Introduction

ANA-III is a course on analysis of functions of more than one variable. This section of the course will be held in English (a Slovenian section is available separately). Topics include: metric spaces and continuity, functions of several variables, partial derivatives and Jacobians, and integration of real functions of more than one variable.

Prerequisite

Analysis I and II, and Algebra I and II.
Note that multivariate analysis makes a fair bit of use of linear algebra (hence the algebra prerequisite).

Textbook and literature

The textbook for this course will be Real mathematical analysis, by Charles Pugh. This clearly-written book does a great job of covering metric spaces. For differentiation and integration it is missing some topics, so we will supplement with other texts such as Serge Lang's Undergraduate analysis, and Walter Rudin's Principles of mathematical analysis. Two more books that you may like are Protter and Morrey's Intermediate calculus, and James Callahan's Advanced calculus: a geometric viewpoint.

At various times in the semester, we may supplement the textbook with handouts or course notes posted to the e-classroom.

Topics

We'll begin by discussing the topology of metric spaces in some detail, including connectedness and compactness. The main example of interest for us will be Rⁿ, but we'll also examine other metric spaces. We'll go on to discuss the calculus and analysis of functions of several variables. We'll define differentiability and partial derivatives of functions Rⁿ → R^m, discuss their relationship, and give applications. We'll go on to discuss double integrals and the extension to triple (and higher) integrals.

At the end of the course, you should have a good understanding of the theory of analysis of several variables. You should also gain some facility with the calculus of several variables.

Exams and grading

Homeworks will be posted weekly to the e-classroom. Homework assignments will comprise 20% of your grade. The remaining 80% of your grade will be determined by exams.

There will be 2 equally weighted midterm examinations (kolokviji). The first will be roughly halfway through the semester, and the second will be around the last week of the semester, at mutually convenient times to be determined. I highly recommend that you study for and take both midterms, even if you don't feel quite ready for them.

You may replace your midterm exam grades with your grade on the final examination (izpit). If you are highly successful on the midterm exams, you need not take the final. (If you are successful on one midterm exam but not the other, you may make up the exam portion of the grade with 30% of the successful midterm and 50% of the final.)

In equation form, your grade will be determined by

20%(homework) + max( 40%(midterm1) + 40%(midterm2),  30%(midterm1_or_2) + 50%(final),  80%(final) ).

In accordance with university policy, there will be at least 4 opportunities to take an instance of the final. However, you may only take the final once more after you have attained a passing grade.

Midterm and final exams will be in-person.

You may not use a calculator or other computational aid, but you may bring one hand-written A4-sized sheet of paper with you to each exam.

Pandemic adjustments

I do not expect the lingering effects of the COVID-19 pandemic to affect us very much. We will follow university regulations and guidance. Please do your part in keeping everyone healthy.

Collaboration

I expect much of your learning to take place in working out the homework problems. While you may collaborate with other students on the homework, I expect you to have thought hard about the problems on your own first. If you do collaborate substantially, then you should so indicate on your homework paper.
Your write-up should in any case represent your own solutions, written in your own words. Copying the solutions from another student or from an internet source is a form of academic dishonesty, and will not be tolerated.

Obviously, exams will be strictly your own work.