Module MATH2811: Mathematical Methods II

Department: Mathematical Sciences

MATH2811: Mathematical Methods II

Prerequisites

• One of:
• Calculus I (Maths Hons) (MATH1081) OR Calculus 1 (MATH1061)
• AND one of:
• Linear Algebra I (Maths Hons) (MATH1091) OR Linear Algebra 1 (MATH1071)

Corequisites

• None

Excluded Combination of Modules

• Mathematical Methods in Physics (PHYS2611)

Aims

• To provide an understanding of vector calculus concepts and their application in terms of practical computations, such as in multivariable integration and coordinate transformations.
• To understand the analysis of linear initial and boundary value problems (BVPs) for ordinary differential equations (ODEs).

Content

• Line integrals: parametrised curves; line integrals of scalar and vector fields; curvilinear coordinates.
• Surface and volume integrals: parametrised surfaces; surface integrals of scalar and vector fields; volume integrals.
• Differential operators: gradient; divergence; curl; product rules; expressions in curvilinear coordinates.
• Integral theorems: Fundamental Theorem of Line Integrals; Divergence/Gauss Theorem; Stokes’ Theorem and Green’s Theorem; conservative fields.
• Index notation: for scalar products; vector products; derivatives; second derivatives.
• Higher-dimensional functions: Jacobian matrix, inverse functions, Implicit Function Theorem.
• Separation of variables: review separation of variables for partial differential equations as motivation for studying boundary value problems.
• Eigenfunction methods for boundary value problems: generalisation of Fourier series to arbitrary linear BVPs with homogeneous/inhomogeneous forcing; Sturm-Liouville theory for counting/signing eigenvalues.
• Green’s functions for boundary value problems: equivalence of eigenfunction solutions to Green’s functions; Dirac delta/distributions as a computational tool.

Learning Outcomes

• Be able to solve seen and unseen problems on the given topics;
• Be able to explain and solve basic problems using concepts from vector calculus such as differentiating and integrating scalar and vector fields, vectorial operators, and theorems generalising the fundamental theorem of calculus;
• Be able to solve initial and boundary value problems for linear ordinary differential equations using eigenfunctions and Green’s function methods;

• In addition students will have enhanced mathematical skills in the following areas: mathematical methods, problem solving, computation.

• Students will have basic mathematical skills in the following areas: problem solving, modelling, computation.

Modes of Teaching, Learning and Assessment and how these contribute to the learning outcomes of the module

• Lectures demonstrate what is required to be learned and the application of the theory to practical examples.
• Problem classes show how to solve example problems in an ideal way, revealing also the thought processes behind such solutions.
• Tutorials provide active problem-solving engagement and immediate feedback to the learning process.
• Formative assessments provide feedback to guide students in the correct development of their knowledge and skills in preparation for the summative assessment.
• Summative assignments test achievement of learning outcomes and provide feedback to students about their mastery of the topics.
• The end-of-year examination assesses the knowledge acquired and the ability to solve predictable and unpredictable problems.

Teaching Methods and Learning Hours

Summative Assessment

Formative Assessment:

There will be at most one formative or summative assignment each week.

■ Attendance at all activities marked with this symbol will be monitored. Students who fail to attend these activities, or to complete the summative or formative assessment specified above, will be subject to the procedures defined in the University's General Regulation V, and may be required to leave the University