Free guide
Math exams in Italy
For international and Erasmus students taking a mathematics or physics exam at an Italian university: what the course names mean, how exams and grades work, and a glossary of the Italian terms used in lectures, textbooks and exam papers, with their English equivalents.
Italian course names and their international equivalents
The first-year and second-year mathematics courses of Italian engineering, science and economics degrees have names that do not translate literally. The table gives the usual international equivalent and what each course covers in Italy. The study material of this site is written for these courses and is available in English.
| Italian course | Usual equivalent | What it covers in Italy |
|---|---|---|
| Analisi matematica 1 | Calculus 1 / Mathematical Analysis 1 | Single-variable calculus with proofs: sequences, limits, derivatives, integrals, numerical series and first differential equations. |
| Analisi matematica 2 | Multivariable calculus / Mathematical Analysis 2 | Functions of several variables, multiple and line integrals, vector calculus, sequences and series of functions, Fourier series. |
| Geometria / Algebra lineare e geometria | Linear algebra and analytic geometry | Matrices, linear systems, vector spaces, eigenvalues, inner products, lines and planes in space, conics and quadrics. |
| Fisica 1 | Physics 1 | Mechanics of particles and rigid bodies, oscillations, gravitation, fluids and thermodynamics. |
| Fisica 2 | Physics 2 | Electromagnetism, from Coulomb’s law to Maxwell’s equations, often with an introduction to optics. |
| Matematica generale | Mathematics for economics / business calculus | The single math course of economics programs: functions, limits, derivatives, integrals, some linear algebra and financial mathematics. |
| Precorso | Precalculus | A short course held before the first semester to review high-school algebra, trigonometry and analytic geometry. |
| Metodi matematici | Mathematical methods for engineering / physics | Complex analysis, Fourier and Laplace transforms, distributions and partial differential equations, usually in the third year. |
| Fisica matematica | Mathematical physics | Analytical mechanics (Lagrangian and Hamiltonian) and the partial differential equations of physics. |
| Analisi numerica / Calcolo numerico | Numerical analysis | Numerical solution of equations, linear systems, interpolation, quadrature and differential equations, usually with MATLAB or Python. |
| Probabilità e statistica | Probability and statistics | Probability spaces, random variables, common distributions, estimation and hypothesis testing. |
| Calcolo delle probabilità | Probability theory | The probability course of mathematics programs, more theoretical than Probabilità e statistica. |
| Ricerca operativa | Operations research | Linear and integer programming, network optimization and the simplex method. |
| Meccanica razionale | Analytical mechanics / Rational mechanics | A mathematical treatment of mechanics: kinematics and dynamics of rigid bodies, Lagrangian mechanics and stability, typical of engineering programs. |
| Matematica discreta | Discrete mathematics | Combinatorics, graphs, modular arithmetic and basic algebraic structures, common in computer science programs. |
Study material in English on this site: Precalculus · Mathematical Analysis 1 · Mathematical Analysis 2 · Linear algebra and analytic geometry · Physics 1 · Physics 2 · Mathematics for Economics.
How a mathematics exam works at an Italian university
Each course ends with an exam (esame) that is held several times a year, in the exam sessions (sessioni d’esame): typically January–February, June–July and September, often with one extra date in spring or autumn. Every date on which the exam is held is an appello; a student can register for any appello of a session and, after a failure, try again at a later one.
A mathematics exam usually has a written part (prova scritta) with exercises and, in many courses, an oral part (prova orale) on definitions, theorems and proofs. Some courses replace the single written exam with two partial tests during the semester (esoneri or prove intercorso); a student who passes both skips the written exam.
Grades are given out of thirty (in trentesimi): 18 is the minimum passing grade and 30 the maximum; 30 e lode is 30 with honors. A student who passes can refuse the grade (rifiutare il voto) and take the exam again at a later appello. Each course is worth a number of credits (CFU, crediti formativi universitari), equivalent one to one to ECTS credits; the final degree mark is a weighted average of the exam grades, converted to a scale of 110.
