65926 - Mathematical Institutions 1

Learning outcomes

Upon completion of this course, the student knows the basic tools of mathematical analysis and linear algebra. He/she is able to use mathematical tools for the study of other disciplines.

Course contents

Analysis. Sets. Relations. Maximum, minimum, lower and upper extremes of a set. Functions. Even, odd, periodic, monotonic (increasing, decreasing), injective, suriective functions. Composition of functions. Invertible functions and their inverse. Fundamental functions.

Real functions of a real variable: limits and their theorems, calculation of limits. Continuity of a function and its theorems: Bolzano's theorem on intermediate values, zeros theorem, Weierstass theorem. Discontinuity. Asymptotes.

Incremental ratio. Derivative. Rules for the calculation of derivatives. Points of non-derivability. Continuity and derivability. Fermat's theorem. Rolle's theorem. Lagrange's intermediate value theorem. Monotony test. Search of relative and absolute extremes. Theorem of de l'Hospital. Higher order derivatives. Taylor polynomials and local approximation of functions. Concavity and convexity, bending, study of a function and its graph.

Integral according to Riemann: integrability and integral. Fundamental theorems of integral calculus. Integration of elementary functions. Integration of fraternal algebraic functions, method of simple fractions. Integration by parts. Integration by substitution (or with change of variable).

Complex numbers: algebraic, Cartesian, trigonometric, exponential representation. Operations in C. N-hex roots of complex numbers. Solving equations in C, geometric places.

Linear Algebra. Matrices and their operations. Elementary operations on the rows of a matrix. Gauss reduction method for the rank of a matrix and for the resolution of linear systems. Homogeneous linear systems. Determinant of a matrix. Inverse matrix, Gauss-Jordan method.

Vectors. Vector spaces. Linear dependence between vectors. Generators of a vector space. Base of a vector space. Scalar product. Orthogonality between vectors. Orthogonal basis. Vector product. Eigenvectors. Eigenspaces.

Analytic geometry in space. Equation of a plane, line equation in parametric and Cartesian form. Orthogonality and parallelism between planes and lines. Point to line distance. Point-plane distance.

Readings/Bibliography

• J. Stewart, Calculus - Early Trascendentals, 8th Edition, Cencage Learning, 2016

• Daniele Ritelli. Lectures in Mathematical Analysis 3rd Edition. Esculapio 2019. ISBN: 9788874888870

• Daniele Ritelli, Massimo Bergamini, Anna Trifone, Fundamentals of Mathematics, Zanichelli
  
  
• M. Barnabei,F. Bonetti, Linear systems and matrices, Pitagora Editrice, Bologna

Teaching methods

Classroom lectures with occasional use of video projector. Individual work assignments. Availability of notes with exercises.

 

During lectures, the focus is primarily on the applications of mathematics and statistics, with special emphasis on environmental sciences.

Topics are presented along with many examples and exercises.

 

Pre-course: A summarizing Pre-course of Mathematics is offered for students of interest

Assessment methods

Written test divided into two parts: Part 1 and Part 2, composed of a set of exercises comparable (by type and level of difficulty) with those carried out during classroom exercises and with the supplementary exercises made available by the teacher during the course, in addition one or two questions concerning more theoretical aspects of the course.

Possible further optional oral test.

Students who complete the Pre-course of Mathematics get 3 extra points to the mid-term exam results

Please, consult the dedicated file in Virtuale

Teaching tools

Various material provided in electronic format (exercise sheets, etc.)

Students who need compensatory tools for reasons related to disabilities or specific learning disorders (SLD) can directly contact the Service for Students with Disabilities (disabilita@unibo.it) and the Service for Students with learning disabilities (dsa@unibo.it) to agree on the adoption of the most appropriate measures.

Office hours

See the website of Martin Huska