Advanced Differentiation and Integration 1 - NMMA437
Title: Derivace a integrál pro pokročilé 1
Guaranteed by: Department of Mathematical Analysis (32-KMA)
Faculty: Faculty of Mathematics and Physics
Actual: from 2023
Semester: winter
E-Credits: 4
Hours per week, examination: winter s.:2/0, Ex [HT]
Capacity: unlimited
Min. number of students: unlimited
4EU+: no
Virtual mobility / capacity: no
State of the course: taught
Language: Czech, English
Teaching methods: full-time
Teaching methods: full-time
Guarantor: RNDr. Daniel Cameron Campbell, Ph.D.
Class: M Mgr. MA
M Mgr. MA > Povinně volitelné
Classification: Mathematics > Real and Complex Analysis
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Annotation -
Last update: T_KMA (02.05.2013)
Real-analytic properties of Sobolev functions. Change of variables in integral for Lipschitz transformations – area and coarea formula. Differentiation of convex functions. Recommended for master students of mathematical analysis.
Course completion requirements -
Last update: prof. RNDr. Jan Malý, DrSc. (29.10.2019)

The exam is oral. The required knowledge corresponds to the sylabus at the presented extent

Literature -
Last update: prof. RNDr. Jan Malý, DrSc. (29.10.2019)

L. C. Evans, R. F. Gariepy: Measure theory and fine properties of functions. CRC Press, Boca Raton, FL, 1992

W. P. Ziemer: Weakly differentiable functions. Sobolev spaces and functions of bounded variation. Springer-Verlag, New York, 1989

Requirements to the exam -
Last update: prof. RNDr. Jan Malý, DrSc. (29.10.2019)

The exam is oral. The required knowledge corresponds to the sylabus at the presented extent

Syllabus -
Last update: doc. Mgr. Petr Kaplický, Ph.D. (09.06.2015)

1. Real-analytic properties of Sobolev functions

Riesz potential estimate

Beppo Levi's characterization

Embedding theorems

Approximate differentiability, a.e. differentiability

Examples concerning discontinuity and non-differentiability

2. Change of variables in integral for Lipschitz transforms

Area formula

Coarea formula

Sard type theorems

Luzin's (N) condition

3. Differentiation of convex functions

Lipschitz estimates

Zajíček's theorem (informatively)

Alexandrov's theorem
Entry requirements -
Last update: prof. RNDr. Jan Malý, DrSc. (02.05.2018)

Measures, Radon-Nikodym theorem, Lebesgue integral, Radon measures, convolution, smoothing by convolution, strong, weak and weak* convergence in Banach spaces, elements of theory of distributions, Lipschitz functions and mappings, Hahn-Banach theorem, Hausdorff measure, L^p spaces and spaces of continuous functions