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 1-, 2-, 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
o Linear combinations of vectors
o Linearly dependent vectors
o Vector basis

• Matrices and Linear systems
o Matrix product
o Linear systems
o Matrix determinant
o Cramer Theorem
o Eigenvalues and eigenvectors

• Vector derivatives

• Operators
o Gradient
o Divergence
o Rotor

CALCULUS

• Integration
o Integral definition
o Integral properties
o Fundamental theorem of calculus
o anti-derivatives

• Multi-variable functions
o Partial derivatives
o Limits

• Differential equations
o First order linear differential equations.
o Separable equations
o Examples of multi-variable differential equation

Course material provided through the moodle online learning platform

The written exam consists of exercises and questions aimed at evaluating the comprehension of the course content (duration: 120 minutes). The exam is closed-book. Samples of past exams are available in the course material.

- Front and online lectures;
- Exercise sessions in class;
- Exercise sessions using smartphone applications allowing student-teacher interactions;
- Self-evaluation tests in moodle online learning platform

Italian

written