Architecture

Parametric Curves: Definitions, support of a curve, concatenation of curves, equivalent curves, tangent vector and unit tangent vector, regular curve, length of a curve. Functions of Several Variables: Elements of topology in R^N, domain of definition of multivariable functions, limits and continuity, partial derivatives, differentiability, tangent plane, gradient, directional derivatives, gradient formula, differentiation of composite functions, second derivatives, Schwarz’s theorem. Optimization: Absolute extrema and Weierstrass’s theorem, relative extrema, unconstrained extrema and Fermat’s theorem, Hessian matrix associated with a function of two variables, study of maxima and minima using the Hessian matrix, constrained optimization. Integral Calculus for Multivariable Functions: Definition of double integral, normal domains, reduction formulas for double integrals, polar coordinates, change of variables in double integrals, brief overview of triple integrals. Ordinary Differential Equations: Definitions and examples, general integral, Cauchy problems, first-order equations with separable variables, first-order linear homogeneous and non-homogeneous equations, formula for the general integral, second-order linear homogeneous equations, structure of the general integral, linear equations with constant coefficients (homogeneous), characteristic equation, linear equations with constant coefficients (non-homogeneous), structure of the general integral, method of undetermined coefficients. Line Integrals and Vector Fields: Definitions and examples, line integrals of the first and second kind, work, conservative and potential fields, characterization of conservative fields via work, irrotational fields in R^2, simply connected domains, relationship between irrotational and conservative fields, extension to the R^3 case (curl), brief overview of surfaces.

Istituzioni di Matematica I

G. Crasta, A. Malusa, "Matematica 2: teoria ed esercizi" N. Fusco, P. Marcellini, C. Sbordone, “Elementi di Analisi Matematica 2” M. Bramanti, C. Pagani, S. Salsa, “Analisi Matematica 2”

Attendance is strongly recommended.

The exam consists of a written exam (also with theoretical questions) and an oral exam (at the professor's discretion).

G. Crasta, A. Malusa, "Matematica 2: teoria ed esercizi" N. Fusco, P. Marcellini, C. Sbordone, “Elementi di Analisi Matematica 2” M. Bramanti, C. Pagani, S. Salsa, “Analisi Matematica 2”

The course consists of 75 hours of lectures divided in classes of 2,5/3 hours twice a week.

Curves Planar and Spatial Curves with Related Properties. Tangent Vector. Equivalent Curves. Piecewise Regular Curves. Length of a Curve. Differential Calculus for Multivariable Functions Elements of Topology in Rn. Multivariable Functions: Domain, Graph, and Image. Definition of Limit and Continuity. Level Curves. Partial Derivatives. Gradient and Differentiability with Geometric Interpretation. Directional Derivative and Gradient Formula. Optimization Fermat's Theorem. Study of the Nature of Critical Points with the Hessian Matrix. Absolute Maximums and Minimums on Closed and Bounded Sets, Weierstrass Theorem. Constrained Extrema on a Regular Curve: Direct Method. Double Integrals Definition of Double Integral according to Riemann and Geometric Interpretation. Reduction Formulas for normal domains. Examples of Changes of Variables for Integral Calculation. Polar coordinates. Line Integrals, Vector Fields, and Differential Forms Line Integrals of Continuous Functions. Vector Fields. Line Integrals of Vector Fields (of the Second Kind). Work of a Vector Field. Conservative and Irrotational Vector Fields. Simply connected domains. Necessary and Sufficient Condition for a field to be Conservative.

Mathematical Institutions I

G. Crasta, A. Malusa, Matematica 2, Amazon KDP G. Catino, F. Punzo, Esercizi svolti di Analisi Matematica e Geometria 2, Amazon Books

Class attendance is not mandatory but highly recommended

Written test with exercises and theory questions

electronic blackboard and/or traditional blackboard and/or writing tablet