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# Vector & Complex Calculus - MAT00047I

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• Department: Mathematics
• Module co-ordinator: Dr. Francesco Cosentino
• Credit value: 20 credits
• Credit level: I
• Academic year of delivery: 2023-24

## Module summary

This module encompasses two subjects. Vector calculus is about differentiation and integration of scalar and vector fields in 2 and 3 dimensions. Complex analysis concerns the differentiation and integration of functions of a complex variable.

## Related modules

• None

### Prohibited combinations

• None

Pre-requisite modules:

• Foundations & Calculus
•  Multivariable Calculus & Matrices

## Module will run

Occurrence Teaching cycle
A Semester 1 2023-24

## Module aims

This module aims to deepen and extend knowledge from first year Calculus, with a more mature look at the fundamental concepts of infinitesimal calculus from the viewpoint of vector-valued functions of many variables. The three variable case underpins all continuous processes in the three space dimensions of our world, and which is therefore essential to the application of mathematics in the natural sciences.

The two variable case leads to the notion of differentiable functions of a complex variable which have very many properties beyond those of the differentiable functions of real variables that were studied previously. These underpin the important technique of contour integration, which is useful for evaluating real integrals.

## Module learning outcomes

By the end of this module, students will be able to:

1. Solve a variety of vector calculus problems in 2 and 3 dimensions using the Stokes theorem, Gauss’ theorem, and Greens’ theorem.

2. Apply gradient, divergence, curl using different coordinate systems such as spherical polar, cartesian, and cylindrical polar.

3. Compute flux and surface integrals in different coordinate systems.

4. Apply the Cauchy-Riemann equations and discuss when a function is analytic.

5. Compute the Laurent series and identify the poles of a complex function.

6. Use the Residue Theorem to compute the value of a definite integral

## Module content

In the first part of this module, we discuss the three essential ingredients of calculus—continuity, differentiability, and integrability—bringing out the distinctive flavour of each theory, and describing their inter-relationships and applications; we gain a deeper understanding of the three famous differential operators of classical vector calculus: div, grad and curl; we describe the theorems of Stokes and Gauss which link these topics together.

The second part of this module will extend the ideas developed about real functions to complex functions, develop the theory of holomorphic functions, and apply this theory to understand problems arising in real analysis or calculus. This lays the groundwork for future study of Fourier analysis which is widely used in real-world applications. These techniques are used by many graduates in their day-to-day work.

## Assessment

Task Length % of module mark
Closed/in-person Exam (Centrally scheduled)
Vector & Complex Calculus
3 hours 100

### Special assessment rules

None

There will be five formative written assignments, with marked work returned in the seminars

### Reassessment

Task Length % of module mark
Closed/in-person Exam (Centrally scheduled)
Vector & Complex Calculus
3 hours 100

## Module feedback

Current Department policy on feedback is available in the student handbook. Coursework and examinations will be marked and returned in accordance with this policy.