Multivariable Analysis, Integrals and Series

Published

18 October 2026

Passport

  • Where and when. Semester II in MIPT’s curricula for the 2025 intake for programs 01.03.02 (profile β€œMathematics”) and 09.03.01 (profile β€œApplied Mathematics, Computer Science and Engineering”).
  • Workload. For 01.03.02: 270 hours, 6 credits: 120 contact hours (60 lectures, 60 seminars), 120 hours of independent work and 30 hours for the exam. For 09.03.01: 225 hours, 5 credits: 120 contact hours (60 lectures, 60 seminars), 75 hours of independent work and 30 hours for the exam.
  • Department. Department of Higher Mathematics.
  • Lecturers in the published 2025/26 programs. A. L. Lukashov (FPMI stream) and A. I. Tyulenev (FPMI and VSHPI).
  • Final assessment. Exam in semester II.

Contents of the published program

Below is the complete list of sections from A. L. Lukashov’s published exam program for 2025/26. The wording is slightly condensed, but no item of the program is omitted.

  1. Continuous maps in \(\mathbb R^n\): compactness, limits along a direction, along sets, and iterated limits; uniform continuity, connectedness and path-connectedness; the Bolzano–Cauchy criterion in \(\mathbb R^n\).
  2. Differentiability of functions of several variables: partial derivatives, tangent plane and gradient; mixed derivatives and the theorems of Schwarz and Young; Taylor’s formula in the Lagrange and Peano forms.
  3. The Riemann integral: Darboux sums and integrability; integrability of continuous and monotone functions; the Newton–Leibniz formula, integration by parts and change of variables; mean value theorems, Bonnet’s theorem and the integral remainder in Taylor’s formula.
  4. Improper Riemann integrals: the Cauchy criterion, comparison tests, absolute and conditional convergence, the Dirichlet, Abel and Hardy tests.
  5. Numerical series: the necessary condition, series with nonnegative terms, comparison and the integral test; the d’Alembert and Cauchy tests, CesΓ ro regularity, absolute and conditional convergence, the Leibniz, Dirichlet and Abel tests; rearrangements, Riemann’s theorem, products of series and Mertens’ theorem.
  6. Function series: the uniform Cauchy criterion and the Weierstrass test, the Dirichlet and Abel tests; passage to the limit, Dini’s theorem, termwise integration and differentiation, van der Waerden’s example; power series, radius of convergence, Abel’s second theorem and conditions for expansion in a Taylor series.
  7. Measure: countably additive volume on the ring of elementary sets, Lebesgue outer measure and Jordan measure, inner measure and measurability; algebras of sets, finite and countable additivity, continuity of measure; structure of open and measurable sets in \(\mathbb R^n\), the Jordan criterion, the Cantor set and an example of a non-measurable set.
  8. Measurable functions: definitions and inequalities, arithmetic operations and composition; an example of a non-measurable composition, passage to the limit, approximation by simple functions, convergence in measure and almost everywhere.
  9. Areas and volumes; Schwarz’s method; area of a surface of revolution.
  10. Implicit function theorems, for one equation and for a system, and the local inverse function theorem.
  11. Bounded linear operators: operator norm, the strong derivative and its uniqueness, the chain rule and higher derivatives.

Grading scheme (spring 2026)

For the discipline β€œMultivariable Analysis, Integrals and Series”, year 1, semester 2, Department of Higher Mathematics, a grading scheme of 120 points has been published:

  • three tests on the assignments, 0–6 each; three assignments, 0–2 each; theory check, 0–3; seminar work, 0–3; productive work in the practicum, up to 3 points, with the semester total capped at 30;
  • written exam paper, 0–30; oral exam, 0–60 (a multiple of 6);
  • scale: 112–120 β€” 10, 103–111 β€” 9, 94–102 β€” 8, 85–93 β€” 7, 76–84 β€” 6, 67–75 β€” 5, 54–66 β€” 4, 41–53 β€” 3, 27–40 β€” 2, 0–26 β€” 1. If the oral answer is 12 or less, special rules apply for grades 1 and 2.

Sources and currency

FPMI program (Lukashov), 2025/26, Tyulenev’s version, grading scheme 2026, curriculum 01.03.02 and curriculum 09.03.01. The grading scheme applies to the stated academic year and may be replaced by the department.