Some exams can be taken only after passing others (propedeuticità): Analisi matematica 2 usually requires Analisi matematica 1, and Fisica 1 often requires it as well. The syllabus of each course is published by the professor (programma del corso) and the exams of previous years (tracce d’esame or compiti) are the usual preparation material: the topics of this site follow those syllabi.
Italian–English glossary
246 terms grouped by subject. The English term is the one used in standard textbooks; the note points out the differences in notation (for instance the Italian sen and tg for the sine and tangent).
University and exams
| Italian | English | Note |
|---|---|---|
| appello | exam sitting / exam date | One of the scheduled dates on which a given exam can be taken; each course usually offers several per year and students choose which one to register for. |
| sessione d’esame | exam session | A period of a few weeks (typically January-February, June-July and September) in which lectures stop and exams are held. |
| prova scritta | written exam | The written part of an exam, usually a set of exercises to be solved in two or three hours; in math courses it is often the main part. |
| prova orale | oral exam | A face-to-face examination with the professor, often on definitions, theorems and proofs, that may follow a passed written exam. |
| voto in trentesimi | grade out of 30 | Italian university exams are graded on a scale from 18 (pass) to 30, so a grade is read as a number out of thirty. |
| 30 e lode | 30 cum laude (30 with honors) | The highest possible grade, awarded for an outstanding exam; it counts slightly more than 30 in the final degree average. |
| 18 (voto minimo) | 18 (minimum passing grade) | The lowest passing grade; a student who scores below 18 fails and must retake the whole exam at a later sitting. |
| rifiutare il voto | to refuse the grade | A student who passes with a low grade may reject it and retake the exam later to improve it; the refused grade is not recorded. |
| CFU (credito formativo universitario) | university credit (ECTS credit) | A unit measuring student workload, equivalent to one ECTS credit; a first-year calculus course is typically worth 6 to 12 CFU. |
| esonero / prova intercorso | midterm exam | A partial test held during the course; students who pass the midterms are exempted from the corresponding part of the final exam. |
| propedeuticità | prerequisite requirement | A rule stating that a given exam must be passed before another one can be taken, for example Analisi 1 before Analisi 2. |
| ricevimento | office hours | The weekly time slot in which a professor is available to meet students and answer questions. |
| laurea triennale | bachelor’s degree (three-year degree) | The first-cycle degree, lasting three years and worth 180 credits. |
| laurea magistrale | master’s degree | The second-cycle degree, lasting two years and worth 120 credits, taken after the laurea triennale. |
| corso di laurea | degree program | The whole study program leading to a degree, for example the degree program in Civil Engineering. |
| matricola | freshman / student ID number | Both a first-year student and the identification number assigned to every student at enrollment. |
| fuori corso | beyond the standard duration | The status of a student who has not completed all exams within the official length of the degree program and keeps enrolling to finish. |
| compito | exam paper | The sheet of exercises given in a written exam; past exam papers are the standard training material for students. |
| traccia | exam question / problem statement | The text of an exercise or of the whole exam as handed out to the students. |
| svolgimento | worked solution | The complete, step-by-step solution of an exercise, as the student is expected to write it in the exam. |
| esercitazione | problem session / tutorial | A class, often led by a teaching assistant, devoted to solving exercises rather than presenting theory. |
| dispensa | lecture notes | Notes written by the professor and distributed to students, often used in place of or alongside a textbook. |
| libro di testo | textbook | The reference book adopted for a course. |
