00674 - Mathematics

Learning outcomes

At the end of the course, the student is able to think in an analytic-mathematical way, and possesses enough background for a conscious application of mathematics. In particular, the student is able to: -study real functions of one real variable -compute derivatives and integrals -approximate a function with polynomials -apply basic results about systems of linear equations, euclidean vector spaces of tuples of real numbers, matrix algebra over real numbers.

Course contents

First part. (Daniele Morbidelli) Real functions of one real variable; powers, exponential, logarithm, sine and cosine. Limits. Indeterminate forms. Monotone functions. Global and local minima and maxima. Definition of derivative. Computation of derivatives. Derivative of a function and straight line tangent to the graph of the function. Rolle's and Lagrange's theorems, De L'Hopital's rule. Link between derivative sign and monotony. Taylor's formula of the second order. Integrals. Geometric meaning of integrals. Fundamental theorem of calculus. Torricelli's formula. Computation of elementary primitives. Integration by parts and by substitution. Definition of generalized integrals.

Other informations will be available at: http://www.dm.unibo.it/~morbidel/rimini.html

Second part. (Francesco Regonati) Linear systems of m equations in n unknowns; geometric interpretation for n=2,3. Matrices associated to a linear system; triangularization process. Non-singular matrices. Matrix multiplication; matrix representation of a linear system. Non-singular matrices as invertible matrices. Gauss-Jordan algorithm. Eigenvectors and eigenvalues; application to computation of matrix powers. Matrix algebra. Determinants, Laplace's expansions, characteristic properties. Cramer's rule. Non-singular matrices as matrices with non zero determinant. Eigenvalues and characteristic polynomial. Vector space R^n; geometric interpretation for n=2,3. Vector representation of a linear system. Linearly independent sets and bases for R^n. Euclidean inner product, norm and orthogonality in R^n; geometric interpretation for n=2,3. Pythagora's theorem. Fourier coefficients; orthogonal projection of a vector on a subspace. Least-squares solution of a linear system.

Other informations will be available at: http://www.dm.unibo.it/~regonati/rimini.html

Readings/Bibliography

Teaching material will be available in the web pages

http://www.dm.unibo.it/%7Emorbidel/rimini.html

http://www.dm.unibo.it/~regonati/rimini.html

Assessment methods

Written and oral exams.

Office hours

See the website of Francesco Regonati

See the website of Daniele Morbidelli