MATHEMATICAL ANALYSIS 2
- Overview
- Assessment methods
- Learning objectives
- Contents
- Full programme
- Bibliography
- Teaching methods
- Contacts/Info
Mathematical analysis I, Linear algebra.
The exam consists of two parts
Written exam: duration 3 hours with exercises (4/5) on the topics discussed in the course in order to verify the level of skills acquired.
Oral exam: after passing the written test to assess the level of knowledge reached
The course is the natural continuation of the first course in Mathematical Analysis and it aims to expand the study of classical and modern analysis.
At the end of the course, the student will be able to:
1. Understand the methods of mathematical analysis,
2. state and prove the main theorems
3. solve exercises, also of a theoretical nature, related to the topics covered.
4. independently demonstrate the results linked to those presented during the course.
1) Metric spaces, complete metric spaces, sequentially compact sets and their properties, continuous functions.
2) Contraction theorem.
3) Normed spaces, linear operators between normed spaces. Equivalent norms.
4) Functions from R ^ n in R ^ m. Continuity and differentiation. Partial derivatives, gradient and Jacobian matrix. Sufficient condition for differentiation. Chain rule.
5) Mean value theorem. Functions with null differential on connected sets.
6) Second partial derivatives.
7) Taylor’s theorem in several variables.
8) Implicit functions. Local theorem of existence and uniqueness.
9) Maximum and minimum. First order conditions.
10) Hessian matrix and sufficient conditions for extrema.
11) Extrema with constraints. The Lagrange Multipliers method.
12) Peano measure in R ^ n and measurable sets.
13) Integration on rectangles. Iterated integrals. Riemann integral on measurable sets.
14) First order differential equations and systems. Solutions.
15) The local existence and uniqueness theorem for the Cauchy problem.
16) Maximal solutions.
17) Sufficient conditions for existence on a interval.
18) Differential Equations of order n. Linear equations of order n. Independent solutions and solution space.
19) Linear equations of order n with constant coefficients.
1) Metric spaces, complete metric spaces, sequentially compact sets and their properties, continuous functions.
2) Contraction theorem.
3) Normed spaces, linear operators between normed spaces. Equivalent norms.
4) Functions from R ^ n in R ^ m. Continuity and differentiation. Partial derivatives, gradient and Jacobian matrix. Sufficient condition for differentiation. Chain rule.
5) Mean value theorem. Functions with null differential on connected sets.
6) Second partial derivatives.
7) Taylor’s theorem in several variables.
8) Implicit functions. Local theorem of existence and uniqueness.
9) Maximum and minimum. First order conditions.
10) Hessian matrix and sufficient conditions for extrema.
11) Extrema with constraints. The Lagrange Multipliers method.
12) Peano measure in R ^ n and measurable sets.
13) Integration on rectangles. Iterated integrals. Riemann integral on measurable sets.
14) First order differential equations and systems. Solutions.
15) The local existence and uniqueness theorem for the Cauchy problem.
16) Maximal solutions.
17) Sufficient conditions for existence on a interval.
18) Differential Equations of order n. Linear equations of order n. Independent solutions and solution space.
19) Linear equations of order n with constant coefficients
W. Rudin, Principles of Mathematical Analysis, Mc Graw Hill.
De Marco, Analisi Due, Zanichelli
E. Giusti, Analisi Matematica 2, Boringhieri
Frontal lessons (presentation by projection or blackboard) and exercises to be carried out at home, which, in general, will be corrected by both the teacher and the exerciser.
The teacher receives the students for clarifications and insights by appointment to be fixed by writing to the institutional email address.
Professors
Borrowers
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Degree course in: Physics