(in Polish) Analiza matematyczna II 2400-L1PPAM2
The course includes a supplement to the knowledge acquired during the first course in mathematical analysis, as well as – in its main part – the study of the theory of differentiability of multivariable functions, the associated optimization methods, the theory of multidimensional manifolds, and multiple integrals, including a range of applications in an economic context.
In the initial part of the course, content related to the differential calculus of single-variable functions will be expanded. This includes higher-order derivatives, the concept of Taylor polynomials, the Taylor formula with remainder in the Peano and Lagrange forms, along with practical applications for expanding functions into power series. A general theorem characterizing local extrema and inflection points of n-times differentiable functions will be presented, along with examples of analyzing function behavior, particularly the characterization of convexity/concavity of twice-differentiable functions. Within convexity theory, the course will cover theorems on local Lipschitz continuity, monotonicity of difference quotients, existence of one-sided derivatives, and Jensen’s inequality, illustrated with economically interpreted examples – e.g., analyzing utility functions and risk aversion.
The main and longest part of the course is devoted to developing the theory of differentiability of multivariable functions with values in Euclidean space. As preparation, key facts about the structure of Euclidean spaces – and more generally, normed spaces – will be reviewed, including: properties of norms, inner products, convergence of sequences in Euclidean spaces, and limits of functions defined on subsets of such spaces.
In the differential calculus of multivariable functions, the course will first introduce the concepts of directional derivatives, partial derivatives, gradient, and the Fréchet differential, all illustrated with examples and geometric interpretations – including the gradient theorem, which identifies the direction of steepest ascent of a function. Key differentiability results will include: the increment theorem, necessary conditions for differentiability, existence and continuity of partial derivatives as a sufficient condition, the chain rule and inverse function theorem, and the Jacobian and Hessian matrices, with numerous examples – including polar and spherical coordinate transformations. Higher-order differentials and the multivariable Taylor formula will also be introduced.
In the next part, Fermat’s theorem will be discussed, along with critical points and methods for finding global extrema of functions defined first on compact subsets, and then on more general subsets of Euclidean space. The Sylvester criterion for determining the definiteness of the quadratic form defined by the Hessian matrix will be presented, along with its use in identifying local extrema and saddle points of multivariable functions.
Next, the course will cover: the concept of diffeomorphism, the local diffeomorphism theorem, manifolds, the determination of tangent and normal spaces, and the implicit function theorem with application examples to implicitly defined functions. This will be followed by Lagrange’s theorem on constrained extrema; the Lagrange multipliers method will be illustrated with many practical examples. Additionally, the Kuhn-Tucker theorem will be introduced (for informational purposes) and demonstrated with economic examples involving inequality constraints in optimization problems.
The next part of the course will discuss the geometric interpretations and applications of the Riemann integral, including its use in calculating curve lengths, surface areas, and volumes of solids of revolution. The final topic will cover double and triple integrals over normal regions, methods for evaluating them, including the change of variables theorem.
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Estimated student workload: 6 ECTS x 25h = 150 hours
• Contact hours (K), Independent work (S):
o Seminar (classes): 60h (K), 0h (S)
o Consultations: 14.5h (K), 0h (S)
o Midterms: 3h (K), 0h (S)
o Final exam: 2.5h (K), 0h (S)
o Preparation for midterms: 0h (K), 8h (S)
o Preparation for exam: 0h (K), 12h (S)
o Preparation for seminars: 0h (K), 30h (S)
o Work with additional materials on the Moodle platform: 0h (K), 20h (S)
Type of course
Course coordinators
Learning outcomes
Learning outcomes (codes): K_W04, K_U03, K_K01
After completing the course, the student:
KNOWLEDGE
• Knows the Taylor formula for functions of one and several variables.
• Understands the concept of convex/concave functions and their key properties.
• Is familiar with the main properties of normed spaces, the concept of the scalar product, and the notions of continuity and limit for functions defined on subsets of Euclidean space.
• Knows the definitions and interrelations of directional derivative, gradient, Jacobian matrix, Hessian matrix, and the Fréchet differential for multivariable functions, as well as the necessary and sufficient conditions for differentiability.
• Knows the conditions characterizing local extrema and inflection points of single-variable functions, as well as local extrema and saddle points of multivariable functions, including Sylvester’s criterion.
• Knows theorems concerning the differentiation of composite and inverse mappings; understands the concept of diffeomorphism and the local invertibility theorem.
• Understands the concept of manifolds, tangent and normal spaces to a manifold, and methods of determining them.
• Is familiar with the implicit function theorem.
• Knows the Lagrange multipliers method for determining constrained extrema.
• Understands the Riemann integral and its interpretations and geometric applications; knows methods for evaluating multiple integrals over normal regions in Euclidean space, including the change of variables theorem.
SKILLS
• Can analyze the behavior of single-variable functions, including determining convexity/concavity and providing Taylor series expansions.
• Can apply Jensen’s inequality to derive estimates and inequalities; identifies convex/concave functions in optimization problems.
• Can compute limits of sequences in Euclidean space, scalar products, vector norms, and determine the existence and values of limits of multivariable functions defined on subsets of Euclidean space.
