# AMA Level 1 Modules

**MTH1001 Analysis and Calculus (Full year)**

*Pre-requisite*: This module is intended for students at stage 1 of either an MSci or a BSc Mathematics pathway, and a mathematical knowledge and ability commensurate with this stage is assumed.

*Lecturers*: Dr C Ramsbottom, Dr B McMaster, Dr G Gribakin

**Introduction**

The module begins with a revision of basic calculus and elementary functions. It then lays the foundations of analysis and explores the rigorous definitions of ‘sequence limit’ and ‘function limit’ into which the historical concept of ‘infinitesimal’ evolved. This is a basis for investigating properties of functions using derivatives, integrals and power series. Differential equations and calculus in several variables are introduced in the last part of the module, providing most useful tools for applications in mechanics, fluid dynamics, electromagnetism, etc.

**Contents**

Review of A-level calculus: functions and their graphs, trigonometric functions, derivatives and differentials, integration.

Sequences of real numbers, the notion of convergence of a sequence, completeness, the Bolzano-Weierstrass theorem, limits of series of non-negative reals and convergence tests.

Analytical definition of continuity, limits of functions and derivatives in terms of a limit of a function. Properties of continuous and differentiable functions. L’Hopital’s rule, Rolle’s theorem, first mean-value theorem. Integration of functions through limits of approximations by simple areas. Fundamental theorem of calculus. Taylor series of functions in terms of a limit of a series. Maclaurin series of basic elementary functions. Euler’s formula.

Differential equations (DE); first-order DE: variable separable, linear; second-order linear DE with constant coefficients: homogeneous and inhomogeneous.

Functions of several variables, partial derivatives, Taylor expansion, total differential, gradient (nabla operator). Lines and surfaces. Spherical and cylindrical coordinates. Scalar and vector functions, div and curl operators. Integrals in 2D and 3D. Green’s theorem, Gauss’s theorem, Stokes’s theorem.

**Assessment**

Class tests 15% (x 2)

Computer-based test 10%

Final Exam 60%

** **

**AMA1021 Mathematical Modelling (2nd semester)**

*Pre-requisite*: This course is intended for students at stage 1 of either an MSci or a BSc Mathematics pathway, and a mathematical knowledge and ability commensurate with this stage is assumed.

*Lecturer*: Dr A Brown

**Introduction**

The module begins with an introduction to computer programming with the python programming language, using the online service repl.it as a development environment. We use the idea of the computer program, and in particular the way a computer uses logic, to address the idea of modelling a problem: how do we break up a large problem into constituent parts? how do we use mathematical tools- numerical or analytical- to solve problems of dynamics, mechanics etc.

The skills required to model a problem are intimately connected with the skills of communication- if you can articulate a problem and its solution then you are better able to develop, implement and analyse models to describe it. Thus a large part of the course is given to mathematics communication, in particular writing reports. For this you will use the LaTeX markup language which is the standard medium for writing mathematics, and will complete a series of writing exercises to learn both the technical and mathematical skills involved in effective report writing.

The final part of the module concerns building a model to solve a particular problem of mechanics. This will require you to implement a numerical approximation to an analytical problem and thereby use and explore a mathematical model of the physical system.

**Contents**

The Python programming language. Basic calculations, vectors and matrices, function definitions, basic plotting, python libraries, logical flow, for loops, if statements.

The LaTeX markup language,using the overleaf online environment. Mathematical report writing.

Differential equations for solving real life problems. Numerical solution of ordinary differential equations.

**Assessment**

Computer programming assignments 20%

Writing exercises 40%

Final Project 40%