Mathematics BSc
Course description
2013.
Course description
2013.
Analysis 3
Hours lect+pc 
Credits lect+pc 
Assessment  Specialization  Course code lect/pc 
Semester  Status 

4 + 3  4 + 4  exam + term grade 
pure math.  mm1c1an3m mm1c2an3m 
3  compulsory 
Course coordinator
Strong  Weak  Prerequisites  

Practice class  
Strong:
 
Weak:
Algebra2L
(mm1c1al2)
 
Lecture  
Weak:
practice class

Prerequisites
Analysis 12 (or Calculus) and linear algebra.
Literature
 Walter Rudin: Principles of Mathematical Analysis. 3rd ed., McGrawHill, 1976.
Syllabus
 Sequences and series of functions: uniform convergence, continuity, differentiability and integrability of the limit and the sum, AbelDirichlet 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, meanvalue 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.
 RiemannStieltjes integral: functions of bounded variation, calculation of the total variation, RiemannStieltjes integral, integration by parts, sufficient conditions of integrability.