MATHEMATICS AND EXERCISES-2

The course belongs to the core educational activities on Mathematics, Physics and Statistics. The course is designed to develop and consolidate knowledge and skills in the theoretical and applied foundations of integral calculus, elementary Differential Equations and Linear Algebra, which are at the basis of all models and tools used in various quantitative courses of the Environmental Sciences curriculum. This course contributes to the achievement of the program's educational objectives, particularly the development of mathematical skills for the analysis of environmental phenomena.

Regular and active participation in the teaching activities offered by the Course, together with independent learning activities, will enable students to:
1. (knowledge and understanding)
1.1 acquire knowledge regarding basic mathematical elements of the continuum and understand deductive reasoning.
1.2 acquire knowledge concerning the main vector operators.
1.3 acquire knowledge about the fundamentals of Calculus 2.and Linear Algebra.
1.4 acquire knowledge about the fundamentals of Linear Algebra.

2 (applying knowledge and understanding)
2.1 be able to recognize and solve simple ordinary differential equations.
2.2 be able to apply the fundamental elements of differential calculus in the study of two varible functions.
2.3 be able to apply the fundamental elements of integral calculus to solve indefinite and definite integrals.
2.4 be able to set up and solve a linear system.

3. Judgment skills
3.1 Ability to express a physical problem using mathematical notation and identify the most suitable tool for solving it.
3.2 Ability to formulate hypotheses for setting up a linear system based on a real problem.
3.3 Ability to check the accuracy of the mathematical model in relation to the constraints of a real problem.

4. Communication skills
4.1 Ability to communicate the results of a quantitative study clearly and effectively, both in writing and orally.
4.2 Ability to interact with the teacher and other students during lessons and exercises.

5. Learning skills
5.1 Ability to take notes and interact with the teacher through active participation in lectures and during exercises.
5.2 Ability to assess one's progress in learning the subject at various stages through self-assessment tests.

Good theoretical and operational knowledge of the fundamentals of Algebra, Geometry and Trigonometry for high school and of Differential Calculus (limits, derivatives) dealt in module 1 of the course.
If case of knowlegde gaps, it is useful to attend the Basic Mathematics course.

LINEAR ALGEBRA

• Vector Space

• Vectors
o Vectors in the 2 or 3-dimensional space
o Vectors analytical and geometrical representation
o Sum and difference among vectors
o Vectors in the n-dimensional space
o Vector modulus
o Vector products


• Matrices and Linear systems
o Matrix product
o Linear systems
o Matrix determinant
o Rank of a matrix
o Cramer Theorem
o Eigenvalues and eigenvectors



CALCULUS

• Integration
o Integral definition
o Integral properties
o Fundamental theorem of calculus
o anti-derivatives
o Integration by parts
o Integration by substitution.

• Multi-variable functions
o Partial derivatives
o Stationary points
o Hessian matrix
o Characterization of the extremal points

• Double integrals
o Integraztion on rectangles;
o Integration on trapezoids.

• Differential equations
o First order linear differential equations.
o Separable equations
o Cauchy problem

We recommend consulting the text: Mathematics for Science - With Fundamentals of Probability and Statistics by Bramanti, Confortola, and Salsa, Zanichelli, 2024.
A wide range of materials will be made available through the Moodle online teaching platform.

Examination Module 2: it consists of a written test (duration: two hours) with open-ended problems aimed at verification of all the course contents in order to evaluate the ability of students in solving differential equations, analyzing extremal points of a two-variables function, computing one or two dimensional integrals, solving simple linear systems and basic problems related to space vectors. No mid-term exams are planned. During the written test, the students are allowed to consult theory notes. The grade is given by the sum of the scores assigned to the single problems: some problem consists of some questions whose score is proportional to the intrinsic difficulty. The exam will be passed if and only if the minimum grade is 18. In the moodle page of the course, some written tests of the previous years will be posted.

Indications for the final grade.
Only the students that already passed the exam of Module 1 can take the exam of Module 2. It is possible to take both the modules 1 and 2 in the same day, in this rigorous succession. The final grade of Calculus is the average of the grades of the two modules, with approximation by excess. The maximum grade of Calculus, 30 summa cum laude, is assigned with the full agreement of both the teachers only. The grade of the first module is valid until the second module is passed: however, students are strongly advised to pass both modules in the same academic year.

written

Assessment grid:

A. range 18-22
- sufficient knowledge and understanding of the program;

B. range 23-26
- fair knowledge and understanding of the program;
- fair rigor in conducting the exercises;

C. range 27-30
- good knowledge and understanding of the program,
- good rigor in conducting the exercises;

D. honors will be awarded in the presence of excellent knowledge and understanding, high mathematical rigour and outstanding presentation skills.

- Front lectures;
- Exercise sessions in class;
Lectures and classroom exercises aim to develop theoretical knowledge and practical skills in solving applied mathematical problems.
Teaching materials are made available on Moodle to support individual study and independent learning.