This module covers some of the most important areas of mathematical physics. Vector calculus provides the mathematics required to understand and use scalar and vector fields, providing the tools essential to allow interpretation of electric, magnetic and gravitational fields in 2nd and 3rd year courses. By deriving expressions for curl, div, grad in different coordinate systems and relating these to physical examples the laws of Gauss, Ampère and Faraday are expressed mathematically in both their macrophysical and differential forms. Finally the basic properties of both real and complex matrices and tensors are studied, how to manipulate them and use them in various physical situations. These have application in special relativity, quantum mechanics, and other areas of physics.
Module learning outcomes
Understand the distinction between scalar and vector fields and how they are represented.
Calculate the gradient of a scalar field and understand its physical meaning.
Find the rate of change of a scalar field in any direction.
Define and understand the physical meaning of a conservative field, giving examples.
Calculate the amount of work done in moving along any path in a vector field.
Understand the concept of divergence of a vector field expressing it mathematically.
Define and understand the physical meaning of a solenoidal field, giving examples.
Evaluate the flux of a vector field through any surface.
Apply the divergence theorem and evaluate both the integrals involved.
Understand the concept of the curl of a vector field expressing it mathematically.
Understand the concept of circulation of a vector field expressing it mathematically.
Evaluate the circulation of a vector field round any closed loop.
Apply Stokes's theorem and evaluate both the integrals involved.
Apply the Laplace operator 2 to a function of the form f(x,y,z).
Derive and use expressions for curl, div, grad and 2 in different coordinate systems.
Express the laws of Gauss, Ampère and Faraday mathematically in both their macrophysical and differential forms.
Define various types of matrix and be able to multiply, divide, add and subtract matrices.
Solve linear simultaneous equations using matrix methods.
Solve sets of homogeneous and non-homogeneous equations.
Define the rank of a matrix and understand its significance.
Understand the concept of ill-conditioning and be able to relate it to simple simultaneous equations.
Be able to define linear dependence and independence and be able to test for these conditions.
Set up matrix equations and apply matrix methods to solve problems in various branches of classical physics.
Apply matrix methods to effect rotations and translations of coordinates/axes/objects.
Understand the concepts and properties of eigenvalues and eigenvectors.
Diagonalise a simple matrix and be able to extract the eigenvalues and eigenvectors.
Normalise the eigenvectors.
Define what is meant by orthogonal Hermitian, Unitary and Normal matrices and be able to demonstrate their properties.
Appreciate that cartesian tensors are expressed in terms of components referred to rectangular cartesian coordinate systems.
Understand the meaning of a transformation matrix.
Be able to define what is meant by the rank or order of a tensor.
Be able to calculate the product of tensors.
Understand what is meant by a scalar invariant.
Be able to apply the principles of contraction.
Understand what is meant by a symmetric and an anti-symmetric tensor.
Be able to apply the quotient rule to a tensor.
Appreciate that tensors are of primary importance in connection with coordinate transforms.
Fields: scalar and vector fields, definitions and representations.
Gradient: definition of gradient;.grad in Cartesian coordinates, vector differential operator , calculation of gradient, directional derivative; use of gradient in physics - electric field, heat conduction, gravitational field; measure of work - line integral; conservative field and scalar potential; examples of conservative fields, level surfaces, evaluation of line integrals.
Divergence: vector area and measure of flux; element of vector area on a sphere and a cylinder, surface integral and its evaluation; volume integral; concept of divergence of a vector field; mathematical expression for divergence; calculation of divergence; divergence theorem, differential form of Gauss’s Law; physical dimensions in div and surface and volume integrals; magnetic and solenoidal fields.
Combination of operators: Laplace operator, Laplace’s equation.
Curl: concept of curl of a vector field, mathematical expression for curl; conservative field revisited, calculation of curl; rigid body rotation and comparison with water down a plug hole; Stokes’s theorem; differential form of Ampère’s Law; summary of integral theorems and field properties.
Polar coordinates: grad div curl and 2 in spherical polars and cylindrical polars.
Applications: time as an extra variable - the continuity equation, examples in fluid flow, electricity and heat conduction, the heat conduction equation; the diffusion equation, Ampère’s Law revisited, differential form of Faraday’s law of electromagnetic induction.
Matrices: basic operations, addition, multiplication, division. Determination of the inverse matrix via Gauss elimination and calculation of the cofactor matrix. Determinants of a 2x2 and 3x3 linear matrix, the rank of a matrix, linear dependence, and definitions of orthogonal and orthonormal matrix, normalisation.
Matrix applications: linear simultaneous equations, matrix operators, translation, rotation, and reflection. Eigenvalues and eigenvectors of a matrix. Definitions of orthogonal Hermitian, unitary and normal matrices.
Tensors: basic algebra of tensors; symmetry and anti-symmetry of tensors; the contraction of a tensor; the quotient rule applied to tensors; the basic transformation of tensor components
% of module mark
Online Exam Mathematics II
Special assessment rules
% of module mark
Online Exam Mathematics II
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Stroud K A; Further engineering mathematics (MacMillan)***
Boas M L; Mathematical methods in the physical sciences, 3rd Ed 2002 (Wiley)***
Kreyszig E; Advanced engineering methods, 8th Ed (John Wiley)**
D.W.Jordan and P.Smith – Mathematical Techniques: An Introduction for the Engineering, Physical and Mathematical Sciences 4th edition (Oxford)**
D.E. Bourne and P.C. Kendall. Vector analysis and cartesian tensors 2nd edition or later. 1977 (Reinhold, New York)**
Jordan D.W. and Smith P.– Mathematical Techniques: An Introduction for the Engineering, Physical and Mathematical Sciences 4th edition (Oxford)**