The
course presents the classical tools of infinitesimal and differential calculus,
and integral calculus for functions in one and two variables, in addition to
geometry and linear algebra, illustrating wherever possible their applications
in other disciplines.
The topics will be covered with a view to:
-
Acquiring analysis skills: i.e.
introducing and becoming accustomed to rigorous discussion and analytical
reasoning.
- Attaining, through practice, command of an exact and
apposite language.
- Applying the tools of calculus to prepare for better
managing the use of quantitative methods.
Students
must be conversant with the following topics:
-
Arithmetic and algebra.
-
Analytical geometry and trigonometry on a plane.
-
Functions: in particular, a knowledge of the
definition, graphs and principal properties of the fundamental functions
(power, exponential, logarithmic, sine, cosine, tangent). They must also be able to solve algebraic, exponential, logarithmic and
trigonometric equations and inequalities.
Syllabus
1. Numerical
sets:
- Sets and logic symbols.
- Numerical sets and their algebraic and topological
structure (natural numbers, integers, rational numbers, real numbers).
- Subsets of the real numbers: definition of upper bound and lower bound, supremum and infimum, maximum
and minimum.
-
Complex numbers
-
Correspondences and functions between sets.
2. Sequences and series.
-Definition of sequence and
calculation of the limits.
- Numerical series and their characteristics.
- Geometric series.
- Harmonic and generalised harmonic series.
- Series with terms of alternating sign. Absolute convergence and Leibniz criterion.
-
Series of powers.
3. Functions of a real variable.
-
Definition of univocal function of a real variable.
-
Symmetric, increasing, decreasing, periodic, limited functions.
- Maximum and minimum of a function.
- Definition of compound and inverse functions.
- Limits of functions.
- Continuity.
- Derivative and differential.
-
Elasticity
- Rules of differential calculus.
- Searching for local or global maxima and minima.
- Higher order derivatives.
- Convex or concave functions.
- Study of a function.
- Taylor - Mac Laurin series.
4. Integrals.
-
Definition of primitive and the indefinite integral.
-
Rules of integration: by
decomposition, by parts, by substitution.
-
Definite integral.
-
Geometric applications.
- Integral function.
- Generalised integrals
5. Elements of geometry and linear algebra.
-
Vectors on a plane and in 3-space. Scalar
and vector product.
-
n –dimensional vectors: the Rn
space. Vector space.
-
Linear dependence and independence, basis and dimension.
-
Scalar product in Rn.
-
Lines and planes in three-dimensional space.
- Matrices and matrix calculus.
- Determinant.
- Inverse matrix.
-
Rank of a matrix.
- Linear systems.
- Cramer's theorem.
- Rouchè-Capelli theorem.
-
Linear transformation and representation theorem.
-
Eigenvalues and eigenvectors. Matrix
diagonalization.
6. Real
functions of one or more real variables.
- Functions in R2: domain, level curves.
- Continuity. Partial
derivatives, gradient vector, tangent plane and differentiability.
-
Directional derivatives.
-
Implicitly defined functions.
- Free or constrained optimisation of functions in two
variables.
7.Differential
equations.
- Equations with separable variables.
- First and second order linear equations.
- Homogeneous and non-homogeneous second order
equations with constant coefficients.
-
Euler equations.
8.Line integral and double integral.
-
Smooth curves and line integrals in R2 and R3.
-
Definition of double integral. Geometric
interpretation.
- Calculation of double integral using Cartesian and
polar coordinates.
-
Gauss-Green formula in the plane.
Examinations
There
will be a written exam, held during the sessions scheduled in the academic
calendar, and possibly completed with an additional oral exam.
There
will also be four intermediate written tests during the course.
Students
who earn a grade of eighteen or higher on each of the four intermediate tests can
be exempted from sitting the final written exam.
Reading list
Bramanti M., Pagani C.D., Salsa S., Matematica,
Zanichelli.
Salsa S., Squellati A., Esercizi di
Matematica, volumi: 1o 2o, Zanichelli.
Lecture
notes and exercises prepared by the lecturer.