58048 - Mathematics with Exercises

Learning outcomes

On successful completion of the course, students will have acquired the basic knowledge of one-variable calculus, vector calculus and linear algebra, the first elements of multivariable calculus, complex numbers and the most elementary methods for solving ordinary differential equations. They will have expertise on numerical methods for solving some classes of mathematical problems. In particular, students will be able to represent data or functions graphically, to apply one-variable and multivariable calculus and to perform operations on vectors and matrices. They will have acquired the knowledge of some basic concepts of scientific computing, such as error analysis, approximation of experimental data, interpolation, numerical integration, nonlinear equations, and systems of linear equations.

Course contents

Real numbers, inequalities, absolute value.
Elementary real functions: power functions, roots, exponential and logarithm, circular and hyperbolic functions and their inverses.
Elements of linear algebra:
Systems of linear equations, coefficient matrix and augmented matrix of a system of linear equations, (Gauss-Jordan) row reduction, rank of a matrix, Rouché-Capelli theorem, solving systems of linear equations by reducing the system to row echelon form (Gaussian elimination), determinant of a square matrix, Cramer's rule.
Vector space structure of R^n, linear dependence and independence of vectors, connection with the rank of suitable matrices, bases of subspaces, dimension of subspaces, linear transformations from R^n to R^m, kernel and image, matrix of a linear transformation, linear transformations from R^n to itself, eigenvalues and eigenvectors, eigenbases, positive definite, negative definite, and indefinite matrices.
Limits and continuity, basic theorems.
Derivatives, basic theorems and applications: tangents to curves, increasing and decreasing functions, convexity, graphs of functions, Taylor's formula.
Integrals in one variable, primitives, integration of rational functions, integration by substitution and by parts.
Ordinary differential equations (ODEs), methods to solve first order ODEs, of linear type or separate variables type, and linear ODEs of higher order whith constant coefficients.
First elements of differential calculus of several variables, partial derivatives, gradient and Hessian matrix, maxima and minima.
Double integrals: geometric meaning, computing double integrals as iterated integrals, change of variables, use of polar coordinates.

Readings/Bibliography

M. Bramanti, C. D. Pagani, S. Salsa : Matematica. Calcolo infinitesimale e algebra lineare. 2a ed., Zanichelli, Bologna, 2004.
S. Salsa, A. Squellati: Esercizi di Analisi matematica 1, 2 (two volumes), Zanichelli, Bologna, 2011.
M.R. Spiegel: Theory and Problems of Advanced Calculus,Schaum's Outline Series, McGraw-Hill, 1974.
P. Negrini: Equazioni differenziali. Pitagora editrice, Bologna, 1999.

Teaching methods

Lessons accompanied by exercise classes with tutor.

Assessment methods

Written and oral examination (both on problem solving).

Teaching tools

Blackboard (and sometimes video projector). Exercises for homework and course material are available at http://www.dm.unibo.it/~achilles/ and http://www.unibo.it/docenti/paolo.negrini.

Links to further information

http://www.dm.unibo.it/~achilles/

Office hours

See the website of Hans Joachim Rudiger Achilles

See the website of Paolo Negrini