00679 - Mathematics

Course Unit Page

Academic Year 2018/2019

Learning outcomes

See the related section in Italian

Course contents

0. Basic knowledge of the topics dealt with in the Preparation Course in Mathematics is essential (see those contents)

1. Numbers. The language of set theory: unions, intersections, relations, functions. Natural numbers (induction),   factorials, binomial coefficients.

2. Linear systems and matrices. The euclidean spaces. Matrices and the related algebra. Square and symmetric matrices. Determinants and their algebraic properties: Laplace's formula, Binet's theorem. Nonsingular matrices and inversion. Elementary row transformations, Gauss-Jordan algorithm.
Generalities on linear systems, their solutions: theorems of Rouché-Capelli and Cramer. Homogeneous systems.
Eigenvalues and eigenspaces. Characteristic polynomial of a matrix.

3. Real sequences and series.
Completeness of the real field. greatest lower and least upper bounds, maxima and minima of subset of R. real sequences: limits, convergence, divergence. Partial sums and series: geometric progressions and series. Series with positive terms, convergence criteria (root and ratio tests).

4. Differential calculus in one real variable.
4A. Natural domain. Elementary functions. Even, odd, bounded or monotone functions. Composition, inversion of functions. Graphs and their symmetries: elementary tranformations.
4B. Limits: convergence, divergence, algebra of limits. Indeterminate forms. Limits of special interest: the number "e". Continuity: elementary properties; Bolzano's and Weierstrass' theorems. Inversion of continuous functions.
4C. Derivatives and elasticities, elementary derivatives. Algebra of derivation, chain rule; derivative of inverse functions. Critical points, maxima and minima: Fermat's theorem. Monotonicity and derivative, mean value. Lagrange's theorem and applications. De l'Hospital's rules. Higher-order derivatives: convexity. Plotting graphs. Taylor's (McLaurin's) formula.

5. Differential calculus in two or more variables.
Euclidean spaces: inner product, norm, metrics; bounded, open, closed subsets. Limits and continuity in many variables. Partial derivatives: their geometrical (graphic) meaning. Gradients, Hessian matrices. Critical points, maxima and minima. Constrained extrema and Lagrange's method of multipliers.

6. Integration.
Areas and integrals. Riemann integrals: continuous or monotone functions are Riemann integrable; main properties: linearity, additivity, positivity. Mean value theorem. Primitives and the fundamental theorem of Calculus. Integration by parts and by substitution. Primitives of some elementary or rational functions. Generalized Riemann integrals.
Ordinary first-order differential equations, linear or separable (hints).

Further details in the Italian section; please read carefully the initial caveat about basic requirements.

Readings/Bibliography

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Teaching methods

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Assessment methods

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Teaching tools

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Office hours

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