MATHEMATICAL ANALYSIS 2
Mathematical analysis I, Linear algebra and geometry.
The exam consists of two parts:
a three-hour long written test with 4/5 exercises on the topics discussed in the course in order to verify the level of skills acquired.
An Oral exam to assess the level of knowledge reached. Only students who pass the written test can access the oral exam.
The students who reach a grade of 16/30 or higher are admitted to the oral exam.
During the semester a partial test will be held. The students who reach a grade of 16/30 or higher in the partial test will be exempted from having to solve the corresponding part of the final written test. The admission grade for the oral exam will be the weighted average of the partial test and of the written test. This partial exemption is only limited to the first two exam sessions.
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 differentiability. Partial derivatives, gradient and Jacobian matrix. Sufficient condition for differentiability. 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) Maxima and minima. First order conditions.
9) Hessian matrix and sufficient conditions for extrema.
10) Implicit functions. Local theorem of existence and uniqueness.
11) Extrema with constraints. The Lagrange multipliers method.
12) Parametric curves. Arclength. Integration on curves. Differential forms. Integration of differential forms. Exact and closed forms. Necessary and sufficient conditions.
13) Peano's measure in R^n and measurable sets.
14) Integration on rectangles. Iterated integrals. Riemann integral on measurable sets.
15) First order differential equations. Solutions.
16) Differential Equations of order n with constant coefficients.
Frontal lectures to expose the theoretical contents; exercise sheets to be carried out at home, which will then be solved by both the teacher and the tutor.
The teacher receives the students for clarifications and insights by appointment to be fixed by writing to the email address marco.magliaro@uninsubria.it
Professors
Borrowers
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Degree course in: Physics