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Subject: ADVANCED CALCULUS (A.A. 2021/2022)

degree course in PHYSICS

Course year 2
Teaching units Unit Complementi di analisi matematica
Mathematics and Information Technology (lesson)
  • TAF: Basic compulsory subjects SSD: MAT/05 CFU: 6
Teachers: Maria MANFREDINI
Exam type oral
Evaluation final vote
Teaching language Italiano
Contents download pdf download




The lectures aim at giving to the students suitable knowledge and skills about integration theory on spaces of dimension Greater than one and integration on manifolds.

Admission requirements

Basic calculus in one and several variables, basic linear algebra

Course contents

Teaching takes place in the first semester of the second year, for a total of 48 hours of frontal teaching including theory and exercises.
After each topic you will find an indication in terms of teaching hours.
It is to be understood as purely indicative and could
undergo changes in the course of teaching that takes into account the feedback
of the students

Lebesgue's measure on R^n (10 ore)

Lebesgue's integral on R^n
Definition and properties of integrable functions.
Comparison in R between the Riemann integral and the Lebesgue integral.
The reduction theorem and the variable change theorem for the integral. (12 ore)

Curvilinear integrals of I and II species.
Definition of a regular curve.
and of rectifiable curve, and of oriented curve.
The Gauss-Green theorem in the plane. Application: calculation of the area of ​​a subset of R^2. (8 ore)

Vector fields or differential forms
Definition of conservative vector fields. Fundamental theorem of the work of a conservative field. Sufficient condition for a closed field to be conservative. (8 ore)

Differentiable varieties (surfaces) (10 ore)
Definition of smooth surface with border. Area of ​​a surface. Surfaces oriented with edge. Flow of a vector field through a surface. The divergence theorem and the Stokes's theorem.

Teaching methods

Lectures aimed at the presenting both the theoretical aspects of the topics and their applied aspects, illustrated by several examples and exercises.

Assessment methods

The final exam consists of a written test and an oral test. The written exam requires the resolution of3/4 exercises similar to those proposed during the lesson and which will be evaluated based on the presentation, clarity and depth of reasoning. Access to the oral exam is allowed only to those who have a sufficient written test. The oral test consists of at least two questions that concern the whole theory. Moreover, during the course problems will be proposed, to be carried out personally or in small groups (maximum 4 people) and with care, to be delivered and will contribute to the final evaluation with the modalities that will be explained in the classroom. Passing the exam will be guaranteed to students who will demonstrate mastery and operational capacity in relation to the key concepts illustrated in the course. A higher score will be given to students who demonstrate that they have understood and are able to use all the contents of the course, illustrating them with language skills and solving even complex problems. The tests could be carried out in presence or remotely depending on of the evolution of the COVID19 health emergency situation.

Learning outcomes

1) Knowledge and understanding: At the end of the course the student will have the basic knowledge of integral calculus in several real variables.
2) Ability to apply knowledge and understanding: At the end of the course the student will be able to apply the techniques and knowledge acquired to the study of problems related to integral calculus.
3) Independent judgment: at the end of the course the student will have perfected their ability to manage theoretical arguments and recognition of formal correctness.
4) Communication skills: at the end of the course the student will be able to orally present the topics presented in the course with an appropriate technical language and a correct mathematical formalism.
5) Learning skills: the tools assigned during the course will allow to improve the learning ability and deepen in an autonomous way topics associated with those presented during the course.


Pagani-Salsa: "Analisi Matematica 2" ed. Liguori

Lanconelli " Lezioni di Analisi Matematica 2, prima e seconda parte" ed. Pitagora

Wheeden, Zygmund "Measure and integral: an introduction to real analysis" Chapman & Hall/CRC Pure and Applied Mathematics

Nelson, "A User-Friendly Introduction to Lebesgue Measure and Integration" ams

Bramanti: “Esercitazioni di analisi matematica 2” ed. Esculapio