75376 - Calculus

Learning outcomes

At the end of the course the student has comprehensive, in-depth coverage of the topics required in a one-term course in advanced calculus. The course is thought for students who have had a course in introductory algebra and undergraduate calculus. As for calculus, the student acquires a solid foundation in several variables calculus, vector analysis, linear algebra, systems of differential equations, vector fields, extremum problems. Furthermore, learns how advanced calculus can model and solve authentic real-world problems: thus tapping heavily into applied mathematics.

Course contents

Preliminaries

Topologies and Metric Spaces

Measuring Sets

The Lebesgue Integral

Measurable Functions

Spaces of Continuos Functions

Lebesgue Spaces

Absolute Continuity

Differential Equations

Readings/Bibliography

 

Recommended texts:

[1] A Probability Path

Sidney Resnick

https://www.amazon.it/Probability-Path-Sidney-I-Resnick/dp/0817684085

[2]

Measures, Integrals and Martingales

Rene' Schilling

https://www.amazon.it/gp/product/1316620247/ref=ox_sc_saved_title_4?smid=A11IL2PNWYJU7H&psc=1

 

[3] Probability 1

Albert Shiryaev

https://www.springer.com/it/book/9780387722054

Teaching methods

Traditional black-board based classes and every lecture will be made available through online platforms.

Assessment methods

Written ( and optional oral ) exams. The written exam is articulated in a series of 6 exercises each with a maximum grade of 5 points. Every exercise attains to elements of the syllabus and the relevant Bibliography covered during the course lectures or otherwise hinted at during classes. The (optional ) oral exam may cover the same range of topics as well, while keeping into accont the candidate's perfomance in the previously held written test.

Some more details may be in order:

[1] Grades are expressed on a scale from 0 to 30 cum Laude (30L), where 18
is the passing threshold.

[2] It is highly recommended that students attend to classes, where a
number of exam-like problems will be solved and presented completely.

[3] Written exams may last from 1 to 3 hours,
depending on the stage and the session they are attached to. Written
exams are open books.

[4] Written exams will be mainly made up by a
collection of exercises, some of them basic run of the mill stuff,
some more challenging.

[5] The method of execution, the precision of
presentation and, needless to say, the correctness and the accuracy of
the results will constitute the main factor in establishing the grade.

[6] The optional oral exam will be graded in a similar manner and an
arithmetic mean of the (i.e. written and oral) grades will yield the
final mark.

Teaching tools

Supplementary notes and material may be suggested and handed out during the course.

Office hours

See the website of Enrico Bernardi