc ,
ab + ac + bc ab + ac + bcf ab + ac + bc( ab + ac + bc ^
Compose the matrix:
M= {{3x,x*f <3X#Z *},
{3y,**,
{dZ/X5, 3Z(y 8, 3ZfZ5}};
M / / MatrixForm
' 2 a 0 0
0 2 b 0 ,0 0 2 c
Al = Det[M]
8 a b c
m2 = (2*a °
\ 0 2 * b /
A2 = Det[m2]
4 a b
dX(XS
2 a
As a > 0, b > 0, c > 0, then dx,x$ > 0, Al > 0, 2 > 0.
Therefore (xO, yO, zO) is the point of conditional minimum, where
be a c a b
xO = ---------------; y0 = --------*-----; zO =
bc+ac+ab bc+ac+ab bc+ac+ab
Find the volume of the function u at the point (x0,y0,z0):
uO = u /. x-»xO/. y-»yO/. z -♦ zO // Simplify
a b c b c + a (b + c)
abc
Result: —----—-----
bc + a(b + c)
Example 2
Problem
taken from the point of its intersection with the plane z = 0 to the point of its intersection with the plane z = a. Solution
Construct the spiral at the segment [0,2Pi], if a=l:
3 t
where L - is the arch of the spiral x = R Cos t, y = R Sin t, z = ——
ParametricPlot3D[(Cos[t] , Sin[t], ------}, {t, 0, 2 Pi}
11 2 Pi J
1
0.11 0.! 0.2
1
a t
t = . ; x : = R Cos [t] ; y : = R Sin [t] ; z : = --------------------------
2 7T
(y z dt x + z VR2 - y2 dt y + x y dt z) dlt // Simplify
'o
Result: — a^R2
Example 2
Problem
Find the coordinates of the center of gravity of the spiral x=a*cos t, y= a*sin t, z=b*t, if the density is constant.
Solution
Construct the plot of the function x=a cost, y= a sint, z=b t
Clear[х, у, z];
Parametric PIot3D [(Cos [t], sin[t], t}, {t, 0, 2* Pi}]
Denote the coordinates of the center of gravity by (#,t7,£)
fxdl Jo
s PV dl s Г z dl;
Jo Jo
— / Л = ------------------/ 5 =
МММ
Denote the mass of the part of the spiral by M. It equals to J p (x, y) ds, where p (x, y) is the density of the line, P (x, y) = const = 1, L is the the way of integration.
x : = a Cos[t] у := a Sin[t] z : = b t p : = 1
s' = Simplify [V1 + (at x) л2 + (aty) л 2 + (<9t z) л2 ] ;
м = Г /
Jo
p s' dlt;
f7rx*s,dlt pyts'dlt f7Tz*s'dlt f Jo Jo J Jo i
I 3 ' 3 ' w )
„ , „ 2a Ья
Result: f = 0, tj = —, ( = —
7Г 2 .
M
M
