Module Catalogue 2025/26

Pre Requisite Comment

A-level Mathematics or equivalent.

Co Requisite Comment

N/A

Aims

To introduce calculus of functions of several variables.

Module Summary
This module, which continues and extends the work of MAS1612, develops many of the ideas that are needed when constructing mathematical models of phenomena in the real world.

The world where we live is multi-dimensional - three-dimensional if we consider spatial dimensions alone, or four-dimensional if we treat time as another variable. It is therefore essential to develop tools to describe and model objects and processes that occur in multi-dimensional spaces. In order to do this we require multidimensional calculus.

This module introduces the partial derivative, and the multiple integral, as well as power series in two or more variables.

Outline Of Syllabus

Introduction to functions of several variables: continuity and differentiability, partial differentiation, gradient, chain rule and Jacobian matrices.

- Sketching multivariable functions and level sets - by hand and using software such as Python.
- Taylor series in two (or more) variables, classification of stationary points.
- Multiple Integrals: double and triple integrals.
- Change of variables (including use of polar, cylindrical and spherical coordinates) The inverse and implicit function theorems.
- Exact differentials.

Intended Knowledge Outcomes

Students will be able to differentiate and integrate functions of two or more variables.

Students will be able to state and use some of the important results in multivariable calculus.

Intended Skill Outcomes

Students will develop some confidence working with functions in several variables. Students will learn techniques for solving calculus problems in dimension two and higher.

Students will develop skills across the cognitive domain (Bloom’s taxonomy, 2001 revised edition): remember, understand, apply, analyse, evaluate and create.

Teaching Rationale And Relationship

The teaching methods are appropriate to allow students to develop a wide range of skills, from understanding basic concepts and facts to higher-order thinking.

Lectures are used for the delivery of theory and explanation of methods, illustrated with examples, and for giving general feedback on marked work. Problem Classes are used to help develop the students’ abilities at applying the theory to solving problems.

Assessment Rationale And Relationship

A substantial formal unseen examination is appropriate for the assessment of the material in this module. The format of the examination will enable students to reliably demonstrate their own knowledge, understanding and application of learning outcomes. The assurance of academic integrity forms are a necessary part of the programme accreditation.

Examination problems may require a synthesis of concepts and strategies from different sections, while they may have more than one way for solution. The examination time allows the students to test different strategies, work out examples and gather evidence for deciding on an effective strategy, while carefully articulating their ideas and explicitly citing the theory they are using.

The coursework assignments allow the students to develop their problem solving techniques, to practice the methods learnt in the module, to assess their progress and to receive feedback; these assessments have a secondary formative purpose as well as their primary summative purpose.

General Notes

N/A