- Offered for Year: 2022/23
- Module Leader(s): Dr Toby Wood
- Lecturer: Professor Anvar Shukurov
- Owning School: Mathematics, Statistics and Physics
- Teaching Location: Newcastle City Campus
Semesters
|
Semester 1 Credit Value:
|
10
|
|
Semester 2 Credit Value:
|
10
|
|
ECTS Credits:
|
10.0
|
Aims
To introduce the mathematics needed to formulate and solve problems involving vector and scalar fields, and ordinary and partial differential equations.
Module Summary
Many applications of mathematics involve objects and quantities that exist in multiple dimensions, as well as their rates of change in space and/or time. This course shows how calculus can be applied to such problems, and introduces techniques for solving the resulting differential equations.
The first half of this module explains how we can mathematically define curves and surfaces in three-dimensional space, and how we can calculate their properties, such as tangent, length and area. We also introduce the concepts of scalar fields (e.g. temperature, pressure, density) and vector fields (e.g. velocity and electromagnetic fields). To describe these objects and quantities we must generalize the principles of calculus to multi-dimensions. This part of the course introduces the mathematical language and concepts that are needed to study continuous media, fluids, and electromagnetism.
The second half of this module continues the exploration of differential equation that started in Stage 1, with emphasis on methods to solve them, both exact and approximate. The essential elements in the theory of ordinary and partial differential equations, and their methods of solution, introduced in this module, provide the basis for specific studies in other modules. Fourier series and transform will be introduced in this context. The methods that will be introduced, justified and practiced apply to a wide range of ordinary and partial differential equations.
Outline Of Syllabus
Semester 1:
• Scalar and vector fields;
• double and triple integrals;
• parametric representations of curves and surfaces;
• tangent vector and line integrals;
• normal vector and surface integrals;
• differential operators (gradient, divergence, curl, and Laplacian);
• subscript notation and the summation convention;
• operators in spherical and cylindrical coordinates;
• Gauss', Stokes' and Green's theorems.
Semester 2:
• A review of ordinary and partial differential equations;
• Series solutions.
• Fourier series and Fourier transforms.
• Wave equation: fundamental properties (standing and travelling waves).
• Second-order partial differential equations. Separation of variables in Cartesian coordinates: application to the wave, heat and Laplace’s equations.
Teaching Methods
Teaching Activities
| Category |
Activity |
Number |
Length |
Student Hours |
Comment |
|---|
| Scheduled Learning And Teaching Activities | Lecture | 40 | 1:00 | 40:00 | Formal Lectures – Present in Person |
| Scheduled Learning And Teaching Activities | Lecture | 4 | 1:00 | 4:00 | Revision Lectures – Present in Person |
| Scheduled Learning And Teaching Activities | Lecture | 10 | 1:00 | 10:00 | Problem Classes – Synchronous On-Line |
| Guided Independent Study | Assessment preparation and completion | 106 | 1:00 | 106:00 | Preparation time for lectures, background reading, coursework review |
| Scheduled Learning And Teaching Activities | Drop-in/surgery | 10 | 1:00 | 10:00 | Synchronous On-Line |
| Guided Independent Study | Independent study | 30 | 1:00 | 30:00 | Completion of in course assessments |
| Total | | | | 200:00 | |
Jointly Taught With
| Code |
Title |
|---|
| PHY2035 | Vector Calculus & Differential Equations, Transforms & Waves |
Teaching Rationale And Relationship
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 Methods
The format of resits will be determined by the Board of Examiners
Exams
| Description |
Length |
Semester |
When Set |
Percentage |
Comment |
|---|
| Written Examination | 150 | 2 | A | 80 | N/A |
Exam Pairings
| Module Code |
Module Title |
Semester |
Comment |
|---|
| PHY2035 | Vector Calculus & Differential Equations, Transforms & Waves | 2 | N/A |
Other Assessment
| Description |
Semester |
When Set |
Percentage |
Comment |
|---|
| Prob solv exercises | 1 | M | 10 | Problem-solving exercises |
| Prob solv exercises | 2 | M | 10 | Problem-solving exercises |
Assessment Rationale And Relationship
A substantial formal unseen examination is appropriate for the assessment of the material in this module. The coursework assignments allow the students to develop their problem solving techniques, to practise 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.
In the event of on-campus examinations not being possible, an on-line alternative assessment will be used for written examination 1.
Reading Lists
Timetable
