The course aims to furnish the basic mathematical knowledge needed to understand operation and management of naval means and modern navigation.
Knowledge and understanding: Students must know the theories of linear systems and of real functions.
Ability to apply knowledge and understanding: Main focus is on problem solving, examples, and exercises, in order to apply the acquired theoretical knowledge in the field of conduction of naval means.
Judgment autonomy: Students must be capable of solving exercises.
Communicative skills: Students must be able of using a correct mathematical language to relate with specialists in other branches of technical sciences.
Learning skills: Students will be able to understand the mathematical aspect of a scientific or technical text in view of, first, completing their curriculum, and then constantly update.
Basic mathematical skills
Number sets, spherical and cartesian coordinates, trigonometry. (4h)
Complex numbers: algebraic and trigonometric form, powers, roots, and equations. (8h)
Linear algebra: vectors, vector spaces, and subspaces, linear dependence. Matrix operations, determinant, inverse matrix, rank. General linear systems: Cramer rule, Gauss reduction, Rouché-Capelli Theorem. (12h)
Analytic geometry: lines, planes, and conics. (8h)
Functions of a single variable: domain and range, equalities, and inequalities. Basic functions: modulus, powers, exponential, logarithmic and trigonometric functions, sequences. (8h)
Infinitesimal calculus. Definition, properties, and evaluation of limits, Uniqueness and Comparison Theorems, one-sided limits, the limit of compositions, limits of indeterminate forms, asymptotes. Continuity: definition and examples, Weierstrass’, Bolzano’s, and intermediate values Theorems. (12h)
Differential calculus. Tangent line and derivative, properties and examples, rules of differentiation. Application to the study of functions: critical values and points, Fermat’s, Rolle’s, Lagrange’s and Cauchy’s Theorems, the derivative tests for increasing/decreasing functions and concavity, optimization. Limits and asymptotic rates via differential calculus: the de L’Hopital’s Theorem and Taylor polynomials. (12h)
Integral calculus: antiderivative and integral function, integration rules. Definite integral: Riemann sums and geometrical interpretation. First and Second Fundamental Theorem of Calculus. (8h)
The course provides basic knowledge of pre-calculus, calculus, linear algebra and geometry.
It deals in particular with:
Numbers and sets.
Vector, matrix and linear systems.
Euclidean geometry (lines, planes, conics).
Real functions: basic knowledge, limits, continuity, differentiability, qualitative properties.
Fundamental Theorems on continuous and differentiable functions.
Riemann Integral and area of plane surfaces.
3 lectures per week, plus 1 exercise
1. Crasta-Malusa: “Elementi di Analisi Matematica e
Geometria con prerequisiti ed esercizi svolti”. Edizioni La Dotta.
2. Bramanti-Pagani-Salsa: "Analisi matematica 1 con elementi di algebra lineare e geometria" Edizioni
3. Lang: “Short Calculus” Undergraduate Texts in
Written and oral test
two written tests (including both exercises and quiz) during the term.