
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 threedimensional volumes
 use the theorems of Green, Gauss and Stokes to calculate integrals of vector fields


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 threedimensional 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 threedimensional volumes
 the theorems of Green, Gauss and Stokes




 Assumed previous knowledgeContents of Analysis 1 and Analysis 2 of the AM Bachelor. 
Bachelor Applied Mathematics 
  Required materialsRecommended materialsInstructional modesLecturePresence duty   Yes 
 Tutor sessionsPresence duty   Yes 
 TutorialPresence duty   Yes 

 TestsAnalysis 3


 