Course teached as: B027608 - ANALISI MATEMATICA II 3-years First Cycle Degree (DM 270/04) in CIVIL, BUILDING AND ENVIRONMENTAL ENGINEERING Curriculum AMBIENTE
Teaching Language
Italian
Course Content
Differential and integral calculus for functions pdf several variables. The n-dimensional Euclidean space. Curves, curvilinear integrals. Differential calculus for real-valued functions of several variables; free and constrained optimisation. Vector valued functions of several variables; parametric surfaces. Integral calculus for functions of several variables. Vector fields and related path integrals. Surface integrals.
Suggested textbook:
Theory
Marco Bramanti, Carlo D. Pagani, Sandro Salsa, Analisi matematica 2, Ed. Zanichelli
Problems:
Boris P. Demidovic , Esercizi e Problemi di Analisi Matematica, Editori Riuniti, 2010
Esercitazioni di Matematica, secondo volume (parti prima e seconda), Ed. Liguori.
Further references:
Walter Rudin Principles of Mathematical Analysis, McGraw-Hill, 1976.
Mariano Giaquinta, Giuseppe Modica, Note di analisi matematica. Funzioni di piu' variabili, Pitagora Ed., 2006.
Enrico Giusti, Analisi Matematica 2, Bollati Boringhieri, 3^ edizione, 2003.
Learning Objectives
KNOWLEDGE ACQUIRED
Real-valued and vector-valued functions defined on the n-dimensional Euclidean space
Regular curves; length of a curve; curvilinear integrals
Derivatives of functions of several variables
Local, absolute and constrained extrema for functions of several variables
Integral calculus for functions of several variables
Parametric surfaces; area of a surface; surface integrals
COMPETENCE ACQUIRED
Compute the derivatives of a given function of several variables
Compute the length of a curve, and curvilinear integrals
Find local, absolute and constrained extrema of a function of several variables
Compute multiple integrals
Compute the area of a surface, and surface integrals
Prerequisites
Differential and integral calculus for real valued functions of one variable.
Teaching Methods
Standard frontal lectures
Further information
The teacher will use Moodle platform as a help for the course, and all students will be invited to register to the Moodle course page. On this page the teacher will upload, along the course, exercises that will be subsequently solved and explained during the lectures.
Type of Assessment
The exam is made of two part. In the first the student is asked to solve some exercises on the topic of the course. Examples are: find the extrema of a function of several variables; compute integrals of various types: multiple integrals; line integrals; surface integrals. The second part concerns the theory: the student is asked to present definitions, or statements of results exposed in the course, possibly with the proof of the statement. This part can be written or oral.
Course program
The n-dimensional Euclidean space. Norm; scalar product; Cauchy-Schwartz inequality.
Curves in the two- and three-dimensional space. Notion of regular curve; equivalent curves. Rectifiable curves; length of a curve. Curvilinear integrals of first and second type.
Differential calculus for real-valued functions of several variables. Limits and continuity. Partial derivatives of first order; gradient; differentiability. Partial derivatives of second order; the Hessian matrix. Local extrema for functions of several variables; sufficient and necessary conditions based on the gradient and on the hessian matrix. Constrained extrema: the notion of regular constraint; the Lagrange multipliers theorem.
Integral calculus for functions of several variables - Riemann integration. Double integrals over rectangles; Peano-Jordan measure of a subset of the plane. Double integrals over normal domains. Elementary properties of the integral. Reduction formulas. Change of variables formula. Multiple integrals.
Parametric surfaces. Notion of regular surface in the three-dimensional space. Local coordinates. Tangent plane, normal vector. Area of a surface; surface integrals.