math 1 ipsp 2022/23

the course is first-year, first semester mathematics for physicists

lectures (in-person) : Mondays and Thursdays 9:15 THS Linnestrasse 5

in-person tutorials: Mondays 11:15. there is a possibility of a second tutorial group at Mondays at 15:15. in case it is needed, please contact David Hruska via moodle

online tutorials, Mondays at 13:15. details in moodle (‘math1 ipsp’)

course materials:

see moodle (the course is called ‘math1 ipsp’). most of the lectures are streamed and recorded via bbb:

https://conf.fmi.uni-leipzig.de/b/jan-qm5-snl-kbs

books/scripts, which fragments can be useful as further reads (I am not following a single source, and your main source should be lecture): in order of difficulty, from the most advanced:

W. Rudin, Principles of mathematical analysis

T. Tao Analysis I

A. Schuler Calculus

S. Lang Undergraduate Analysis

rules: 50% of points from homeworks admits to the exam

course overview:

  1. Basics: logic and set theory
    • proposition, Boolean algebra, implication, structure of proof, proof by contradiction
    • sets, operation on sets, connection with propositions, complement, inclusion, Cartesian product, planar sets, refreshing basic planar functions
    • quantifiers, order of quantifiers
  2. Numbers
    • natural numbers, Peano axioms, induction, rational numbers
    • gaps in rationals, order, field, mentioning construction of reals
    • sup and inf, Archimedean property, density of rationals, countability vs incoutability, extended reals
    • complex numbers as pairs, i representation, modulus, polar representation, rational roots of complex numbers,
    • inequalities (most for reals): Young, Cauchy-Schwarz, bertween means, (discrete, finite) Holder and Minkowski
  3. Series and sequences
    • convergent/divergent sequences, 1/n, convergent implies bounded, arithmetics of limits, squeeze test
    • monotonic sequences, examples (including e)
    • limsup, liminf, subsequences, Bolzano-Weierstrass
    • Cauchy sequences, completeness
    • metric, convergence beyond real sequences (finite dimension)
    • practical limits, ways of computing them