# MAS2804 : Vector Calculus & Differential Equations, Transforms & Waves

• Offered for Year: 2020/21
• 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

#### 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. 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.
?       Elements of the Sturm-Liouville theory.
?       Fourier series and Fourier transforms.
?       Wave equation: simple derivation in one spatial dimension, generalisation to three dimensions, fundamental properties (D’Alembert’s solution, phase speed, plane waves, superposition, standing and travelling waves), wave packet.
.Second-order partial differential equations. Separation of variables in Cartesian coordinates: application to the wave, heat, Laplace’s and Poisson’s equations.

#### Teaching Methods

Please note that module leaders are reviewing the module teaching and assessment methods for Semester 2 modules, in light of the Covid-19 restrictions. There may also be a few further changes to Semester 1 modules. Final information will be available by the end of August 2020 in for Semester 1 modules and the end of October 2020 for Semester 2 modules.

##### Teaching Activities
Category Activity Number Length Student Hours Comment
Guided Independent StudyAssessment preparation and completion301:0030:00N/A
Scheduled Learning And Teaching ActivitiesLecture91:009:00Present in person *
Scheduled Learning And Teaching ActivitiesLecture91:009:00Synchronous online Material
Structured Guided LearningLecture materials361:0036:00Non-Synchronous Activities
Structured Guided LearningStructured non-synchronous discussion181:0018:00N/A
Scheduled Learning And Teaching ActivitiesDrop-in/surgery41:004:00Office hour or discussion board activity
Guided Independent StudyIndependent study941:0094:00N/A
Total200:00
##### Jointly Taught With
Code Title
PHY2035Vector Calculus & Differential Equations, Transforms & Waves
##### Teaching Rationale And Relationship

Non-synchronous online materials are used for the delivery of theory and explanation of methods, illustrated with examples, and for giving general feedback on assessed work. Present-in-person and synchronous online sessions are used to help develop the students’ abilities at applying the theory to solving problems and to identify and resolve specific queries raised by students, and to allow students to receive individual feedback on marked work. Students who cannot attend a present-in-person session will be provided with an alternative activity allowing them to access the learning outcomes of that session. In addition, office hours/discussion board activity will provide an opportunity for more direct contact between individual students and the lecturer:  a typical student might spend a total of one or two hours over the course of the module, either individually or as part of a group.

Alternatives will be offered to students unable to be present-in-person due to the prevailing C-19 circumstances.
Student’s should consult their individual timetable for up-to-date delivery information.

#### Assessment Methods

Please note that module leaders are reviewing the module teaching and assessment methods for Semester 2 modules, in light of the Covid-19 restrictions. There may also be a few further changes to Semester 1 modules. Final information will be available by the end of August 2020 in for Semester 1 modules and the end of October 2020 for Semester 2 modules.

The format of resits will be determined by the Board of Examiners

##### Exams
Description Length Semester When Set Percentage Comment
Written Examination1202A80Alternative assessment - in class test
##### Other Assessment
Description Semester When Set Percentage Comment
Written exercise1M10written exercises
Written exercise2M10written exercises
##### Assessment Rationale And Relationship

A substantial formal examination is appropriate for the assessment of the material in this module. The course assessments will will 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.