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16954 - Mathematics 2

Academic Year 2026/2027

  • Teaching Mode: In-person learning (entirely or partially)
  • Campus: Bologna
  • Corso: First cycle degree programme (L) in Materials Science (cod. 6830)

Learning outcomes

At the end of the course, the student possesses basic knowledge of differential and integral calculus for functions of multiple real variables and is acquainted with the main methods for solving differential equations. The student develops a habit for scientific reasoning and sensitivity in analyzing mathematical models. Moreover, they are capable of conducting a detailed study of functions, sequences, and series, both numerical and functional.

Course contents

  • Real-valued functions of several variables. Partial derivatives and directional derivatives for functions of several variables.
  • The differential. The gradient and its properties. Higher-order derivatives. The Hessian.
  • Differential calculus for vector-valued functions of several variables. The Jacobian.
  • Critical points. Lagrange multipliers.
  • Integration for functions of several variables. Polar, spherical, and cylindrical coordinates.
  • Parametric curves and line integrals. Parametric surfaces and surface integrals.
  • Vector fields. Line integral of a vector field. Potential of a vector field.
  • Flow. Divergence theorem. Stokes' theorem.
  • Differential forms. Closed forms and exact forms. Generalized Stokes' theorem.
  • Elements of probability. Sample space. Draws with and without replacement.
  • Random variables and their cumulative function. Examples. Discrete and absolutely continuous random variables. Expected value and variance. The central limit theorem.

    • Readings/Bibliography

    • Marco Bramanti, Carlo D. Pagani, Sandro Salsa: Analisi matematica 2 (Zanichelli)
    • Marco Bramanti, Fulvia Confortola, Sandro Salsa: Matematica per le scienze (Zanichelli)
    • Simonetta Abenda: Analisi Matematica (Esculapio)
    • Simonetta Abenda: Esercizi di Analisi Matematica (Esculapio)

    Teaching methods

    Lectures and exercises on the blackboard. Slides might be used in exceptional circumstances.

    Assessment methods

    A closed book (no notes allowed) written examination. The exam can be taken at the end of the course, or it may be divided into four partials, covering each of the four parts of the course.

    This will be a closed-book examination. Students will not be allowed to access their notes and/or textbooks or other aids.

    The use of AI is not permitted. Any attempts to use AI in the examination will be considered a breach of academic integrity.

    Teaching tools

    Blackboard and textbooks.

    Office hours

    See the website of Nicola Tito Pagani