# MAS1607 : Multivariable Calculus & Differential Equations

• Offered for Year: 2020/21
• Module Leader(s): Dr Paul Bushby
• Lecturer: Dr Magda Carr
• Owning School: Mathematics, Statistics and Physics
• Teaching Location: Newcastle City Campus
##### Semesters
 Semester 2 Credit Value: 20 ECTS Credits: 10

#### Aims

To develop an understanding of ordinary differential equations and a familiarity with relevant solution methods. To introduce the calculus of functions of several variables.

Module summary

This module, which continues and extends the work of MAS1605, develops many of the ideas that are needed when constructing mathematical models of phenomena in the real world. Many such models are formulated in terms of ordinary differential equations, and this module introduces the methods that are needed to solve problems of this type. 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 ordinary differential equations (ODEs): terminology and examples.

First-order ODEs: separable equations, homogeneous equations, integrating factor. Existence and uniqueness of the initial value problem for first-order ODEs, singular points and integral curves of first-order ODEs.

Second-order ODEs: homogeneous equations with constant coefficients, particular integrals for inhomogeneous equations, method of reduction of order.

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

Taylor series in two (or more) variables, classification of stationary points.

Multiple Integrals: double and triple integrals, change of variables (including polar coordinates).

#### 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
Scheduled Learning And Teaching ActivitiesLecture91:009:00Synchronous On Line Material
Structured Guided LearningLecture materials361:0036:00Non Synchronous Activities
Guided Independent StudyAssessment preparation and completion301:0030:00Completion of In Course Assessment
Scheduled Learning And Teaching ActivitiesSmall group teaching91:009:00Present in Person
Structured Guided LearningStructured non-synchronous discussion181:0018:00Non Synchronous Discussion to Support Learning
Scheduled Learning And Teaching ActivitiesDrop-in/surgery41:004:00Office Hour or Discussion Board Activity
Guided Independent StudyIndependent study941:0094:00Preparation Time for Lectures, Background Reading, Coursework Review
Total200:00
##### 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.
Student 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 Examination1202A60In Class Test
##### Other Assessment
Description Semester When Set Percentage Comment
Written exercise2M20N/A
Written exercise2M20N/A
##### Assessment Rationale And Relationship

The course assessments 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.