MATHEMATICS AND EXERCISES-2

The course belongs to the core educational activities on Mathematics, Physics and Statistics. The Course aims to provide the students with theoretical and applied fundamentals about Integral Calculus, basic Differential Equations and Linear Algebra, which are at the basis of all models and tools used in the following quantitative courses of the Environmental Sciences curriculum.

Knowledge and understanding
- First principles on vectorial and matrix calculus;
- Methods for solving definite and indefinite integrals;
- Study of function of two variables;
- Methods for solving differential equations.

Applying knowledge
- Describe a physical problem through a mathematical formulation and identify the suitable tool to solve it;
- Use of vectorial calculus to solve physical and geometrical problems;
- Apply principles of linear algebra and matrix calculus to characterize and solve linear systems of equations;
- Solve indefinite and definite integrals of function of one and two variables;
- Find and characterize stationary points of functions of two variables;
- Solve first order ordinary differential equation;
- Write the differential equation that describes a physical problem.

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

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 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

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

Course material provided through the moodle online learning 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. In any case, students must pass the exams of both modules in the same academic year. Otherwise, any passing grade achieved in Module 1 will be cancelled.

- Front lectures;
- Exercise sessions in class;

Italian

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