# Advanced Calculus and Mathematical Methods

## Course summary

Course code: MATH1040
Level: 4
Credits: 30
School: Architecture, Computing and Hums
Department: Mathematical Sciences
Course Coordinator(s): Yvonne Fryer

## Specification

None

### Aims

This course will review the student's mathematical background and extend it to provide a firm foundation to underpin all other courses in mathematics. It will introduce the student to advanced calculus and to the mathematical methods which form an essential element of the expertise of an applied mathematician. The emphasis is on methods relevant for applications in science and technology.

### Learning outcomes

On successful completion of this course a student will be able to:

1. Consolidate their knowledge of elementary calculus;
2. Solve simple first order differential equations and second order differential and difference equations with constant coefficients;
3. Solve problems involving partial differentiation;
4. Use simple numerical techniques to approximate solutions to equations and approximate solutions to problems in differential and integral calculus;
5. Use multiple integrals and apply multiple integration to physical problems;
6. Apply methods and techniques of calculus to solve real-world problems in fields such as mechanics.
7. Use appropriate mathematical language

### Indicative content

Introduction to complex numbers: manipulation, polar form, exponential form.
Limits, continuity and differentiability.
Taylor's Series for functions of one and two independent variables.
Partial differentiation and applications, including Lagrange multipliers and finding extrema of functions of two independent variables.
Double integrals, improper integrals.
Solution of 1st order differential equations by separation of variables and integrating factors. Solution of 2nd order differential equations with constant coefficients with appropriate particular integrals. First and second order difference equations.
Iterative methods for solving equations: bisection, secant, fixed-point, Newton-Raphson.
Lagrange interpolating polynomials.
Mathematical methods: forward and backward difference, Eulerâ€™s method, Runge Kutta.
Relevance to practical applications and employability will be built into both the course material and course delivery.

### Teaching and learning activity

The material will be delivered through a flexible mix of lectures, tutorials and independent study.

### Assessment

Summative Assessment:
Coursework 1 - weighting 25%, outline details: coursework covering term 1 early material, LO - 1, 3, 4, 7.

Coursework 2 - weighting 25%, outline details: coursework covering material from term 1 & 2, LO - 1-7.

Examination - weighting 50%, outline details: an examination of 3 hours in duration covering all learning outcomes, LO - 1-7.

Formative assessment: Weekly tutorial exercises.