B026296 - MATHEMATICAL ANALYSIS AND GEOMETRY

Italian

Italian.

Matrices and determinants. Linear systems. Coordinates and vectors. Plane analytic geometry. Analytic geometry in space. Vector spaces and eigenvalues. Functions of one variable. Limits and continuity. Differentiation. Integration. Functions of two or more variables, mentioned. Calculus in several variables, mentioned. Multiple integrals, mentioned.

Giuseppe Anichini, Giuseppe Conti. Geometria analitica e algebra lineare. Pearson, Milano, 2009. ISBN 9788871925714

Giuseppe Anichini, Giuseppe Conti. Analisi matematica I. Pearson, Milano, 2015. ISBN 9788865189559

Giuseppe Anichini, Giuseppe Conti. Analisi matematica 2. Pearson, Milano, 2010. ISBN 9788871925929

Andrea Ratto, Antonio Cazzani. Matematica per le Scuole di Architettura. Liguori, Napoli, 2010. ISBN: 9788820752422

The course aims at providing basic tools in geometry and analysis. These tools are prerequisites for subsequent courses in science and technology.

Basic notions of algebra and geometry taught in high school. Trigonometry. Logarithms and exponentials. Solution of algebraic, trigonometric, logarithmic and exponential equations.

Lectures and problem sessions in class and online.

Written and oral examination. Tests passed throughout the academic year can substitute for the written exam.

Matrices and determinants: matrices; operations on matrices; matrix product; determinants; linear combinations; characteristic and rank of a matrix.
Linear systems: linear equations; linear systems; Rouché-Capelli’s theorem; Cramer’s rule; Gauss’ method; homogenous linear systems; inverse of a matrix.
Coordinates and vectors: Cartesian coordinates in plane and in space; vectors; scalar product; vector product; triple product; linear combinations.
Plane analytic geometry: polar coordinates; change of frame; equation of a straight line; parallel or perpendicular lines; angle between two lines; equation of a straight line in explicit form; distance of a point from a straight line; circumference; ellipse; hyperbola; parabola; conic sections in general.
Analytic geometry in space: parametric equation of a straight line; equation of a plane; parallel or perpendicular planes; equations of a straight line, sheaf of planes; parallel or perpendicular lines; lines parallel or perpendicular to a plane; angles in space; distance of a point from a straight line or from a plane; skew lines; curves and surfaces in space, mentioned; quadrics.
Vector spaces and eigenvalues: vector spaces; vector subspaces; base and dimension; linear maps; eigenvectors and eigenvalues; matrix diagonalization; spectral theorem for symmetric matrices.
Functions of one variable: review of real numbers; review of functions; topology of the real line; examples of real functions of one real variable and their graphs.
Limits and continuity: limit of a function; continuous functions; properties of continuous functions; asymptotes.
Differentiation: derivative of a function; higher derivatives; differentiation rules; local maxima or minimum points; some properties; differentiability and monotonicity; convexity and concavity; applications of derivatives.
Integration: definition of integral; main properties of the integral; fundamental theorem of calculus; some integration techniques; areas, volumes, center of mass.
Functions of two or more variables, mentioned: domain; graph; contour lines.
Calculus in several variables, mentioned: partial derivatives; gradient; critical points.
Multiple integrals: definition of double integral; double integral calculus; change of variables; applications; centroids.