Mathematical analysis 2
Scientific Disciplinary Sector (SSD)
MAT/05 - MATHEMATICAL ANALYSIS
Primo semestre dal Oct 3, 2022 al Jan 27, 2023.
The aim of the course is to provide students with the fundamental notions of differential and integral calculus in many variables, generalizing and mastering the notions learned in the course “Mathematical Analysis I” and employing, if needed, the notions of the other courses attended during the first year of the Bachelor in Computer Science. At the end of the course the student must prove: - to know and to be able to understand the tools and the advanced notions of the mathematical analysis and to use such notions for the solution of problems; - to be able to use the notions learned in the course for the comprehension of the topics of further courses, not necessarily in the mathematical area, where the knowledge of mathematical analysis can be a prerequisite; - to be able to choose which mathematical tool or theoretical result can be useful for the solution of a problem; - to be able to appropriately use the language and the formalism of the mathematical analysis; - to be able to broaden the knowledge in Mathematics, Computer Science or in any scientific area using, when needed, the notions of the course.
Prerequisites and basic notions
Key concepts from previous courses (Analisi I and Algebra Lineare).
Cauchy problem for first order differential equations (ODE). Separable equations and the theorem of existence and local uniqueness.
Structure of the space of the solutions of a homogeneous linear ODE (first and second order). Solutions in the non-homogeneous case. The variation of constants method in the case of an equation of order 2. Slope fields and graphical analysis of an ODE in very simple cases.
Differential calculus in several real variables: limits and continuity. Directional derivatives and the gradient of a scalar function. Differentiability and the differential, the theorem on the total differential. Higher order derivatives, the Hessian matrix and Schwarz's theorem. Taylor's formula with Lagrange and Peano remainders. Unconstrained optimization: necessary and sufficient conditions for having local extrema. Constrained optimization: Lagrange multipliers. Dini's theorem.
The Riemann integral over the cartesian product of real intervals: definition and techniques for the calculation. The Riemann integral on admissible domains. Change of variables and special coordinate systems: polar, cylindrical and spherical. Parametric curves. Line integrals of scalar functions. Parametric surfaces in space, area of a surface, surface integrals and Gauss-Green formula. Line integrals and vector fields.
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Lectures, classroom and online activities.
Learning assessment procedures
The final exam consists of a written test including a series of exercises to be solved related to the academic program (specific instructions will be communicated throughout the course).
The final exam could be substituted by two ongoing tests, the former scheduled around the end of November and the latter coinciding with the first exam date in February. In this case, the exam grade will be given by the sum of the two partial assessments, with a maximum of 16 points each.
Exams will be held in presence.
The exam aims to verify the candidate's ability to solve program-related problems, their possession of adequate analytical skills, as well as the ability to synthetize and abstract, starting from requests formulated in natural or specific language.