Lecturer: Marco Luigi Bernardi   Carlo Lovadina  

Course name: Calculus (Mathematical analysis)
Course code: 500446
Degree course: Ingegneria Civile e Ambientale
Disciplinary field of science: MAT/05
University credits: CFU 12
Course website: n.d.

Specific course objectives

The course is aimed at providing the basic knowledge of calculus (differential, integral) for real and vector-valued functions of one or several real variables, together with the fundamentals of the theory of series and with an introduction to ordinary differential equations. Lectures will be mainly focused on the comprehension of notions (definitions, results), although some proofs will still be detailed. Examples and exercises will be presented. By the end of the course the Students are expected to be able to handle correctly and without hesitation limits, derivatives, function graphs, integrals (also multiple, line and surface integrals), series, linear differential equations, and the corresponding theoretical facts.

Course programme

Functions, Limits, Continuity.
Recalls and complements on real numbers. Complex numbers. Functions: definitions, graphs; invertible functions; odd and even functions; monotone functions; periodic functions; operations on functions; nested functions. Elementary functions and corresponding graphs. Limits of functions: definitions, operations on limits. Continuous functions. Discontinuity points and their classification. Global properties of continuous functions.

Differential Calculus in one real variable and Applications.
Derivative of a function: definition and properties, applications in Geometry and Physics. Derivation rules and calculus. Fundamental theorems of differential calculus. Primitives and indefinite integrals. Successive derivatives. Function study: extrema, monotonicity, convexity. De l'Hopital rules.

Integral Calculus in one real variable and Applications.
Definite integrals: definitions and basic properties, applications in Geometry and Physics. Fundamental theorems of integral calculus. Integration techniques. Improper integrals.

Series.
Limits of real sequences. Real series: definitions and basic examples; series with positive terms (and convergence tests); absolute and simple convergence. Fundamentals of real power series. Taylor polynomials and Taylor formulas. Taylor series; Taylor series of some elementary functions.

Differential Equations.
Introduction to ordinary differential equations. The Cauchy problem. Linear ordinary differential equation of the first order. Linear ordinary differential equation of the second order with constant coefficients.

Differential Calculus in several real variables.
Real functions of several real variables: definitions, graphs; limits and continuity; partial derivatives, gradients, and directional derivatives. Successive derivatives. Differentiability. Partial derivatives of nested functions (chain rules). Free extrema of real functions of several real variables; critical points and their classification. Fundamentals of differential calculus for vector-valued functions; jacobian matrices.

Multiple Integrals.
Double integrals: definitions and basic properties, application in Geometry and Physics. Integration techniques: iteration formulas; change of variables; double integrals in polar coordinates. Fundamentals of volume integrals.

Line Integrals and Surface Integrals.
Curves: tangent vectors; rectifiable curves and arc length. Surfaces: tangent planes; surface area. Line integrals with respect to the arc length. Line integrals of vector fields, and applications in Physics. Gradient fields, potentials, and path independence. The operators curl and div. Surface integrals, and applications in Physics. Green's theorem and divergence theorem in two variables. Stokes' theorem and divergence theorem in three variables.

Course entry requirements

Mathematics: the required prerequisites for enrollment into the Engineering Faculty.

Course structure and teaching

Lectures (hours/year in lecture theatre): 60
Practical class (hours/year in lecture theatre): 60
Practicals / Workshops (hours/year in lecture theatre): 0

Suggested reading materials

M. Bramanti, C.D. Pagani e S. Salsa. Matematica. Calcolo infinitesimale e Algebra lineare (seconda edizione). C.E. Zanichelli, Bologna, 2004. (Recommended book).

M. Bramanti, C.D. Pagani e S. Salsa. Analisi matematica 1 (prima edizione) e Analisi Matematica 2 (prima edizione) . C.E. Zanichelli, Bologna, 2008-2009. (Suggested books for a more refined study, and for various complements).

Testing and exams

Finals consist in a written (exercises) and an oral test (theory plus possibly exercises). Both written and oral tests have to be passed within the same final session. In order to be admitted to the oral test, a specific minimum of points has to be obtained in the written test.