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## PH10007: Mathematical methods for physics 1

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:
• Coursework (CW 20%)
• Other (OT 20%)
• Examination (EX 60%)
Supplementary Assessment:
PH10007 - Reassessment Examination (where allowed by programme regulations)
Requisites:
Description: Aims:
The aim of this unit is to introduce mathematical techniques required by physical science students, both by showing the application of A-level 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, arithmetico-geometric 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; nth roots of unity; solving polynomial equations. Complex logarithms and complex powers. Applications to differentiation and integration. Hyperbolic functions: definitions; hyperbolic-trigonometric 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 many-variable functions; stationary values of many-variable 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 anti-Hermitian; unitary; normal. Eigenvectors and eigenvalues of normal, Hermitian and anti-Hermitian, 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.
First-order ordinary differential equations (4 hours): General form of solution. First-degree first-order equations: separable-variable equations; exact equations; inexact equations, integrating factors; linear equations; homogeneous equations; isobaric equations; Bernoulli's equation; miscellaneous equations. Higher-degree first-order equations: equations soluble for p; for x; for y; Clairaut's equation.
Normal modes (2 hours): Typical oscillatory systems; symmetry and normal modes.
Higher-order 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 second-order equations. General ordinary differential equations: dependent variable absent; independent variable absent; non-linear exact equations; isobaric or homogeneous equations.
Programme availability:

#### PH10007 is a Designated Essential Unit on the following programmes:

Department of Physics
• USPH-AFB01 : BSc(Hons) Physics (Year 1)
• USPH-AAB02 : BSc(Hons) Physics with Study year abroad (Year 1)
• USPH-AKB02 : BSc(Hons) Physics with Year long work placement (Year 1)
• USPH-AFM02 : MPhys(Hons) Physics (Year 1)
• USPH-AFM04 : MPhys(Hons) Physics with Research placement (Year 1)
• USPH-AAM03 : MPhys(Hons) Physics with Study year abroad (Year 1)
• USPH-AKM03 : MPhys(Hons) Physics with Professional Placement (Year 1)
• USPH-AKM04 : MPhys(Hons) Physics with Professional and Research Placements (Year 1)
• USPH-AFB10 : BSc(Hons) Physics with Astrophysics (Year 1)
• USPH-AAB10 : BSc(Hons) Physics with Astrophysics with Study year abroad (Year 1)
• USPH-AKB10 : BSc(Hons) Physics with Astrophysics with Year long work placement (Year 1)
• USPH-AFM10 : MPhys(Hons) Physics with Astrophysics (Year 1)
• USPH-AFM11 : MPhys(Hons) Physics with Astrophysics with Research placement (Year 1)
• USPH-AAM10 : MPhys(Hons) Physics with Astrophysics with Study year abroad (Year 1)
• USPH-AKM10 : MPhys(Hons) Physics with Astrophysics with Professional Placement (Year 1)
• USPH-AKM11 : MPhys(Hons) Physics with Astrophysics with Professional and Research Placements (Year 1)

#### PH10007 is Compulsory on the following programmes:

Programmes in Natural Sciences
• UXXX-AFB01 : BSc(Hons) Natural Sciences (Biology with Physics stream) (Year 1)
• UXXX-AAB02 : BSc(Hons) Natural Sciences (Biology with Physics stream) with Study year abroad (Year 1)
• UXXX-AKB02 : BSc(Hons) Natural Sciences (Biology with Physics stream) with Year long work placement (Year 1)
• UXXX-AFM01 : MSci(Hons) Natural Sciences (Biology with Physics stream) (Year 1)
• UXXX-AAM02 : MSci(Hons) Natural Sciences (Biology with Physics stream) with Study year abroad (Year 1)
• UXXX-AKM02 : MSci(Hons) Natural Sciences (Biology with Physics stream) with Professional Placement (Year 1)
• UXXX-AFB01 : BSc(Hons) Natural Sciences (Chemistry with Physics stream) (Year 1)
• UXXX-AAB02 : BSc(Hons) Natural Sciences (Chemistry with Physics stream) with Study year abroad (Year 1)
• UXXX-AKB02 : BSc(Hons) Natural Sciences (Chemistry with Physics stream) with Year long work placement (Year 1)
• UXXX-AFM01 : MSci(Hons) Natural Sciences (Chemistry with Physics stream) (Year 1)
• UXXX-AAM02 : MSci(Hons) Natural Sciences (Chemistry with Physics stream) with Study year abroad (Year 1)
• UXXX-AKM02 : MSci(Hons) Natural Sciences (Chemistry with Physics stream) with Professional Placement (Year 1)
• UXXX-AFB01 : BSc(Hons) Natural Sciences (Physics with Biology stream) (Year 1)
• UXXX-AAB02 : BSc(Hons) Natural Sciences (Physics with Biology stream) with Study year abroad (Year 1)
• UXXX-AKB02 : BSc(Hons) Natural Sciences (Physics with Biology stream) with Year long work placement (Year 1)
• UXXX-AFM01 : MSci(Hons) Natural Sciences (Physics with Biology stream) (Year 1)
• UXXX-AAM02 : MSci(Hons) Natural Sciences (Physics with Biology stream) with Study year abroad (Year 1)
• UXXX-AKM02 : MSci(Hons) Natural Sciences (Physics with Biology stream) with Professional Placement (Year 1)
• UXXX-AFB01 : BSc(Hons) Natural Sciences (Physics with Chemistry stream) (Year 1)
• UXXX-AAB02 : BSc(Hons) Natural Sciences (Physics with Chemistry stream) with Study year abroad (Year 1)
• UXXX-AKB02 : BSc(Hons) Natural Sciences (Physics with Chemistry stream) with Year long work placement (Year 1)
• UXXX-AFM01 : MSci(Hons) Natural Sciences (Physics with Chemistry stream) (Year 1)
• UXXX-AAM02 : MSci(Hons) Natural Sciences (Physics with Chemistry stream) with Study year abroad (Year 1)
• UXXX-AKM02 : MSci(Hons) Natural Sciences (Physics with Chemistry stream) with Professional Placement (Year 1)

#### PH10007 is Optional on the following programmes:

Department of Chemistry

 Notes: This unit catalogue is applicable for the 2017/18 academic year only. Students continuing their studies into 2018/19 and beyond should not assume that this unit will be available in future years in the format displayed here for 2017/18. Programmes and units are subject to change in accordance with normal University procedures. Availability of units will be subject to constraints such as staff availability, minimum and maximum group sizes, and timetabling factors as well as a student's ability to meet any pre-requisite rules. Undergraduates: Find out more about these and other important University terms and conditions here. Postgraduates: Find out more about these and other important University terms and conditions here.