## MATHEMATICAL ANALYSIS AND GEOMETRY

Some familiarity with basic pre-university mathematical concepts is required (operations, algebraic equations and inequalities, properties of powers and logarithms, elements of trigonometry) which will be used and refreshed during the course.

Evaluation will take place through a written test, which may possibly be replaced by partial written tests carried out during the course, followed by a mandatory oral test.

In the written test, through the proposed exercises, students will have to demonstrate that they have the basic theoretical and practical knowledge related to the course material. In the oral exam, consisting of a discussion of the written test and simple theoretical questions, including the proofs of the theorems demonstrated in class, students will have to demonstrate that they have logical deductive and argumentative skills, using mathematical language appropriately.

The objectives of the course are not limited to the simple acquisition of calculation tools, but want to emphasize a deeper critical understanding of ideas and the method of mathematical thinking. At the end of the course, the student must therefore have acquired basic knowledge and skills of advanced mathematics, starting from the algebraic and geometric structure of Euclidean space to arrive at the differential and integral calculus for scalar and vector functions of real variables, with the aim of being able to apply this knowledge, critically in other areas of knowledge. In particular, students must be able to: (knowledge and ability to understand) master the algebraic and metric structure of the most used numerical sets in the world of applications (real and complex numbers); know the fundamentals of Linear Algebra and Geometry, knowing how to apply them to the manipulation of vectors and matrices, to the calculation of determinants, to the resolution of linear systems and simple exercises of analytical geometry in space, with particular regard to planes and lines, conics and quadratics, to the study of quadratic forms and to search for eigenvalues and eigenvectors of linear transformations; demonstrate that they have learned the basic concepts of integral and differential calculus for scalar and vector functions of one or more real variables, showing that they are able to use them to solve simple optimization problems, integration of linear differential forms, calculation of vector field flows through surfaces and for the solution of the most common differential equations; (ability to apply knowledge and understanding) through the exercises carried out in the classroom relating to the topics of the program, learn how to apply the abstract knowledge acquired to simple cases and only later be able to connect different concepts in order to solve more complex exercises autonomously and independently; use the mathematical method to break down complex problems into more easily attachable sub-problems; (independent judgment) to evaluate the consistency and correctness of the results obtained and to analyze the appropriate solution strategies for the proposed exercises; (communication skills) learn to use a formally correct language that allows them to communicate both the contents of the program and the logical steps used in solving the exercises with clarity of presentation and thought. Lectures and direct confrontations with the teacher will favor the acquisition by the student of a specific and appropriate scientific lexicon; (learning ability) autonomously deepen their knowledge, starting from the basic ones provided in the course, in order to be able to properly and effectively manage the use of additional mathematical tools and concepts. These will be important both in the remaining courses of the Degree Course and in subsequent training courses.

The course aims to provide students with the basic mathematical elements that allow to face, with the appropriate tools, the subsequent technical / scientific courses.

Introductory: Basics of set theory. Algebraic structures of numerical sets with operations. Real numbers. Complex numbers.

Geometry: Vector spaces, normed spaces. Systems of generators and vector subspaces (Span (S)). Linear independence of vectors, bases and dimension of a subspace. Rn spaces as coordinate spaces. Rn spaces as geometric spaces: free vectors, applied vectors and points in cartesian coordinates. Scalar product, norm, distance, orthogonality, vector projection in Rn; the Cauchy-Schwarz inequality; the triangular inequality; angle between vectors of Rn; the vector product in R3. Matrices and operations between matrices. Inverse matrix. Transpose of a matrix. The determinant of a square matrix. Properties of the determinant. Rank for minors. Linear systems and matrices. Reduced matrices and systems. Set of solutions of a reduced system. Solvability of linear systems: Rouché-Capelli theorem and Gauss algorithm. Linear maps and associated matrix: image subspace and kernel of a linear map, Eigenvalues and eigenvectors. Parametric and Cartesian equations of lines and conics in the Cartesian plane R2, planes and quadratics in the Cartesian space R3. Orthogonality and parallelism between lines and planes. Skewed lines. Quadratic forms and associated matrix. Diagonalization and canonical form of a quadratic form.

Mathematical Analysis: Lower and upper bound, maximum and minimum of a scalar function.The limit operation: heuristic motivation, rigorous definition and its properties. Indeterminate, infinite and infinitesimal forms. Asymptotic behaviors. Continuous functions. Operations between continuous functions. Zeros theorem and study of the sign of a scalar function. Weierstrass theorem for scalar functions. Directional derivative of a scalar function. Gradient vector and Jacobian matrix. Differentiability for scalar and vector functions. Mean value theorem (or Lagrange's theorem) for scalar functions. Relationships between continuity and differentiability; examples of non-differentiable functions. Derivation of elementary functions and rules of derivation. Search for points of relative and absolute minimum and maximum for scalar functions of one real variable: link between the monotony of a function and the sign of its first derivative, convexity of a function and the sign of its second derivative. Main differential operators: curl, divergence, Laplacian. Indefinite integration of linear differential forms and search for scalar potentials. Elementary integrals. Integration rules. The line integral of a vector field and the Fundamental Theorem of Integral Calculus. The double and triple integral. The surface integral for scalar fields and flow for vector fields. Green's formulas, Stokes' theorem and divergence theorem. Improper integrals. Ordinary differential equations (ODE): separable variables and linear variables. The Cauchy problem. The Laplace transform method for linear ODEs with constant coefficients.

Materials produced relating to lectures is largely sufficient to take the exam with complete success.

As a complement to the course, the following texts are recommended:

E. Acerbi, G. Buttazzo: Basic pre-university mathematics, Pitagora Editrice, Bologna, 2003.

E. Acerbi, G. Buttazzo: First Course in Mathematical Analysis, Pitagora Editrice, Bologna, 1997.

Abate M., Geometry, McGraw - Hill, 1996

Courant R., Robbins H., What is Mathematics? Boringhieri (TO), 1961

Nelsen R.B., Proofs without Words I and II, Classroom Resource Materials, The Mathematical Association of America, 2000

Each of them deals with different aspects of the course material; please contact the teacher for suggestions and advice in this regard, and for English written materials.

The course consists of 12 credits which correspond to 96 hours of lessons. Given the blended modality with which the course is delivered, learning objects on the topics from a formal point of view will be made available in advance, in video and/or paper format, in order to use the lesson time for clarification alternating with significant examples, applications and exercises. The course will give particular emphasis to the application and calculation aspects, without neglecting the rigorous theoretical aspects, aimed at a greater understanding of the phenomena at stake. In order to foster a systematic, deep and concrete understanding of the course topics, lecture notes with exercises to be carried out in parallel with the study of the theoretical topics will be made available on the Moodle portal. The detailed program of the topics covered in the classroom will also be uploaded on a weekly basis, in support of both attending and non-attending students. This program will ultimately constitute the index of contents in view of the preparation for the final exam.

Contacts:

Department of Chemistry and Pharmacy

Via Piandanna 4 Sassari

Tel. 079 229486; Cell. 347 116 1591; Fax 079 229482; e-mail: pensa@uniss.it

Supervision hours: Please contact the teacher by @mail. Available to offer individual assistance in a foreign language to incoming students (English). Materials prepared by the teacher will be made available to students in the online Moodle platform and in a dedicated DropBox file. Available to accept examination of incoming students also in foreign language (English).

In order to acquire greater skills in the topics considered as basics and improve understanding of the arguments

presented in the course, the use of the following website is recommended: https://www.khanacademy.org/