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
Having successfully completed this module you will be able to:
- Evaluate line, surface and volume integrals in simple coordinate systems
- Appreciate how complex methods can be used to prove some important theoretical results such as the Fundamental Theorem of Algebra
- 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?)
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.
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
Definition of electric field; Coulomb law; Superposition principle; electric dipole; Electric flux and Gauss law; electric potential; electric potential energy; Applications.
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
|Total study time||150|
Resources & Reading list
Lecturers of MATH2044 (Oscar Dias & Andreas Schmitt) (Slightly updated every year). Lecture Notes of MATH2044 (Applications of Vector Calculus). Southampton University. Pdf available on Blackboard & printed version distributed at no cost..
Batchelor G.K.. An Introduction to Fluid Dynamics. Cambridge University Press.
Parker D.F. (2003). Fields, Flows and Waves. Springer Undergraduate Mathematic Series.
Acheson D. J.. Elementary Fluid Dynamics. Oxford University Press.
Grant I. S. & Phillips W. R. (1990). Electomagnetics. Wiley.
Cottingham W. & Greenwood D.A (1991). Electricy & Magnetism. CUP.
Jackson J. D. (1998). Classical Electrodynamics. Wiley.
Summative assessment description
Referral assessment description
Repeat type: Internal & External