The aim of the course is to give further knowledge and tools of calculus, required for an adequate understanding of mathematical methods and models relevant for engineering, including probability and statistics.
teacher profile teaching materials
Main definitions about ODEs. Order. ODE solvable by direct or subsequent integrations. Initial conditions and Cauchy problem. ODE with separable variables. First order linear ODEs, homogeneous and non homogeneous. Linear ODEs of any order: space of solutions of a homogeneous ODE and general integral of a non homogeneous ODE. Linear ODEs with constant coefficients. Methods for the particular solution of a non homogeneous ODE: method of indeterminate coefficients (similarity method), Wronskian matrix and method of constants' variation. Euler's equation. ED which can be solved by order lowering.
FUNCTIONS OF SEVERAL VARIABLES
Open and closed sets in ℝ^n. Limits and continuity of functions of several variables. Domains of functions of several variables. Partial derivatives. Partial derivatives of higher order. Critical points. Hessian determinant method for determining the nature of critical points. Examples of finding absolute extremes of a function of two variables in a compact set of the plane.
RECALLS OF ANALYTICAL GEOMETRY
Cartesian coordinate systems in the plane and in the tridimensional space. Straight lines in the plane. Bundles of straight lines. Planes in space. Straight lines in space. Direction parameters. Curves in the plane: cartesian and parametric equations. Main characteristics of conic sections. Degenerate conic sections. Parametrization of conic sections. Surfaces in space: cartesian and parametric equations. Rotation surfaces. Curves in space. Main characteristics of the quadric surfaces, with particular reference to quadrics in canonical form. Parametrization of quadrics.
INTEGRALS OF SEVERAL VARIABLE FUNCTIONS
Hints about the measure of a limited set of ℝ^n. Integral of a continuous function on a compact set of ℝ^n. Integral properties. Normal domains in the plan. Reduction formula for double integrals. Normal domains in space. Reduction formula for triple integrals. Coordinate changes: linear transformations, polar coordinates and generalizations, spherical coordinates and generalizations. Geometric and physical applications: calculation of volumes, center of gravity, moments of inertia.
Regular curves. Length of a curve arc. Line integrals of scalar functions and vector fields. Physical meaning of the line integral. Conservative vector fields and scalar potential. Conditions for a vector field to be conservative. Regular domains in the plan. Green's theorem in the plane. Simply connected domains. Study of the conservativeness of a bidimensional vector field in a set of the plane.
Regular surfaces. Area of a surface. Superficial integral of a vector field. Curl of a vector field. Stokes' theorem and applications. Divergence of a vector field. Green's theorem in space and applications.
FOURIER SERIES
Review of the exponential function in the complex field. Expression of sine and cosine functions as combinations of complex exponentials. Orthogonal function families. Expression of a function as a series of orthogonal functions. Fourier series of a periodic function of any period, as a series of goniometric functions or as a bilateral series of complex exponentials. Point convergence theorem for Fourier series.
Notes by the teacher (distributed on web).
Programme
ORDINARY DIFFERENTIAL EQUATIONSMain definitions about ODEs. Order. ODE solvable by direct or subsequent integrations. Initial conditions and Cauchy problem. ODE with separable variables. First order linear ODEs, homogeneous and non homogeneous. Linear ODEs of any order: space of solutions of a homogeneous ODE and general integral of a non homogeneous ODE. Linear ODEs with constant coefficients. Methods for the particular solution of a non homogeneous ODE: method of indeterminate coefficients (similarity method), Wronskian matrix and method of constants' variation. Euler's equation. ED which can be solved by order lowering.
FUNCTIONS OF SEVERAL VARIABLES
Open and closed sets in ℝ^n. Limits and continuity of functions of several variables. Domains of functions of several variables. Partial derivatives. Partial derivatives of higher order. Critical points. Hessian determinant method for determining the nature of critical points. Examples of finding absolute extremes of a function of two variables in a compact set of the plane.
RECALLS OF ANALYTICAL GEOMETRY
Cartesian coordinate systems in the plane and in the tridimensional space. Straight lines in the plane. Bundles of straight lines. Planes in space. Straight lines in space. Direction parameters. Curves in the plane: cartesian and parametric equations. Main characteristics of conic sections. Degenerate conic sections. Parametrization of conic sections. Surfaces in space: cartesian and parametric equations. Rotation surfaces. Curves in space. Main characteristics of the quadric surfaces, with particular reference to quadrics in canonical form. Parametrization of quadrics.
INTEGRALS OF SEVERAL VARIABLE FUNCTIONS
Hints about the measure of a limited set of ℝ^n. Integral of a continuous function on a compact set of ℝ^n. Integral properties. Normal domains in the plan. Reduction formula for double integrals. Normal domains in space. Reduction formula for triple integrals. Coordinate changes: linear transformations, polar coordinates and generalizations, spherical coordinates and generalizations. Geometric and physical applications: calculation of volumes, center of gravity, moments of inertia.
Regular curves. Length of a curve arc. Line integrals of scalar functions and vector fields. Physical meaning of the line integral. Conservative vector fields and scalar potential. Conditions for a vector field to be conservative. Regular domains in the plan. Green's theorem in the plane. Simply connected domains. Study of the conservativeness of a bidimensional vector field in a set of the plane.
Regular surfaces. Area of a surface. Superficial integral of a vector field. Curl of a vector field. Stokes' theorem and applications. Divergence of a vector field. Green's theorem in space and applications.
FOURIER SERIES
Review of the exponential function in the complex field. Expression of sine and cosine functions as combinations of complex exponentials. Orthogonal function families. Expression of a function as a series of orthogonal functions. Fourier series of a periodic function of any period, as a series of goniometric functions or as a bilateral series of complex exponentials. Point convergence theorem for Fourier series.
Core Documentation
B. Palumbo: Integrali di funzioni di più variabili (II edition). Accademica, Roma, 2009.Notes by the teacher (distributed on web).
Reference Bibliography
T.M. Apostol: Calculus, Vol. III. John Wiley and Sons, 1967Type of delivery of the course
Four hours a week, including exercises.Type of evaluation
Written test on the topics covered in the course (finding maximum and minimum of a function of several variables in a compact set, solving a differential equation or a Cauchy problem, calculating integrals of functions of several variables, with applications).