### Mathematical Analysis II NMAI055

Time and place: Monday at 12:20 in S11

Tutorials: Thursday at 14:00 in T10, by Vojtěch Kaluža

Exam: You must obtain credit from the tutorial to be able to take the exam. Exam dates are 12., 22. and 29.6., and there will be one more exam last week of September. You must register in SIS to take the exam. Exam description and list of exam requirements. The exam will have a form similar to the practice exam from previous year.

For recommended study materials see the syllabus, you might also find useful notes by Aleš Pultr.

Covered material:

• 19.2..: Antiderivatives: definition, antiderivatives of elementary functions. Integration per partes. Lecture notes.
• 26.2.: Proof that a function with a primitive function has Darboux property. Substitution. Partial fractions. Lecture notes.
• 5.3.: Riemann integral. Upper and lower Riemann sum and integral. Criterion of integrability. Lecture notes.
• 12.3.: Example: Riemann integral of linear function. Monotonicity implies integrability. Uniform continuity coincides with continuity on a compact interval. Continuity implies integrability. Linearity of Riemann integral. Lecture notes.
• 19.3.: Lebesgue characterisation of integrable functions. Fundamental theorems of calculus, Newton integral. Substitution and per partes for definite integral. Lecture notes.
• 26.3.: Estimate of factorial and harmonic numbers. Integral criterion of convergence of series. Gamma function. Geometric applications of integral: length of a curve, volume of a rotating solid. Lecture notes.
• 9.4.: Euclidean distance, functions of multiple variables - continuity. Multivariate Riemann integral. Multivariate Lebesgue characterisation of integrable functions, definition of sets of measure zero. Fubini theorem and its application (proof will be next time). Characteristic function of a set. Lecture notes.
• 16.4.: Proof of Fubini Theorem. Partial and directional derivatives. Total differential. Lecture notes.
• 23.4.: Connection between total differential and partial derivatives. Geometric meaning of total differential. Continuous partial derivatives imply differentiability. Lecture notes.
• 30.4.: Lagrange theorem for multivariate functions. Zero differential implies constantness. Arithmethic of partial derivatives and differentials. Lecture notes.
• 7.5.: Partial derivatives of higher orders and their interchangeability. Extremes of multivariate functions. Lecture notes.
• 14.5.: Implicit function theorems. Lagrange multipliers. Lecture notes.
• 21.5.: Metric and topological spaces. Compactness. Extremes on a compact set. Lecture notes.