PRINCIPLES OF MATHEMATICAL ANALYSIS
Learning outcomes of the course unit
The course aims at consolidating the knowledge of Mathematical Analysis, acquired by the students during the first year, through the study of ordinary differential equations, of plane and space curves and of the differential calculus fo functions of several real variables. Many of the concepts presented are useful in the technical disciplines of Architecture. The course also proposes to develope precision, capacity of verifying the reliability of the results and space vision.
Prerequisites: Principles of Mathematics Suggested: Geometry and Algebra
Course contents summary
PLANE AND SPACE CURVES: Parametrizations, support, derived vectors, velocity and acceleration, scalar velocity, tangent and normal vector, tangent and normal lines. Lenght of a curve. Curvature. FUNCTIONS OF SEVERAL REAL VARIABLES: Domain, graph, sections and level curves. Paraboloids, cones, spherical and elliptical surfaces. Functions depending on a single variable and radial functions. Identifying and drawing sets in two and three dimensions. Differential calculus: partial derivatives, gradient, tangent plane to the graph of a function. Line integrals. ORDINARY DIFFERENTIAL EQUATIONS: Concept of differential equation and physical models. First-order and second-order linear ODEs with constant coefficients. General integral of homogeneous equations. Direct method to find a particular integral of complete equations. Cauchy and boundary value problems. Study of a model of a column with variable circular section; of a model of the oscillations of a building or of a bridge under the effects of an earthquake or of gusts of wind; of a model of the ideal pin-ended column.
Recommended text: N. Fusco, P. Marcellini, C. Sbordone,Elementi di Analisi Matematica 2, Liguori Editore, Napoli (2001). Course notes with exercises and solutions for the written tests.