# Infinitesimal Calculus IV (MAY411)

### General

School School of Science Department of Mathematics Undergraduate MAY411 4 Infinitesimal Calculus IV Lectures (Weekly Teaching Hours: 5, Credits: 7.5) General Background - Language of Instruction (lectures): Greek Language of Instruction (activities other than lectures): Greek and English Language of Examinations: Greek and English Yes (in English) See eCourse, the Learning Management System maintained by the University of Ioannina.

### Learning Outcomes

Learning outcomes Here, the acronym VFomV stands for Vector Function of multiple Variables. Remembering: The concept of the integral of VFomV. Basic properties of this integral. The concept of improper integral of VFomV. Basic properties of this integral. The concept of integrals of VFomV over paths and surfaces. Basic properties of this integral. The concepts of vector field and gradient field. The concepts of the sequence of VFomVs, of uniform convergence of such sequences, of the series of such sequences and of the Fourier series. Comprehension: Integration of VFomV over a rectangle and over an elementary region. Changing the order of integration. Integration over vector fields and gradient fields. The Stokes, Green and Gauss Theorems. Applying: Finding length of path, area of elementary region, volume of solid body. Finding curvature of surfaces and minimal surfaces. Conservative fields and their applications in Physics. Study of liquid fluids and study of waves. Differential forms and their applications in Differential Geometry. Evaluating: Teaching undergraduate and graduate courses. Creative, analytical and inductive thinking. Required for the creation of new scientific ideas. Working independently. Working in groups. Decision making.

### Syllabus

Definition of multiple integral using lower and upper sums over closed rectangles, set of zero volume, Lebesgue Criterion for Riemann Integrability, Jordan measurable sets and the definition of the integral over such sets, Fubini Theorem, Cavalieri Principle, elementary regions in two and three dimensional spaces, change of variables and their basic applications, evaluation of integrals using the aforementioned methods. Definition of integrals over paths for parametrizes functions an vector fields, definition of path length, parametrizes paths, parametrized transformations, gradient fields and path independent integrals, Green Theorem. Surfaces and parametrization of surface integrals. Definition of surface integral for real functions and for vector fields. Area of surface. Stokes and Gauss Theorems. Uniform convergence of function’s sequences and series. Fourier series.

### Teaching and Learning Methods - Evaluation

Delivery
• Lectures in class.
• Teaching is assisted by Learning Management System.
• Teaching is assisted by the use of online forums where students can participate in order to improve their problem solving skills, as well as their understanding of the theory they are taught.
• Teaching is assisted by the use of pre-recorded videos.
Use of Information and Communications Technology
• Use of Learning Management System, combined with File Sharing and Communication Platform, for
1. distributing teaching material,
2. submission of assignments,
3. course announcements,
4. gradebook keeping for all students evaluation procedures,
5. communicating with students.
• Use of Web Appointment Scheduling System for organising office appointments.
• Use of Survey Web Application for submitting anonymous evaluations regarding the teacher.
• Use of Wiki Engine for publishing manuals regarding the regulations applied at the exams processes, the way teaching is organized, the grading methods, as well as the use of the online tools used within the course.
Teaching Methods
Lectures (13X5) 65
Study and analysis of bibliography 100
Preparation of assignments and interactive teaching 22.5
Course total 187.5
Student Performance Evaluation

Language of evaluation: Greek and English.
Methods of evaluation:

• Weekly written assignments.
• Few number of tests during the semester.
• Based on their grades in the aforementioned weekly assignments and tests, limited number of students can participate in exams towards the end of the semester, before the beginning of the exams period.
• In any case, all students can participate in written exams at the end of the semester, during the exams period.

The aforementioned information along with all the required details are available through the course’s website. The information is explained in detail at the beginning of the semester, as well as, throughout the semester, during the lectures. Reminders are also posted at the beginning of the semester and throughout the semester, through the course’s website. Upon request, all the information is provided using email or social networks.

### Attached Bibliography

See the official Eudoxus site or the local repository of Eudoxus lists per academic year, which is maintained by the Department of Mathematics. Books and other resources, not provided by Eudoxus:

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