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Module Title
LC Mathematics for Physicists 1
School
Physics and Astronomy
Department
Physics & Astronomy
Module Code
PHYS 41135
Module Lead
Dr Rob Smith and Professor Mike Gunn
Level
Certificate Level
Credits
20
Semester
Full Term
Pre-requisites
Co-requisites
Restrictions
None
Exclusions
Description
Mathematics forms an extremely important part of your programme of studies. It is an important discipline in its own right and all students need to develop a significant competence in the subject. Physics has long played an important part in spurring the development of many important areas of Mathematics - Newton's development of the differential and integral calculus was motivated by the need to describe motion in the world around him. In recent years, the study of symmetry has blossomed into the major area of pure mathematics known as Group Theory, and many of the mathematical ideas developed in Group Theory are now part of the normal language of both crystallographers and particle physicists. Mathematics is the natural language in which Physics is expressed and, as in the study of any language, you need to become fluent both actively, so that you can set up and do calculations on your own, and passively, so that you can follow the mathematical arguments set out by other writers.Some of you will choose to study Mathematics more deeply through study of some of the more theoretical and mathematical modules offered either by this School or by the School of Mathematics, others of you will rely on the core material which all physicists must have mastered and which is covered in the core modules in both years 1 and 2. All the Mathematics which you learn in the first two years will be used somewhere - although in some cases you may need to wait a little while to see the relevance to physics of what you are learning. Where possible in this module, examples will always be given to illustrate where and why the mathematics you are studying is relevant to physics.To learn mathematics effectively it is not sufficient just to attend lectures and get a good set of notes, you need to practice the material you are studying in a fairly intensive way by doing lots of problems and by reading about both the mathematics you are studying and about its applications to physics. We will provide you with fortnightly sets of problems which you should attempt and specialist help will be provided for you in your Mathematics Examples Classes, which are sufficiently small in size that you should be able to get individual help from time to time if you get stuck and need help on a particular topic.Attendance at these Examples Classes is compulsory and part of the end of year module mark is made up of a component for attendance at these classes. The non-assessed Examples Sheets and the Examples Classes should help you get the practice and advice you need to cope with the fortnightly assessed Mathematics problems, the marks from which are used towards the assessment of the module. We also supplement the Examples Sheets by a series of computer-based practice problems on a number of basic mathematical topics that align directly with this module. Using this facility, you can test yourself, practice your mathematical skills and access the worked answers.Students on the Theoretical Physics Programme must get 70% or high in Mathematics for Physicists 1 in order to stay on the programme. A maths mark of lower than 70% will result in you being asked to transfer to a straight Physics programme. High marks in Mathematics for Physicists 2 and other theoretical modules may enable you to re-join the Theoretical Physics programme.
Learning Outcomes
By the end of the module students should be able to:
Differentiate with fluency any reasonable function of one variable
Recognise, sketch and differentiate expressions involving trigonometric and hyperbolic functions and their inverses
Sketch functions of one variable, identify turning points and handle asymptotes correctly
Translate a piece of physics narrative involving rates of change into a mathematical expression
Sum an arithmetic or geometrical progression
Handle limits using de l'Hopital's rule or other appropriate methods
Construct a Taylor series for a function of one variable
Handle the algebra of vectors and apply this to geometrical problems
Calculate and understand scalar and vector products and their physical applications
Construct the equations of lines and planes in vector and Cartesian form
Differentiate vector expressions with respect to a parameter such as time
Handle algebra and calculations involving complex numbers in Cartesian and polar form and use the Argand diagram to represent these
Use Euler's formula and de Moivre's Theorem to handle calculations involving trigonometric problems
Use a phasor to represent a simple harmonic oscillation
Integrate any reasonable function of one variable using all the standard techniques such as integration by parts, substitution etc
Resolve an expression into partial fractions
Construct and use a reduction formula
Translate straightforward mechanical, electrical and other problems into differential equations and understand the role of initial and boundary conditions
Solve first order ordinary differential equations by separating variables, substitution or by using an integrating factor
Solve a second order linear and homogeneous differential equation with prescribed initial or boundary conditions and understand the physical significance of both the equation and the solution (where appropriate)
Appreciate the physical need for dealing with scalar functions of several variables (such as temperature or density) and calculate the partial derivatives of a function of several variables
Write down an expression for the total change df in a function of several variables and use it to calculate the relative error df/f when the independent variable are changed
Calculate ?f in Cartesian coordinates
Use the chain rule to handle changes of variables and to calculate ??f in plane polar coordinates
Construct the Taylor expansion of a function of two variables and use it to identify maxima, minima and saddle points
Use Lagrange multipliers to determine the extrema of constrained problems
Integrate a function of two or three variables over a straightforward geometrical domain using Cartesian coordinates and handle the limits correctly
Use plane, spherical and cylindrical polar coordinates
Sketch a simple curve whose equation is given in plane polar coordinates and calculate the area enclosed by the curve
Construct an expression for an elementary area or volume in the standard polar coordinates using simple geometrical arguments
Use plane, spherical or cylindrical polar coordinates to calculate straightforward integrals of functions of two or three variable
