Mathematics BSc
Course description
2013. Analysis 3
Hours
lect+pc
Credits
lect+pc
Assessment Specialization Course code
lect/pc
Semester Status
4 + 3 4 + 4 exam +
pure math. mm1c1an3m
mm1c2an3m
3 compulsory
Strong Weak Prerequisites
Practice class
Strong:
Analysis2L (mm1c1an2) or
Foundations of AnalysisL (mm1c1ap2)
Weak:
Algebra2L (mm1c1al2)
Lecture
Weak:
practice class
Prerequisites
Analysis 1-2 (or Calculus) and linear algebra.
Literature
• Walter Rudin: Principles of Mathematical Analysis. 3rd ed., McGraw-Hill, 1976.
Syllabus
• Sequences and series of functions: uniform convergence, continuity, differentiability and integrability of the limit and the sum, Abel-Dirichlet test, power series, Abel's theorem, Taylor series, examples.
• Basic topology in finite dimensional Euclidean spaces: basic notions of topology in Euclidean and metric spaces (interior, exterior, boundary point, open, closed, compact sets), limit and continuity of multivariable functions, sequential continuity, continuous functions defined on compact sets.
• Differentiation of multivariable functions: differentiability of coordinate functions, partial and directional derivatives, gradient, tangent plane, mean-value inequality, extreme values, n times differentiable functions, Taylor polynomials, symmetry of the mixed derivatives, local extrema, differentiability of the inverse, inverse function theorem, implicit function theorem, Lagrange multipliers.
• Jordan measure and multiple integral: Jordan inner and outer measure, measure of the boundary, Jordan measurable sets, measurability of convex polyhedron and normal domains, translation and rotation invariance, definition of multiple integral, properties, equivalent conditions of integrability, integrability of continuous functions, Jordan measure and the integral of the indicator function, Fubini's theorem, Cavalieri's principle, Volume of normal domains, volume of the n dimensional ball, change of variables in multiple integrals, polar coordinates.
• Parametric integrals: continuity, differentiability and integrability of parametric integrals, improper parametric integrals, uniform convergence, sufficient conditions of uniform convergence.
• Riemann-Stieltjes integral: functions of bounded variation, calculation of the total variation, Riemann-Stieltjes integral, integration by parts, sufficient conditions of integrability.