### Mathematical Analysis II NMAI055

**Exam and consultations in September:** Exam is scheduled in SIS on September 14 (most likely the last exam). I will be available for consultations on September 4, 5 and 13, please write me an e-mail if you want to come, proposing times when you can come.

**Time and place:** Tuesday at 12:20 in S11

**Tutorials (by Debarati Das/ Hans Raj Tiwary):** Tuesday at 14:00 in S11.

You must obtain a "pass" in the tutorial to be able to take the exam for this course, that will be given for scoring at least 60% in the midterm test or scoring at least 40% and submitting homework proportional to the missing percentage in the test. There will be opportunity to retake the test once. The midterm test on integrals will be during the tutorial on April 25. Topics: primitive functions - per partes, substitutions, partial fractions, definite integrals - substitution, improper integrals, applications: area bounded by curves, volume and surface of a rotating body, length of a curve, integral criterion for convergence. Practice test. All written materials will be allowed, no electronics.

Practice problems from tutorials.
**Resit of the midterm test is on May 31 from 9:00, in S1.**

**Exam:** You must register in SIS before taking an exam: two dates are available: 14.6. from 13:30 and 28.6. from 9:00. There will be one mor date in September. Please do not register for other exams in SIS, they are intended only for Czech students and English version of tasks will not be available. Exam description and a list of exam requirements.Practice exam.

For recommended study materials see the syllabus, you might also find useful notes by Aleš Pultr. Scan of my notes. Disclaimer: The notes prepared to be used only by me. There might be parts missing/too brief/not legible and some mistakes (that I hopefully noticed when lecturing and fixed). There are also some parts that I skipped.

**Covered material:**

- 21.2..: Antiderivatives: definition, antiderivatives of elementary functions. Integration per partes.
- 28.2.: Substitution. Partial fractions.
- 7.3.: Riemann integral.
- 14.3.: Upper and lower Riemann integral. Criterion of integrability. Monotonicity implies integrability.
- 21.3.: A function is uniformly continuous on a compact interval iff it is continuous. Continuous function is integrable. The 1st and The 2nd fundamental theorem of calculus (statements). Newton integral.
- 28.3.: Fundamental theorems of calculus - proofs. Relation of Newton and Riemann integral. Per partes and substitution for definite integral. Geometric applications of integral.
- 4.4.: Double lecture: Further applications of definite integrals (integral criterion for convergence of series). Functions of multiple variables. Multidimensional Riemann integral, Fubini theorem. Reading homework: a proof of Fubini Theorem - pages 174-176 in notes by Aleš Pultr
- 11.4.: Double tutorial.
- 18.4.: Partial and directional derivatives. Total differential.
- 25.4.: Continuous partial derivatives imply differentiability. Multivariate Lagrange theorem. Zero differential implies constantness. Arithmethic of partial derivatives and differentials. Relevant parts of lecturenotes of A. Pultr, slightly adapted.
- 2.5.: Differential of composed mapping, chain rule. Geometric meaning of differential. Partial derivatives of higher orders and their interchangeability. Relevant parts of lecturenotes of A. Pultr, slightly adapted.
- 9.5.: Implicit function theorems. Extremes of functions of multiple variables. Relevant parts of lecturenotes of A. Pultr, slightly adapted.
- 16.5.: Metric spaces. Relevant parts of lecturenotes of A. Pultr, slightly adapted.
- 23.5.: Metric spaces. Compactness. Extremes on a compact set.