| ripetizioni | private tutoring | One-to-one or small-group paid lessons taken outside the university to prepare for an exam. |
| Ingegneria | Engineering | Engineering programs in Italy require Analisi 1, Analisi 2, Geometria and Fisica 1 and 2 in the first two years. |
Precalculus and algebra
| Italian | English | Note |
|---|---|---|
| equazione | equation | |
| disequazione | inequality | |
| equazione di primo grado | linear equation / first-degree equation | |
| equazione di secondo grado | quadratic equation | |
| discriminante | discriminant | Called delta in Italian classrooms, from the symbol Δ. |
| equazione irrazionale | radical equation / irrational equation | |
| condizioni di esistenza | domain conditions | The conditions (denominators nonzero, radicands nonnegative, logarithm arguments positive) that an expression requires to be defined; often abbreviated C.E. |
| polinomio | polynomial | |
| grado | degree | |
| prodotti notevoli | special products | |
| scomposizione in fattori | factoring / factorization | |
| regola di Ruffini | Ruffini’s rule | Known in English as synthetic division. |
| radicale | radical | |
| razionalizzazione | rationalizing the denominator | |
| logaritmo | logarithm | |
| logaritmo naturale | natural logarithm | Written ln in both languages; Italian texts sometimes write log for the natural logarithm, English texts for base 10. |
| goniometria | trigonometry | |
| circonferenza goniometrica | unit circle | |
| seno | sine | Italian notation sen x, English notation sin x. |
| coseno | cosine | |
| tangente | tangent | Italian notation tg x, English notation tan x; similarly cotg becomes cot and arctg becomes arctan. |
| formule di addizione | angle addition formulas | |
| formule di duplicazione | double-angle formulas | |
| valore assoluto | absolute value | |
| piano cartesiano | Cartesian plane | |
| retta | line / straight line | |
| coefficiente angolare | slope | |
| parabola | parabola | |
| circonferenza | circle | Italian distinguishes circonferenza (the curve) from cerchio (the disk); English uses circle for the curve and disk for the region. |
| ellisse | ellipse | |
| iperbole | hyperbola | |
| conica | conic section / conic |
Calculus 1
| Italian | English | Note |
|---|---|---|
| insieme | set | |
| estremo superiore | supremum / least upper bound | Written sup in both languages. |
| estremo inferiore | infimum / greatest lower bound | |
| principio di induzione | principle of mathematical induction | |
| dominio naturale / campo di esistenza | natural domain | The largest set on which the formula defining a function makes sense; campo di esistenza is the older school name. |
| codominio | codomain | The target set of the function, not to be confused with the image. |
| immagine | image / range | The set of values actually taken by the function; English textbooks often call it the range. |
| funzione iniettiva | injective function / one-to-one function | |
| funzione suriettiva | surjective function / onto function | |
| funzione inversa | inverse function | |
| successione | sequence | |
| punto di accumulazione | accumulation point / limit point | |
| limite | limit | |
| limite notevole | fundamental limit / standard limit | There is no fixed English name; textbooks speak of special limits or standard limits, for example sin x / x as x goes to 0. |
| teorema del confronto | squeeze theorem | Also called teorema dei carabinieri in Italian. |
| infinitesimo | infinitesimal | In Italian usage, a function that tends to zero; English texts more often say a function that tends to zero than infinitesimal. |
| gerarchia degli infiniti | growth hierarchy | The comparison of logarithms, powers, exponentials and factorials as x tends to infinity. |
| o piccolo | little-o | |
| asintoto verticale | vertical asymptote | |
| asintoto obliquo | slant asymptote / oblique asymptote | |
| continuità | continuity | |
| teorema degli zeri | Bolzano’s theorem / intermediate value theorem for zeros | |
| teorema di Weierstrass | extreme value theorem | English textbooks rarely attach the name Weierstrass to this theorem. |
| derivata | derivative | |
| derivabile | differentiable | |
| punto stazionario | stationary point / critical point | |
