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MX3535: ANALYSIS IV (2021-2022)

Last modified: 20 Oct 2021 15:30


Course Overview

Analysis provides the rigourous, foundational underpinnings of calculus. This course builds on MX3035 Analysis III, continuing the development of multivariable calculus, with a focus on multivariable integration. Hilbert spaces (infinite dimensional Euclidean spaces) are also introduced.

Students will see the benefit of having acquired the formal reasoning skills developed in Analysis I, II, and III, as it enables them to work with increasingly abstract concepts and deep results. Techniques of rigourous argumentation continue to be a prominent part of the course.

Course Details

Study Type Undergraduate Level 3
Session Second Sub Session Credit Points 15 credits (7.5 ECTS credits)
Campus Aberdeen Sustained Study No
Co-ordinators
  • Dr Alexey Sevastyanov

Qualification Prerequisites

  • Either Programme Level 3 or Programme Level 4

What courses & programmes must have been taken before this course?

What other courses must be taken with this course?

None.

What courses cannot be taken with this course?

None.

Are there a limited number of places available?

No

Course Description

Description: Syllabus

1 Multidimensional Riemann integral

1.1 The de nition of the multidimensional Riemann integral over boxes
1.2 Riemann second criterion of integrability
1.3 Properties of the Riemann integral
1.4 Iterated integrals
1.5 Riemann integrals over bounded sets
1.6 Change of variables

2 Path and surface integrals

2.1 Paths and path integrals
2.2 Surfaces and surface integrals
2.3 Gauss divergence theorem
2.4 Stokes's theorem

3 Hilbert spaces

3.1 Scalar products on linear spaces
3.2 The definition of the Hilbert space
3.3 Orthonormal bases in Hilbert spaces and the isomorphism theorem
3.4 Orthogonal decompositions and orthogonal projection operators
3.5 Continuous linear functionals and the Riesz Representation Theorem


In light of Covid-19 this information is indicative and may be subject to change.

Contact Teaching Time

Information on contact teaching time is available from the course guide.

Teaching Breakdown

  • 2 Lectures during University weeks 26 - 30, 32 - 35, 39 - 40
  • 1 Tutorial during University weeks 27 - 30, 32 - 35, 39 - 40

More Information about Week Numbers


Summative Assessments

Alternative Assessment

3x Standard Course Assignments - 33.33% each

Alternative Resit Assessment

Resubmission of failed elements (pass marks carried forward)

 

Formative Assessment

There are no assessments for this course.

Course Learning Outcomes

Knowledge LevelThinking SkillOutcome
FactualUnderstandunderstand integration and theorems about the Riemann integral for multivariable functions;
FactualApplybe able to state the main definitions and theorems of the course;
ConceptualUnderstandbe familiar with the concept of Jordan measurability and understand theorems about Jordan measurable sets;
ConceptualApplybe able to prove most results from the course;
ConceptualApplyapply ideas from Euclidean spaces such as inner products and convergence to the abstract setting of Hilbert spaces.

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