MATH2031: ANALYSIS IN MANY VARIABLES II

• Calculus I (Maths Hons) (MATH1081) or Calculus 1 (MATH1061) and Linear Algebra I (Maths Hons) (MATH1091) or Linear Algebra 1 (MATH1071) and Analysis 1 (MATH1051) [the latter may be co-requisite].

• Analysis 1 (MATH1051) unless taken before.

• Mathematics for Engineers and Scientists (MATH1551), Single Mathematics A (MATH1561), Single Mathematics B (MATH1571), Mathematical Methods in Physics (PHYS2611)

• To provide an understanding of calculus in more than one dimension, together with an understanding of and facility with the methods of vector calculus.
• To understand the application of these ideas to a range of forms of integration and to solutions of a range of classical partial differential equations.

• Functions between multi-dimensional spaces, chain rule, inverse and implicit function theorems, curves, curvature, planar mappings.
• Vector calculus, line and surface integrals and integral theorems, suffix notation.
• Stokes and divergence theorems, conservative field and scalar potential.
• Generalised functions, Dirac delta distribution.
• Hilbert space.
• Sturm-Liouville Theory, Generalised Fourier Series.
• Special functions and orthogonal polynomials.
• Green's functions for ordinary and partial differential equations.
• Method of images for elliptic 2D partial differential equations.

Subject-specific Knowledge:

• By the end of the module students will: be able to solve a range of predictable and unpredictable problems in Vector Calculus.
• Have an awareness of the abstract concepts of theoretical mathematics in the field of analysis in many variables.
• Have a knowledge and understanding of fundamental theories of these subjects demonstrated through one or more of the following topic areas: differential and integral vector calculus.
• The divergence and Stokes' theorems.
• The solution of Partial Differential Equations by separation of variables and relation to special functions.
• Sturm-Liouville theory, Fourier and use of Greens functions to solve ordinary and partial differential equations.

Subject-specific Skills:

• In addition students will have the ability to undertake and defend the use of mathematical skills in the following areas with minimal guidance: Modelling, Spatial awareness.

Key Skills:

• Lecturing demonstrates what is required to be learned and the application of the theory to practical examples.
• Weekly homework problems provide formative assessment to guide students in the correct development of their knowledge and skills.
• Tutorials provide active engagement and feedback to the learning process.
• The end-of-year examination assesses the knowledge acquired and the ability to solve predictable and unpredictable problems.

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