| punto angoloso | corner point | A point where left and right derivatives exist, are finite and differ. |
| cuspide | cusp | |
| punto di flesso | inflection point | |
| concavità | concavity | |
| funzione convessa | convex function | Italian also says concavità verso l’alto; English says concave up. |
| massimo relativo | local maximum / relative maximum | |
| teorema di Rolle | Rolle’s theorem | |
| teorema di Lagrange | mean value theorem | English textbooks say mean value theorem, not Lagrange’s theorem. |
| teorema di de l’Hôpital | l’Hôpital’s rule | |
| formula di Taylor | Taylor’s formula / Taylor expansion | |
| polinomio di McLaurin | Maclaurin polynomial | |
| resto di Peano | Peano remainder | The little-o form of the remainder; English texts mostly use the Lagrange form or the big-O form. |
| studio di funzione | curve sketching / graphing a function | The standard exam exercise: domain, symmetries, sign, limits, asymptotes, derivatives, monotonicity, concavity and graph. |
| primitiva | antiderivative | |
| integrale indefinito | indefinite integral | |
| integrale definito | definite integral | |
| integrazione per parti | integration by parts | |
| integrazione per sostituzione | integration by substitution | |
| fratti semplici | partial fractions | |
| teorema fondamentale del calcolo integrale | fundamental theorem of calculus | |
| integrale improprio | improper integral | |
| serie numerica | numerical series / series | |
| carattere di una serie | behavior of a series | Whether the series converges, diverges or is indeterminate; English has no single word for this. |
| serie geometrica | geometric series | |
| serie a termini di segno alterno | alternating series | |
| convergenza semplice | conditional convergence | Italian semplicemente convergente means convergent but not absolutely convergent. |
| criterio del confronto | comparison test | |
| criterio del rapporto | ratio test | |
| criterio della radice | root test | |
| criterio di Leibniz | alternating series test / Leibniz test | |
| equazione differenziale | differential equation | |
| problema di Cauchy | initial value problem / Cauchy problem | |
| equazione a variabili separabili | separable equation | |
| equazione lineare del primo ordine | first-order linear equation | |
| equazione caratteristica | characteristic equation | |
| integrale generale | general solution | |
| metodo di somiglianza | method of undetermined coefficients |
Multivariable Calculus
| Italian | English | Note |
|---|---|---|
| funzione di più variabili | function of several variables | |
| curva di livello | level curve | |
| insieme aperto / chiuso | open set / closed set | |
| insieme compatto | compact set | |
| derivata parziale | partial derivative | |
| gradiente | gradient | |
| differenziabilità | differentiability | |
| teorema del differenziale totale | total differential theorem | In English texts this is usually stated as: continuous partial derivatives imply differentiability. |
| matrice hessiana | Hessian matrix | |
| punto di sella | saddle point | |
| teorema di Dini | implicit function theorem | English textbooks do not attach Dini’s name to it. |
| massimi e minimi liberi | unconstrained extrema | |
| massimi e minimi vincolati | constrained extrema | |
| moltiplicatori di Lagrange | Lagrange multipliers | |
| curva parametrica | parametric curve | |
| integrale di linea di prima specie | line integral of a scalar field | |
| integrale di linea di seconda specie | line integral of a vector field | |
| campo vettoriale | vector field | |
| campo conservativo | conservative vector field | |
| potenziale | potential function | |
| dominio semplicemente connesso | simply connected domain | |
| teorema di Green | Green’s theorem | |
| integrale doppio | double integral | |
| dominio normale | normal domain | A region bounded by two graphs over an interval; English texts say type I or type II region. |
| cambiamento di variabili | change of variables | |
| flusso | flux | |
| divergenza | divergence | |
| rotore | curl | |
| teorema della divergenza | divergence theorem / Gauss’s theorem | |
| teorema di Stokes | Stokes’ theorem | |
| convergenza puntuale | pointwise convergence | |
