COURSE LANGUAGE: English

YEAR OF THE DEGREE PROGRAMME (I, II, III): I

SEMESTER (I, II, ANNUAL): II

CFU: 9

REQUIRED PRELIMINARY COURSES (IF MENTIONED IN THE COURSE STRUCTURE “REGOLAMENTO”)
Calculus I.

PREREQUISITES (IF APPLICABLE)
None.

LEARNING GOALS
To provide the fundamental concepts, in view of the applications, relating to the differential and integral calculus for real functions of several real variables, and to ordinary differential equations; do acquire conscious operational skills,

EXPECTED LEARNING OUTCOMES (DUBLIN DESCRIPTORS)

Knowledge and understanding
The students will have to demonstrate knowledge of the notions (definitions, statements, proofs if provided by the program) related to infinitesimal, differential and integral calculus for real functions of several real variables and the developed calculation tools and be able to understand related topics by elaborating the notions acquired. 

Applying knowledge and understanding
The students must prove that they are able to apply what they have learned in solving exercises developed by the teacher, related to topics such as: sequences and series of functions, limits and graph of functions of several variables, multiple integration, ordinary differential equations and Cauchy problems.

COURSE CONTENT-SYLLABUS

• (0.5 cfu) Complex numbers. Definition and properties. Sum and product operations. Algebraic form and trigonometric form. Powers and roots of a complex number. Euler's formulas, exponential form.
• (1 cfu) Sequences and series of functions. Pointwise and uniform convergence; pointwise and uniform Cauchy convergence criteria. Theorems on the continuity of the uniform limit, of passage to the limit under the sign of integral and derivative. Absolutely convergent and totally convergent series; Cauchy criteria for series; total convergence and uniform convergence. Continuity theorems of the uniform sum of a series, of integration by series and derivation by series. Power series. Taylor series: developability and remarkable developments. Analytic functions.
• (2.5 cfu) Differential calculus for functions of several variables. Elements of topology. Euclidean distance; definition of neighborhood. Internal, external points, boundary points. Open and closed sets; cluster points and isolated points. Bounded sets; Bolzano-Weierstrass theorem. Compactness and characterization of compacts. Convexity and connected sets. Functions of several variables: limits, continuity and relative properties; Weierstrass theorem. Partial derivatives; differentiability and total differential theorem; directional derivatives and gradient; derivation of composite functions. Functions with null gradient in a connected domain. Higher order derivatives and Schwarz's theorem. Lagrange's theorem. Taylor's formula of the first and second order. Relative extrema: necessary condition of the first order. Relative extrema of functions of two variables: necessary condition of the second order, sufficient condition of the second order. Search for absolute maxima and minima of continuous functions in compact sets of the plane. Relative extrema of functions of three variables: sufficient conditions. Positively homogeneous functions, Euler's theorem.
• (0.5 cfu) Implicit functions. Local equivalence of a plane curve with a graph. Dini's theorem for equations of the type f (x, y) = 0. Constrained maxima and minima of functions of two variables. Lagrange multiplier theorem.
• (0.5 cfu) Curves. Regular and piecewise regular curves: tangent line; oriented curves. Length of a curve, rectifiability of regular curves. Curvilinear abscissa. Curvature of a plane curve. Curvilinear integral of a function.
• (1 cfu) Multiple integrals. Double integrals on normal domains. Integrability of continuous functions. Reduction formulas for double integrals. Change of variables in double integrals. Triple integrals; reduction formulas; change of variables. Solids of rotation and Guldino's Theorem.
• (1 cfu) Surfaces. Smooth surfaces: tangent plane; orientable surfaces; surfaces with boundary; closed surfaces. Area of a surface. Surfaces of rotation and Guldino's Theorem. Surface integral of a function. Flow integrals of a vector field. Divergence theorem in R3.
• (1 cfu) Linear differential forms. Exact differential forms and conservative fields. Curvilinear integral of a linear differential form. Integration criterion of differential forms. Closed differential forms. Poincaré's lemma. Radial forms. Homogeneous forms. Gauss-Green formulas in the plane. Divergence theorem in the plane. Stokes formula in the plane. Differential forms, closed in simply connected open of the plane. Differential forms in space. Irrotational fields. Stokes formula in R3. Differential forms closed in simply connected open spaces of space.
• (1 cfu) Differential equations. Cauchy problem for differential equations of order n: local and global existence and uniqueness theorems. General integrals; singular integrals. Linear differential equations of order n: theorem on the general integral of a homogeneous equation, Wronskian theorem, theorem on the integral of a complete equation. Linear equations of the first order; linear equations with constant coefficients. Method of variation of constants. Equations with separable variables. Equations of the form y '= f (y / x). Bernoulli equations. Equations of the form y '' = f (x, y ').

READINGS/BIBLIOGRAPHY
Check the teacher's website.

TEACHING METHOD
The lessons will be face to face, and about one third of the lessons will be of exercises.

EXAMINATION/EVALUATION CRITERIA 

Exam type:

• Written and oral.

In case of a written exam, questions refer to:

• Numerical exercises.
• Open answers.
• Multiple choice answers.

Evaluation pattern: 
The grade is formulated by the Examination Commission on the basis of the outcome of written test, in particular on the basis of the consistency and accuracy of the exercises performed and the adequacy of the answers provided by the student to the theory questions. The final grade is also suitably motivated to the student.