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Course module: 202300017
202300017
Analysis 3
Course info
Course module202300017
Credits (ECTS)5
Course typeStudy Unit
Language of instructionEnglish
Contact personDr. rer. nat. J. A. Iglesias Martínez
E-mailjose.iglesias@utwente.nl
Lecturer(s)
Examiner
Dr. rer. nat. J. A. Iglesias Martínez
Contactperson for the course
Dr. rer. nat. J. A. Iglesias Martínez
Lecturer
Dr. rer. nat. J. A. Iglesias Martínez
Lecturer
dr. F.L. Schwenninger
Examiner
dr. F.L. Schwenninger
Academic year2023
Starting block
1A
RemarksPart of module 5 AM
Application procedureYou apply via OSIRIS Student
Registration using OSIRISYes
Aims
Upon completion of this course the student is able to:
  • give precise definitions of some important concepts from real analysis, work with them, and give rigorous proofs of some important results related to these concepts
  • explain and work with the concepts related to metric spaces, completeness, compact and connected sets, limit and continuity of functions on metric spaces
  • explain and work with the concepts related to continuity and differentiability of functions on ℝn and apply related results such as Taylor expansion, implicit function and inverse function theorems
  • use the concepts of vector fields, rotation, divergence and gradient, conservative fields
  • integrate vector fields and their related quantities along curves, surfaces and three-dimensional volumes
  • use the theorems of Green, Gauss and Stokes to calculate integrals of vector fields
Content
The topic of this course is multidimensional analysis of functions with a focus on ℝ2 and ℝ3. We first
look into metric spaces, compact and connected subsets, limits and continuity of functions defined
on metric spaces. Next, we consider total differentiability of functions on ℝn together with related concepts
such as Taylor expansion, implicit function theorem and inverse function theorem.

In the second part of this course, we center ourselves on vector fields. The concepts of rotation, divergence
and gradient are introduced, and special attention is given to conservative vector fields. Integrals of vector fields and their related quantities along curves, surfaces and three-dimensional volumes are treated, using the theorems of Green, Stokes and Gauss to establish relationships between these different types of integrals. This provides more insight into the meaning of integrals of vector fields, their use in modelling physical systems, and important theoretical relationships, besides simplifying the calculation of such integrals.

The following topics are covered:
  • metric spaces, completeness, compact and connected subsets
  • limits, continuity and Lipschitz continuity of functions defined on metric spaces
  • differentiability of functions in ℝn
  • multidimensional Taylor Series
  • implicit and inverse function theorems
  • vector fields, gradient, divergence, rotation, and conservative fields
  • integrals of vector fields over lines, surfaces, and three-dimensional volumes
  • the theorems of Green, Gauss and Stokes
Assumed previous knowledge
Contents of Analysis 1 and Analysis 2 of the AM Bachelor.
Module
Module 5
Participating study
Bachelor Applied Mathematics
Required materials
-
Recommended materials
-
Instructional modes
Lecture
Presence dutyYes

Tutor sessions
Presence dutyYes

Tutorial
Presence dutyYes

Tests
Analysis 3

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Kies de Nederlandse taal