| convergenza uniforme | uniform convergence | |
| convergenza totale | normal convergence | Convergence of the series of sup norms, the Weierstrass M-test in English texts. |
| serie di potenze | power series | |
| raggio di convergenza | radius of convergence | |
| serie di Fourier | Fourier series |
Linear algebra
| Italian | English | Note |
|---|---|---|
| vettore | vector | |
| versore | unit vector | |
| prodotto scalare | dot product / inner product | |
| prodotto vettoriale | cross product | |
| matrice | matrix | |
| matrice trasposta | transpose | |
| matrice inversa | inverse matrix | |
| determinante | determinant | |
| sviluppo di Laplace | cofactor expansion / Laplace expansion | |
| rango | rank | |
| riduzione a scala | row reduction / row echelon form | |
| sistema lineare | linear system / system of linear equations | |
| teorema di Rouché-Capelli | Rouché-Capelli theorem | English textbooks state it without a name: a system is consistent if and only if the coefficient matrix and the augmented matrix have the same rank. |
| regola di Cramer | Cramer’s rule | |
| spazio vettoriale | vector space | |
| linearmente indipendenti | linearly independent | |
| base | basis | |
| dimensione | dimension | |
| applicazione lineare | linear map / linear transformation | |
| nucleo | kernel / null space | |
| immagine | image / column space | |
| teorema della dimensione | rank-nullity theorem | |
| autovalore | eigenvalue | |
| autovettore | eigenvector | |
| diagonalizzazione | diagonalization | |
| base ortonormale | orthonormal basis | |
| procedimento di Gram-Schmidt | Gram-Schmidt process | |
| matrice ortogonale | orthogonal matrix | |
| teorema spettrale | spectral theorem | |
| forma quadratica | quadratic form | |
| rette sghembe | skew lines | |
| quadrica | quadric |
Physics
| Italian | English | Note |
|---|---|---|
| legge oraria | equation of motion / position as a function of time | The function x(t); English has no single standard term. |
| moto rettilineo uniforme | uniform motion in a straight line / constant-velocity motion | |
| moto uniformemente accelerato | uniformly accelerated motion / constant-acceleration motion | |
| moto del proiettile | projectile motion | |
| moto circolare uniforme | uniform circular motion | |
| principi della dinamica | Newton’s laws of motion | Italian numbers them primo, secondo and terzo principio; English says Newton’s first, second and third law. |
| reazione vincolare | normal force / constraint force | |
| attrito radente | sliding friction / kinetic friction | |
| forza elastica | spring force / elastic force | |
| lavoro | work | |
| potenza | power | |
| energia cinetica | kinetic energy | |
| energia potenziale | potential energy | |
| quantità di moto | momentum / linear momentum | |
| urto elastico | elastic collision | |
| urto anelastico | inelastic collision | |
| moto armonico | simple harmonic motion | |
| corpo rigido | rigid body | |
| momento di una forza | torque | Italian momento (or momento meccanico) corresponds to torque; English uses moment mainly in statics. |
| momento angolare | angular momentum | |
| momento d’inerzia | moment of inertia | |
| gas perfetto | ideal gas | |
| carica elettrica | electric charge | |
| legge di Coulomb | Coulomb’s law | |
| campo elettrico | electric field | |
| flusso del campo elettrico | electric flux | |
| legge di Gauss | Gauss’s law | |
| potenziale elettrico | electric potential | |
| differenza di potenziale | potential difference / voltage | |
| condensatore | capacitor | |
| capacità | capacitance | |
| corrente elettrica | electric current | |
| resistenza | resistance | |
| forza elettromotrice | electromotive force (emf) | |
| leggi di Kirchhoff | Kirchhoff’s laws | |
| campo magnetico | magnetic field | |
| forza di Lorentz | Lorentz force | |
| teorema di Ampère | Ampère’s law | |
| circuitazione | circulation / line integral around a closed path | |
| induzione elettromagnetica | electromagnetic induction | |
| legge di Faraday-Neumann-Lenz | Faraday’s law of induction | English texts say Faraday’s law and treat Lenz’s law separately for the sign. |
| induttanza | inductance | |
| equazioni di Maxwell | Maxwell’s equations |
No term matches.