*MATH2044 *Applications of Vector Calculus

## Module Overview

Over the last four hundred years progress in understanding the physical world (theoretical physics) has gone hand in hand with progress in the mathematical sciences, so much so that the terms applied mathematics and theoretical physics have come to be almost coterminous. Vector calculus is one of the main mathematical tools to study the world around us. Many physical quantities are described by vector or scalar fields. Examples include not only velocities and forces (particularly useful in fluid mechanics), but also particle displacements (useful in solid mechanics), and electric and magnetic fields (electromagnetism). In this module we use the vector calculus as a tool to understand some basic theories in theoretical physics. We also introduce tensors and the tensor calculus. Tensors extend the idea of a vector. A tensor is a multi-index array (e.g. a matrix) with well-defined transformation rules under coordinate transformations. This module applies vector calculus in fluid mechanics and electromagnetism. We concentrate on fluids which do not have any resistance to flow (inviscid fluid flow) and electromagnetiism in vaccum. The mathematical models we discuss all involve solutions of equations including vector derivatives (i.e. div, grad and curl and their tensor generalisations). A particularly interesting feature of our development is the close mathematical similarity between equations from different branches of theoretical physics.

### Aims and Objectives

#### Module Aims

• Manipulate vector and tensor quantities using the index notation • Have a basic knowledge and understanding of tensors • Have a good knowledge and understanding of inviscid fluid dynamics • Have a good knowledge and understanding of electromagnetism • Solve problems of inviscid fluid dynamics (in particular, of fluid flows using complex potential theory) and electromagnetism

#### Learning Outcomes

##### Learning Outcomes

Having successfully completed this module you will be able to:

- Have a broad detailed understanding of: 1) Tensor Calculus and Index Notation, 2) inviscid Fluid Dynamics and 3) Electromagnetism in vacuum (including deriving and solving the wave equation)
- 1. Calculate grad, div and curl in Cartesian and other simple coordinate systems, and establish vector identities connecting these quantities. Use Gauss’s and Stokes’ theorems to simplify calculations of integrals and prove several results. 2. Solve practical problems of fluid dynamics of interest in real-life experiments (For example: how does a plane wing work? how do we increase the downforce of a sports car? ) 3. Solve practical problems of electromagnetism of interest in real-life experiments (For example: how does the battery of a mobile phone work? how can we make a device that generates an electric current? how are light and electromagnetic waves generated and propagated?)
- Appreciate how complex methods can be used to prove some important theoretical results such as the Fundamental Theorem of Algebra
- Evaluate line, surface and volume integrals in simple coordinate systems

### Syllabus

PART A: TENSOR CALCULUS 1. Index Notation - Vector identities & Cartesian tensors. Einstein Summation Convention; Kronecker delta; Outer product; Gradient, Divergence, Curl; Scalar and Cross products; Levi-Civita symbol and its properties and many uses; use Index Notation to prove any Vector Identity; Fundamental theorems of Calculus and Line integrals. 2. Tensors Definition; Representations of a tensor in a different frame of reference; Properties of rotation matrices; Invariants in a change of frame; Properties of tensors; Diagonal, antisymmetric and symmetric tensors; Decomposition of tensors; Antisymmetric tensors and axial vectors; Examples of tensors PART B: INVISCID FLUID DYNAMICS 3. Basic Definitions Streamlines; rate of change; pressure. 4. Fundamental equations Equation of continuity; Euler’s equation and its boundary conditions; Bernoulli's equation; The many applications of Bernoulli’s equation 5. Complex potentials Basic theory for two-dimensional inviscid, incompressible and irrotational flows; Derivation of complex potentials for particular flows; Fluid vortices; Complex potentials in the presence of boundaries; Force on bodies. PART C: ELECTROMAGNETISM 6. Electrostatics Definition of electric field; Coulomb law; Superposition principle; electric dipole; Electric flux and Gauss law; electric potential; electric potential energy; Applications. 7. Magnetostatics Electric current and current density; Definition of magnetic field; Biot-Savart law; Vector potential; Lorentz force; Ampere law; Applications of these laws. 8. Time-varying electromagnetic fields Faraday’s law of induction and Lenz’s law; Ampere-Maxwell’s law; Applications of these laws 9. Light and electromagnetic waves Maxwell’s theory of electromagnetism (Maxwell equations, continuity equation); Derivation of the wave equation; Solution of the wave equation in vacuum; Plane wave limit; Polarisation of a light wave.

### Learning and Teaching

#### Teaching and learning methods

Lectures, problem class, quizzes, private study

Type | Hours |
---|---|

Teaching | 54 |

Independent Study | 96 |

Total study time | 150 |

#### Resources & Reading list

Parker D.F. (2003). Fields, Flows and Waves.

Lecturers of MATH2044 (Oscar Dias & Andreas Schmitt) (Slightly updated every year). Lecture Notes of MATH2044 (Applications of Vector Calculus).

Cottingham W. & Greenwood D.A (1991). Electricy & Magnetism.

Grant I. S. & Phillips W. R. (1990). Electomagnetics.

Batchelor G.K.. An Introduction to Fluid Dynamics.

Acheson D. J.. Elementary Fluid Dynamics.

Jackson J. D. (1998). Classical Electrodynamics.

### Assessment

#### Summative

Method | Percentage contribution |
---|---|

Coursework | 20% |

Written exam | 80% |

#### Referral

Method | Percentage contribution |
---|---|

Exam | 100% |

#### Repeat Information

**Repeat type: Internal & External**

### Linked modules

#### Pre-requisites

To study this module, you will need to have studied the following module(s):

Code | Module |
---|---|

MATH2045 | Vector Calculus and Complex Variable |

MATH2008 | Introduction to Applied Mathematics |

### Costs

#### Costs associated with this module

Students are responsible for meeting the cost of essential textbooks, and of producing such essays, assignments, laboratory reports and dissertations as are required to fulfil the academic requirements for each programme of study.

In addition to this, students registered for this module typically also have to pay for:

##### Books and Stationery equipment

Absolutely self-contained "Lecture Notes" (approximately 150 pages) are delivered to the students, both as a pdf file in Blackboard that can be downloaded and a printed copy of them. For additional bibliography (always recommended! but not fundamental here), a limited number of course texts are provided by the library and there are no additional compulsory costs associated with the module.

Please also ensure you read the section on additional costs in the University’s Fees, Charges and Expenses Regulations in the University Calendar available at www.calendar.soton.ac.uk.