Unit MATHEMATICAL ANALYSIS
- Course
- Food science and technology
- Study-unit Code
- GP000938
- Curriculum
- Tecnologie agro-alimentari
- Teacher
- Luca Zampogni
- Teachers
-
- Luca Zampogni
- Hours
- 54 ore - Luca Zampogni
- CFU
- 6
- Course Regulation
- Coorte 2022
- Offered
- 2022/23
- Learning activities
- Base
- Area
- Matematiche, fisiche, informatiche e statistiche
- Academic discipline
- MAT/05
- Type of study-unit
- Obbligatorio (Required)
- Type of learning activities
- Attività formativa monodisciplinare
- Language of instruction
- Italian
- Contents
- Integrals and first order differential equations. Linear systems of the first order differential equations. Exponential and logistic models for growth. Vectors, matrices, systems of linear equations. Eigenvalues and eigenvectors of matrices. Functions of two real variables, gradient, directional derivatives, tangent plane and linear approximation .
- Reference texts
- 1. James STEWART CALCOLO Funzioni di una variabile, Maggioli Editore 2013.
2. James STEWART CALCOLO. Funzioni di più variabili", Maggioli Editore 2013. (Original english language edition: Calculus-Concepts and Contexts, 2nd edition.) 3.Lecture notes available online. - Educational objectives
- Knowledge:
1. Integrals and applications.
2. The concept of differential equation. Examples and main properties of the Cauchy problem.
3. Malthus, Verhulst and Gompertz models.
4. Matrices and systems of linear equations. The Rouche-Capelli Theorem.
5. Geometric transformations. Eigenvalues and Eigenvectors of a matrix.
6. Elements of multivariable calculus. Partial derivatives, Gradient, directional derivatives and applications.
Ability: 1. To define and analyze simple mathematical models solvable by means of integrals and differential equations.
2. To compute definite integrals.
3. To solve differential equations with the method the separation of variables. To solve linear and Bernoulli differential equations.
4. To discuss linear systems.
5. Topics related to multivariable functions: traces, level curves, gradient, tangent plane and linear approximation. - Prerequisites
- In order to be able to understand and reach the objectives of the course of Mathematical Analysis, the student must have gained all the knowledge and abilities related to topics of the course program of Mathematics.
- Teaching methods
- The course is organized as follows: lectures on all subjects of the course, classroom exercises to prepare students for the written exams. It is planned a tutor teaching activity.
- Other information
- Attendance: optional but strongly recommended
- Learning verification modality
- The exam is made of both a written and oral test. The written test consists of the solution of four problems and has a duration of at most three hours. Its objectives are the following: - The understanding of the proposed problems;
- The handling of mathematical instruments;
- The interpretation of the results obtained. The oral test consists of a talk of about 30 minutes and is aimed at testing the degree of comprehension reached by the student and his skills in handling mathematical objects, with particular attention to his capacity of finding connections between the topics explained. - Extended program
- Integrals and ordinary differential equations.
Indefinite integrals. Integration by substitution and by parts. Numerical integration. Ordinary differential equation: terminology and notations, solution, order. Initial value problems. First-order equations: separation of variables. The Malthusian growth model. First-order linear equations. The Bernoulli equation. The Verhulst model (the logistic growth model). The Gompertz model. Systems of the first order differential linear equations.
Introduction to Linear Algebra.
Matrices, operations with matrices, determinant and inverse of matrices. The rank of a matrix. Systems of linear equations. Cramer's rule and Rouché-Capelli Theorem.
Real three-dimensional space and Cartesian coordinate system, coordinate axes and coordinate planes. Distance between two points in three-dimensional space. Lines and planes in three-dimensional space. Vectors in 3-D space. Linear independence. Linear transformations (rotation, reflections) and translationn (or "shift"). Eigenvalues and eigenvectors of matrices.
Functions of more real variables.
Functions of two real variables: domain, range, graph. Level curves. Quadratic surfaces. Partial derivatives, gradient. Directional derivatives. Higher order partial derivatives. Tangent plane and linear approximation.