• Can calculate directional and partial derivatives; understands the geometric interpretation of the gradient of scalar functions.
• Can determine global extrema of multivariable functions using Fermat’s theorem.
• Can test for differentiability of multivariable functions, compute Jacobian and Hessian matrices, and apply the chain rule, the differentiation theorem for composite functions, and the inverse function theorem.
• Can apply Sylvester’s criterion to classify critical points of multivariable functions.
• Can determine tangent and normal spaces to manifolds embedded in Euclidean space.
• Can apply the implicit function theorem to examine properties of implicitly defined functions.
• Can model optimization problems by reducing them to problems of constrained extrema and solve them using the Lagrange multipliers method.
• Can compute multiple integrals over normal regions and apply the change of variables theorem, particularly using polar and spherical substitutions.
COMPETENCES
• Demonstrates independence in applying theoretical knowledge from mathematical analysis to solve optimization problems
• Shows responsibility and self-control through the experience of learning in a setting that involves choice in how knowledge is acquired
• Is systematic due to the need to adapt to a course completion structure where the final result includes scores from quizzes, tests, class participation, and the final exam
• Acts with integrity and honesty due to the strict adherence required for completion and examination requirements
Assessment criteria
To pass the course, the student must:
1. Attend the seminar sessions (a maximum of two absences is allowed).
2. The final grade is based on the total number of points earned during the semester (max. 80 points). Points can be earned in the tutorial sessions (max. 20 pts), in two common mid-term tests during the semester (max. 15 + 15 = 30 pts), and on the final exam after the semester ends (max. 30 pts).
To receive a passing final grade, a student must obtain at least 40 points.
Default grade thresholds:
o Satisfactory (3.0): 40 pts
o Satisfactory plus (3.5): 47 pts
o Good (4.0): 55 pts
o Good plus (4.5): 62 pts
o Very good (5.0): 70 pts
o Excellent (5.5): 75 pts
3. Points for the tutorials can be earned through quizzes (6 × 2 pts) conducted in each group and for active participation (max. 8 pts).
4. To take the final exam, the student must pass the tutorials, i.e., earn at least a satisfactory grade (20 pts).
5. The exam is held in two sessions: the main exam session and the retake session. The exam, conducted in-person, is written and consists of solving a number of problems covering the full course material.
6. In the case of a justified absence from a test (i.e., a quiz, mid-term, or final exam) due to important circumstances, the student has the right to take the test at a make-up date.
To do this, the student must submit a request for justification:
o For a quiz or mid-term – to the tutorial instructor
o For the final exam – to the Dean of the Faculty of Economic Sciences (WNE)
If no such request is submitted, the student loses the right to a make-up test.
According to the Detailed Study Rules at the Faculty of Economics, the justification request with proper documentation must be submitted no later than 7 days after the date of the exam or 7 days after the reason for the absence has ceased.
Make-up quizzes and mid-terms are arranged and administered by the tutorial instructors. The make-up exam takes place during the next exam session or is scheduled by the Dean of WNE.
7. All of the above rules also apply to upper-year students with conditional pass status.
8. For first-year MSEM students who want to transfer to the Faculty of Economics during the semester, points and grades earned from quizzes and mid-terms at the MIM Faculty cannot be transferred and counted as part of the tutorial and mid-term assessments at the Faculty of Economics.
Bibliography
Basic literature:
M. Krych Analiza matematyczna dla ekonomistów, Wydawnictwa Uniwersytetu Warszawskiego, 2009
Supplementary literature:
R. Antoniewicz, A. Misztal, Matematyka dla studentów ekonomii. Wykłady z ćwiczeniami, WN PWN, Warszawa 2009.
J. Banaś, S. Wędrychowicz, Zbiór zadań z analizy matematycznej, WNT, Warszawa 2006.
T. Bażańska, I. Karwacka, M. Nykowska, Zadania z matematyki, podręcznik dla studiów ekonomicznych, PWN, Warszawa 1980.
Alpha C. Chiang, Podstawy ekonomii matematycznej, Państwowe Wydawnictwo Ekonomiczne, Warszawa 1994.
W. Dubnicki, J. Kłopotowski, T. Szapiro, Analiza Matematyczna. Podręcznik dla ekonomistów, WN PWN, Warszawa 2010.
W. J. Kaczor, M. T. Nowak, Zadania za analizy matematycznej, część 1, 2 i 3, WN PWN, Warszawa 2005.
W. Kołodziej, Analiza matematyczna, WN PWN, Warszawa 2009.
W. Krysicki, L. Włodarski, Analiza matematyczna w zadaniach, część I i II, WN PWN, Warszawa 2008.
K. Kuratowski, Rachunek różniczkowy i całkowy. Funkcje jednej zmiennej, WN PWN, Warszawa 2008.
W. Rudin, Podstawy analizy matematycznej, WN PWN, Warszawa 2009.A. Ostoja-Ostaszewski, Matematyka w ekonomii. Modele i metody, t. 1 i 2, PWN, Warszawa 1996.
Additional information
Additional information (registration calendar, class conductors, localization and schedules of classes), might be available in the USOSweb system: