Last modified: 14 Nov 2025 12:16
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.
| Study Type | Undergraduate | Level | 3 |
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| Term | Second Term | Credit Points | 15 credits (7.5 ECTS credits) |
| Campus | Aberdeen | Sustained Study | No |
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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
Information on contact teaching time is available from the course guide.
| Assessment Type | Summative | Weighting | 70 | |
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| Assessment Weeks | Feedback Weeks | |||
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| Assessment Type | Summative | Weighting | 15 | |
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| Knowledge Level | Thinking Skill | Outcome |
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| Assessment Type | Summative | Weighting | 15 | |
|---|---|---|---|---|
| Assessment Weeks | Feedback Weeks | |||
| Feedback | ||||
| Knowledge Level | Thinking Skill | Outcome |
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There are no assessments for this course.
| Assessment Type | Summative | Weighting | ||
|---|---|---|---|---|
| Assessment Weeks | Feedback Weeks | |||
| Feedback |
Best of (resit exam mark) or (resit exam mark with carried forward CA marks). |
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| Knowledge Level | Thinking Skill | Outcome |
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| Knowledge Level | Thinking Skill | Outcome |
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| Conceptual | Understand | Understand multivariable Riemann integration and its properties. |
| Procedural | Apply | To use the methods and results of the course to solve problems at levels similar to those seen in the course. |
| Conceptual | Analyse | To state the theorems of the course, to explain their significance, and to give examples to indicate the role of the hypotheses. |
| Procedural | Analyse | To demonstrate knowledge and understanding of proof techniques used in the course. |
| Conceptual | Understand | To state and illustrate the definitions of the concepts introduced in the course. |
| Procedural | Apply | Be able to calculate path and surface integrals and apply Gauss's and Stockes's theorems. |
| Conceptual | Understand | Understand basic Hilbert space theory and examples. |
| Conceptual | Understand | Be familiar with the change of variables theorem for multidimensional integrals. |
| Conceptual | Understand | Be familiar with the notions of the upper and lower Darboux sums and integrals and be familiar with their properties. |
| Procedural | Apply | Be familiar with the notion of linear functionals and be able to apply the Riesz representation theorem for them. |
| Conceptual | Understand | Understand the second Riemann criterion of integrability. |
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