69260 - MATEMATICA

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. In particular, students will be able to represent data or functions in graphical form, to apply one-variable and multivariable calculus and to perform operations on vectors and matrices and will know how to use 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

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 with 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.

Complex numbers. Cartesian form and polar form. Nth roots of a complex number. Newton's binomial. Polynomials. Sum and product of polynomials. Euclidean division. Roots of polynomials and factorization. Ruffini's theorem. Linear systems and matrices. Systems of linear equations and set of solutions. Gauss algorithm (Gauss elimination method). Matrix of a linear system and reduction to the reduced step form. Application of the Gauss algorithm for the inversion of matrices. Vector spaces: sum of vectors and multiplication by a scalar. Linear combinations. Linear subspaces. Examples (polynomials, matrices, sequences). Intersection, sum and direct sum of subspaces. Vector subspace generated by a vector family. Free families, parent families. Bases and dimensions. Coordinates in a fixed base. Linear applications. Definition of linear application. Image and Kernel of a linear application. Rank theorem. Injectivity and surjectivity. Matrix of a linear application from a starting base to a destination base. Eigenvalues and eigenvectors of a linear map. Diagonalizable matrices and diagonalizability of endomorphisms. Determinants and their computation.

Readings/Bibliography

• M. Bramanti, F. Confortola, S. Salsa, Matematica per le scienze, Zanichelli, 2024 [https://www.zanichelli.it/ricerca/prodotti/matematica-per-le-scienze]
• M. Abate, Matematica e statistica. Le basi per le scienze della vita. IV edizione. McGraw-Hill, 2023 [https://www.mheducation.it/matematica-e-statistica-4-ed-con-connect-9788838656842-italy]
  

Teaching methods

Frontal lectures accompanied by exercise sessions with tutor.

Assessment methods

The assessment consists of a written examination, followed by an oral examination. The examination is unique for both modules of the course.

The minimum required to pass a written examination is 15. The validity of a written exam is limited to a single examination session. The final mark, on a 30-point scale, is based on both parts of the examination.

Teaching tools

Alma Mathematica [https://almaorienta.unibo.it/it/almamathematica] is an e-learning course (in italian) with the primary scope of improving the basic mathematical background of those students who want to embrace an academical curriculum in a scientific field.

Office hours

See the website of Jacopo Gandini

See the website of Enrico Fatighenti