Last modified: 20 Oct 2021 15:30
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.
|Session||Second Sub Session||Credit Points||15 credits (7.5 ECTS credits)|
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
Information on contact teaching time is available from the course guide.
3x Standard Course Assignments - 33.33% each
Alternative Resit Assessment
Resubmission of failed elements (pass marks carried forward)
There are no assessments for this course.
|Knowledge Level||Thinking Skill||Outcome|
|Factual||Understand||understand integration and theorems about the Riemann integral for multivariable functions;|
|Factual||Apply||be able to state the main definitions and theorems of the course;|
|Conceptual||Understand||be familiar with the concept of Jordan measurability and understand theorems about Jordan measurable sets;|
|Conceptual||Apply||be able to prove most results from the course;|
|Conceptual||Apply||apply ideas from Euclidean spaces such as inner products and convergence to the abstract setting of Hilbert spaces.|