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:
- 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
- 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 incountability, extended reals
- complex numbers as pairs, i representation, modulus, polar representation, rational roots of complex numbers,
- inequalities (most for reals): Young, Cauchy-Schwarz, between means, (discrete, finite) Holder and Minkowski
- 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
- series, reformulation, reformulation of Cauchy statement for partial sums (‘tails’), boundedness in case of nonnegative summands, necessary vanishing of summands,
- geometric series, 1/n, Cauchy test, 1/n^p
- absolute convergence vs alternating series, Leibniz criterion
- ratio and root tests
- power series, radius of convergence, motivation for Cauchy product, theorem on multiplication (one needs converge absolutely)
- e revisited, irrationality
- rearangements, Riemanns theorem mentioned
- Functions (primarily on intervals), 1d continuity, differentiation:
- function, domain, image and preimage of a set, image of preimage vs set itself etc
- surjection, injection, bijection, inverse function
- limit of a real function (after some initial general remarks, we restrict ourselves to functions on intervals), Heine and Cauchy definitions, their equivalence, examples of limits
- basic properties of limits: uniqueness, algebraic properties, limit of a composition
- improper limits, one-sided limits
- continuity, examples of elementary continuous functions, including definition of a^x via supremum, left/right continuity
- absolute continuity,
- attainment of max/min, intermediate value theorem (‘Darboux’) and their applications: existence of roots of some equations (odd polynomials; g(x)-x for g:[0,1] to [0,1], i.e. ‘baby fixed-point’); proof of existence of continuous inverse for strictly monotone functions; proof that image of interval is an interval
- derivative, definition, velocity/tangent line interpretation
- basic properties of derivatives: algebraic, chain rule
- derivative of inverse to strictly monotone, differentiable function
- derivatives of elementary functions
- mean value theorems: Rolle, Lagrange, Cauchy
- their applications: monotonicity vs derivative, find all f such that f’=f, finding extrema, proving inequalities via bounds on derivative
- smoothness of functions: continuity, uniform continuity, Holder cont, Lipschitz cont, differentiability, C^1 etc; higher-order derivatives
- Taylor expansion with Peano and Lagrange remainder, its applications to extrema; analytic functions
- L’Hospital lemma
Exam admission: 50% of HWs (non-negotiable). The last HW9 consists entirely of ‘extra points’. Admissions should be known around 31.01.
Exam on 22.02.2023: it will consists only of the ‘analysis’ part (as outlined above, the last two weeks of the semester lectures are linear algebra, which will be continued next semester). Exam will last 120 mins, and will be structured similarly to two ‘mock exam assignments’ provided via moodle. You are allowed two pages of own notes and writing utensils, nothing more.