Babes-Bolyai University of Cluj-Napoca
Faculty of Mathematics and Computer Science
Study Cycle: Master
SUBJECT
| MMA1017 | Topics of Mathematical Analysis II (for teachers education) |
| Teaching Staff in Charge |
Prof. MURESAN Marian, Ph.D., mmarian math.ubbcluj.ro |
| Aims |
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A deeper study and the completion of the knowlege of the differential calculus for functions of one and more variables with an opening for certain elements of differential calculus for nonlinear mappings defined between linear normed spaces. |
| Content |
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I)The definition of the Fréchet diffrential for a nonlinear mapping between linear normed spaces. Different properties. The Dini derivative numbers. Continuous functions that not admit derivative in any poit. Generalizations of the classical mean-value theorems of the differential calculus. The mean-value theorem of Danjoy-Bourbaki. Properties of the intermediate point of different mean-value theorems. Applications to the proof of the different inequalities and to the study of the extreme (optimum) points of the functions. Derivatives and differentials of higher order for a real function and for a nonlinear mapping. The proof of some formulas for the n-th derivative for different functions. The formula of Taylor for nonlinear mappings and some applications. The calculus of some limits of sequences and of functions using Taylor’s formula and developments in Taylor series.
II) The set of functions that admit primitives and the relation of this set to the set of continuous functions and the set of the functions with the property of Darboux. Generalized primitives. III) Approximation methods for the solution of some equations in the set of real numbers. The method of inverse interpolation. The separation of the roots of a polynomial equation and the calculus of the approximats of this roots to error evaluations. Newton’s method for equations in the set of real numbers. The existence, the uniqueness and the approximation of the solution. Concrete examples of approximation of a root of some equation ussing Newton’s method. Convergence theorems. The notion of convergence order and the determination of this order in the case of Newton’s method, of the method of chord and of the method of Steffensen. Estimates of the convergence order of the method obtained by inverse interpolation. IV) Extrema and conditioned extrema for functions and functionals. Concrete examples. Different applications of the method of Lagrange multipliers. V) The definition of the exponential function with the help of limits of sequences. Functions defined with the help of the exponential function (the logarithmic function, the power function and their composed functions. The hyperbolic functions, their properties and applications. The exponential function of a complex variable. The trigonometric functions and their inverses. Asymptotic formulas. The coonnection between the trigonometric functions and the hyperbolic functions. Remarkable relations. The definitin of the number π and properties of the trigonometric functions in connection with π. Remarkable functional equations and solutions of these equations. The connection with the studied elementary functions. |
| References |
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[1] MARUSCIAC, I., Analiză matematică, vol. I-II, UBB Cluj-Napoca,1973.
[2] COLOJOARĂ, I., Analiză matematică, Editura didactică şi pedagogică, Bucureşti, 1984. [3] NICOLESCU, M., DINCULEANU, N,. MARCUS, S., Manual de analiză matematică, vol. I-II, Editura didactică şi pedagogică , Bucureşti, 1964. [4] BOBOC, N., Analiză matematică, Univ., Bucureşti, 1988. [5] MEGAN, M., Analiză matematică, vol. 1-2, Mirton, Timişoara, 1999. [6] MURESAN, M., A Concrete Approach to Mathematical Analysis, Springer, New York, 2009. [7] PĂVĂLOIU, I., POP, N., Interpolare şi aplicaţii, Risoprint, Cluj-Napoca, 2005. |
| Assessment |
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Examination in writing and oral in session. |
| Links: | Syllabus for all subjects Romanian version for this subject Rtf format for this subject |