## PH10007: Mathematical methods for physics 1

[Page last updated: 15 October 2020]

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: CW30EX70
Assessment Detail:
• Assessment detail data for this unit is currently being updated as a change has been approved. Updated assessment information will be published here shortly.
Supplementary Assessment:
Like-for-like reassessment (where allowed by programme regulations)
Requisites: You must have A-level Physics (or equivalent) and A-level Mathematics (or equivalent) to take this unit
Description: Aims:
The aim of this unit is to introduce mathematical techniques required by physical science students for first year physics and to enable more advanced study.
The unit also aims to show regularly the application of mathematics to physical problems as well as underpinning mathematical ideas.

Learning Outcomes:
After taking this unit the student should be able to:
* evaluate the derivative of a function and the partial derivative of a function of two or more variables;
* analyse stationary points and apply this for problem solving.
* integrate functions using a variety of standard techniques;
* apply discrete and continuous probabilty distributions to find probabilities of events, expected values and variances.
* write down or derive 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.
* Use mutilpe integrals to find areas, volumes and simple properties of solids,
* find the general solution of first and second order ordinary differential equations and show how a particular solution may be found using boundary conditions;
* solve some first and second order partial differential equations by separation of variables;
* calculate the determinant and inverse of a matrix, and the product of two matrices;
* use matrix methods to solve simple linear systems.

Skills:
Numeracy T/F A, Problem Solving T/F A.

Content:
Preliminary calculus (8 hours, including problems classes)
Differentiation [Limits - continuity, Differentiation from first principles; product, chain, implicit and inverse, logarithmic differentiation; Higher derivatives - Leibnitz' theorem; radius of curvature]
Integration [Integration from first principles; the inverse of differentiation; infinite and improper integrals; parts, tan-half-angle, reduction formulae, integral inequalities, improper integrals;]
Probability and distributions (6 hours, including problems classes)
Definition of probability, Permutations and combinations
Discrete distributions [mean and variance; expectation values; Binomial and Poisson distributions]
Continuous distributions [expectation values, Gaussian distribution including as an approximation for Binomial and Poisson; simple applications, e.g. velocity distributions.]
Central limit theorem
Complex numbers and hyperbolic functions (5 hours, including problems classes)
The need for complex numbers Manipulation of complex numbers [Addition and subtraction; modulus and argument; multiplication; complex conjugate; division]
Polar representation of complex numbers [Multiplication and division in polar form] de Moivre's theorem [Trigonometric identities; finding the nth roots of unity; solving polynomial equations]
Complex logarithms and complex powers Applications to differentiation and integration
Hyperbolic functions [Definitions; hyperbolic-trigonometric analogies; identities of hyperbolic functions; solving hyperbolic equations; inverses of hyperbolic functions; calculus of hyperbolic functions]
Vector algebra (4 hours, including problems classes)
Multiplication of vectors [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; plane polar coordinates]
Reciprocal vectors
Series and limits (4 hours, including problems classes)
Series, Operations with series
Power series [Convergence of power series; operations with power series]
Taylor series [Taylor's theorem; approximation errors; standard Maclaurin series], Evaluation of limits [L'Hopital's rule]
Partial differentiation (6 hours, including problems classes)
Definition of the partial derivative, The total differential and total derivative
Exact and inexact differentials, The chain rule, Taylor's theorem for many-variable functions, Stationary values of many-variable functions, Least squares fits
Multiple integrals (7 hours, including problems classes)
Double integrals, 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
[Change of variables in double integrals; change of variables in triple integrals]
First-order ordinary differential equations (7 hours, including problems classes)
General form of solution
First-degree first-order equations
[Separable-variable equations; exact equations; inexact equations, integrating factors; linear equations; homogeneous equations; Bernoulli's equation; miscellaneous equations]
Higher-degree first-order equations
[Equations soluble for p; for x; for y;]
Higher-order ordinary differential equations (4 hours, including problems classes)
Linear equations with constant coefficients
[Finding the complementary function yc(x); finding the particular integral yp(x); constructing the general solution yc(x) + yp(x); linear recurrence relations;
Matrices and determinants (5 hours, including problems classes)
Linear operators, Matrices, Basic matrix algebra[Matrix addition; multiplication by a scalar; matrix multiplication]
Functions of matrices, The transpose of a matrix
The complex and Hermitian conjugates of a matrix
The trace of a matrix
The determinant of a matrix [Properties of determinants]
The inverse of a matrix
The rank of a matrix
Special types of square matrix [Diagonal; triangular; symmetric and antisymmetric; orthogonal; Hermitian and anti-Hermitian; unitary; normal]
Linear Algebra (5 hours, including problems classes)
Eigenvectors and eigenvalues
[Of a normal matrix; of Hermitian and anti-Hermitian matrices; of a unitary matrix; of a general square matrix]
Determination of eigenvalues and eigenvectors
[Degenerate eigenvalues]
Typical oscillatory systems
Change of basis and similarity transformations
Diagonalization of matrices
[Stationary properties of the eigenvectors; quadratic surfaces]
Simultaneous linear equations
[Range; null space; N simultaneous linear equations in N unknowns; singular value decomposition
Modelling (5 hours, including problems classes)
Mathematisation of systems described by text. Test elements and applying physical laws. e.g. conservation of number, Newton's third law, Rates of change.
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-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-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-AAM12 : 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-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-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-AFB01 : BSc(Hons) Natural Sciences (Environmental Science with Physics stream) (Year 1)
• UXXX-AAB02 : BSc(Hons) Natural Sciences (Environmental Science with Physics stream) with Study year abroad (Year 1)
• UXXX-AKB02 : BSc(Hons) Natural Sciences (Environmental Science with Physics stream) with Year long work 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-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-AFB01 : BSc(Hons) Natural Sciences (Physics with Environmental Science stream) (Year 1)
• UXXX-AAB02 : BSc(Hons) Natural Sciences (Physics with Environmental Science stream) with Study year abroad (Year 1)
• UXXX-AKB02 : BSc(Hons) Natural Sciences (Physics with Environmental Science 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-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-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-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)

 Notes: This unit catalogue is applicable for the 2020/21 academic year only. Students continuing their studies into 2021/22 and beyond should not assume that this unit will be available in future years in the format displayed here for 2020/21. 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.