
Academic Year:  2017/8 
Owning Department/School:  Department of Physics 
Credits:  12 [equivalent to 24 CATS credits] 
Notional Study Hours:  240 
Level:  Certificate (FHEQ level 4) 
Period: 

Assessment Summary:  CW 20%, EX 60%, OT 20% 
Assessment Detail: 

Supplementary Assessment: 

Requisites:  
Description:  Aims: The aim of this unit is to introduce mathematical techniques required by physical science students, both by showing the application of Alevel mathematics content to physical problems in a more general and algebraic form and by introducing more advanced topics. Learning Outcomes: After taking this unit the student should be able to: * sketch graphs of standard functions and their inverses; * evaluate the derivative of a function and the partial derivative of a function of two or more variables; * write down the Taylor series approximation to a function; * represent complex numbers in Cartesian, polar and exponential forms, and convert between these forms; * calculate the magnitude of a vector, and the scalar and vector products of two vectors; * solve simple geometrical problems using vectors. Skills: Numeracy T/F A, Problem Solving T/F A. Content: Preliminary calculus (6 hours): Differentiation: differentiation from first principles; products; the chain rule; quotients; implicit differentiation; logarithmic differentiation; Leibnitz' theorem. Integration: integration from first principles; sinusoidal functions; logarithmic integration; using partial fractions; substitution method; by parts; reduction formulae; infinite and improper integrals; plane polar coordinates; integral inequalities; applications. Probability and Distributions (3 hours): Probability; permutations and combinations. Discrete distributions: mean and variance; expectation values; binomial and Poisson distributions. Continuous distributions: expectation values and moments; Gaussian distribution; simple applications, e.g. velocity distributions. Central limit theorem. Series and limits (3 hours): Summation of series: arithmetic, geometric, arithmeticogeometric series; difference method; series involving natural numbers; transformation of series. Convergence of infinite series: absolute and conditional convergence; alternating series test. Operations with series. Power series: convergence; operations with power series. Taylor series: Taylor's theorem; approximation errors; standard Maclaurin series. Evaluation of limits. Complex numbers and hyperbolic functions (4 hours): Manipulation of complex numbers: addition and subtraction; modulus and argument; multiplication; complex conjugate; division. Polar representation of complex numbers, multiplication and division. De Moivre's theorem: trigonometric identities; n^{th} roots of unity; solving polynomial equations. Complex logarithms and complex powers. Applications to differentiation and integration. Hyperbolic functions: definitions; hyperbolictrigonometric analogies; identities; solving hyperbolic equations; inverses; calculus of hyperbolic functions. Partial differentiation (4 hours): Total differential; total derivative; exact and inexact differentials; useful theorems of partial differentiation; chain rule; change of variables; Taylor's theorem for manyvariable functions; stationary values of manyvariable functions; thermodynamic relations; differentiation of integrals; least squares fits. Multiple integrals (4 hours): Double and triple integrals. Applications of multiple integrals: areas and volumes; masses, centres of mass and centroids; Pappus' theorems; moments of inertia; mean values of functions. Change of variables in multiple integrals. General properties of Jacobians. Vector algebra (2 hours): Basis vectors and components. Magnitude of a vector. Multiplication of vectors: scalar product; vector product; scalar triple product; vector triple product. Equations of lines, planes and spheres. Using vectors to find distances: point to line; point to plane; line to line; line to plane. Reciprocal vectors. Matrices and vector spaces (7 hours): Vector spaces: basis vectors; inner product. Linear operators. Basic matrix algebra. Functions of matrices. Transpose; complex and Hermitian conjugates; trace; determinant; properties of determinants. Inverse of a matrix; rank of a matrix. Special types of square matrix: diagonal; triangular; symmetric and antisymmetric; orthogonal; Hermitian and antiHermitian; unitary; normal. Eigenvectors and eigenvalues of normal, Hermitian and antiHermitian, unitary, and general square matrices. Determination of eigenvalues and eigenvectors: degenerate eigenvalues. Change of basis and similarity transformations. Diagonalization of matrices. Quadratic and Hermitian forms: stationary properties of the eigenvectors; quadratic surfaces. Simultaneous linear equations: range; null space; N simultaneous linear equations in N unknowns; singular value decomposition. Firstorder ordinary differential equations (4 hours): General form of solution. Firstdegree firstorder equations: separablevariable equations; exact equations; inexact equations, integrating factors; linear equations; homogeneous equations; isobaric equations; Bernoulli's equation; miscellaneous equations. Higherdegree firstorder equations: equations soluble for p; for x; for y; Clairaut's equation. Normal modes (2 hours): Typical oscillatory systems; symmetry and normal modes. Higherorder ordinary differential equations (5 hours): Linear equations with constant coefficients: complementary function, particular integral, general solution; linear recurrence relations; Laplace transform method. Linear equations with variable coefficients: The Legendre and Euler linear equations; exact equations; partially known complementary function; variation of parameters; Green's functions; canonical form for secondorder equations. General ordinary differential equations: dependent variable absent; independent variable absent; nonlinear exact equations; isobaric or homogeneous equations. 
Programme availability: 
PH10007 is a Designated Essential Unit on the following programmes:Department of Physics
PH10007 is Compulsory on the following programmes:Programmes in Natural Sciences
PH10007 is Optional on the following programmes:Department of Chemistry

Notes:
