SCHOOL OF MATHEMATICS MATHEMATICS FOR ELECTRONIC AND ELECTRICAL ENGINEERING 1. Introduction/Outline MATH1055 Mathematics is an essential tool for the engineer: in this course you are introduced to some of the mathematical techniques which you will need in the rest of your engineering studies. This is a self-study course, written around a textbook there are no lectures. The syllabus is divided into what we call modules, each of which is designed to cover the same amount of work. (Elsewhere in the University, a module refers to a course running over a whole semester or year; here it is a topic, taking about one week.) You have a list of modules (topics) to study and a loose list of deadlines for the topics. For each module you get a set of instructions detailing which section of the textbook to read, which exercises to do, and a specimen test. The modules are in one booklet; all solutions are in a separate booklet. When you think you are ready, come to the testing room (05/2017) and get a test from the administrator. The test should take about 20-30 minutes. After the test, take your work to the marking room (05/2011 or 05/2015) and sit down with any free marker. They will mark your results and go over them with you, giving you feedback on any errors. The mark you get counts towards the 20% coursework mark for this course. This marking session is really about feedback and one-to-one teaching. After marking, take the test back to the administrator in the testing room to have your mark recorded. That s it. If you passed, go home and study the next module. If you didn t pass, go home and revise this module: you have one more go at passing the test. In January there is a one-hour multiple-choice test worth 10% of your final mark. This test covers the material you should have learned until then, approximately the first half of the syllabus. In May there is a final exam, worth 70% of the mark. This covers the material of the entire year. 2. Details STUDY: Make sure you go through all the work in each module, especially the specimen test, which is similar in length and difficulty to the test you will take. Go through your work with the solutions, mark your work. The module instructions are a minimal amount of work necessary to learn a topic. You would of course benefit from doing more than the minimum: there are many more exercises in the book and online, see section 5 below. HELP: If you are having difficulties, come and get help. For short queries, the markers are glad to help. Come to any of the marking sessions and ask away. If there are many people queuing to get their tests marked, they will get priority from the markers, but if 1
you come at the beginning of a session, 11am (Mon) or 10am (Weds), you will certainly find a free marker. For longer queries, come to the Mathematics workshop: 3-6 pm, Mondays and Wednesdays (including exam weeks) in 05/2017. See also section 8 below. TIMING: The test is not timed, so time yourself: if you can do it in about 25 mins, that s good. If you are taking longer than 35 minutes, this means you probably haven t prepared enough: keep that in mind for the next topic. The formula sheet is available for the test, as for the exam in May, so get familiar with what s in there. See also section 10 below. MARKS: Each test is worth about 1% of your final mark, so don t worry too much if you get a low mark every now and then. Go over the material again, think about the feedback you got from the marker, make sure you know it for the test in January, and especially, for the final exam in May, that s when it counts most. The marks are whole numbers from 0 to 4. Marks 0-2 are failing and you have one more chance at taking a test on the same topic. Much more important is that you do each module: study it, take a test, get marked. See also section 7 below.. IF YOU ARRIVE LATE: If you take a long time for a test and don t finish it or get it marked in one session, give it to the administrator who will give it back to you when you come to the next session to finish the test or get it marked. No tests begin in the last half hour of a session, so don t turn up after 12:15am on Monday or 11:15am on Wednesday. CREDIT: Don t walk off without showing your mark to the administrator: you will not get credit. Look on Blackboard occasionally to check that your marks have been recorded. ORDER OF MODULES: The Modules you study are shown below, the order is chosen by ECS. The list is what you are expected to learn this year, and these will be examined in May/June, but don t let that stop you from looking at other modules. All the modules are available on the course website and you are welcome to try them out for yourself. You will only be tested on the modules in your syllabus, but if you want help on any of the other modules, you can ask a marker or use the Mathematics Workshop, see above, under HELP. Semester 1: 5, 3, 4, 14, 15, 7-11, class test. Semester 2: 16-17, 23, 6, 22, 12-13, 21, 18-19, final exam. Note: modules 5,11,14 and 15, although taught as part of the self-paced course, don t form part of the syllabus, they are taught here merely for convenience. For administrative and examination purposes, they form part of other ECS courses. Module 5, complex numbers is part of and will be examined in ELEC1200. Similarly, module 11 on integration and modules 14-15 on vectors form part of and will be examined in ELEC1206. The tests you take here for these modules will form part of your coursework mark in ELEC1200 and ELEC1206. If in any doubt, please discuss this with your ELEC1200/1206 lecturers. As these four modules are a significant part of semester one, they will be included in the test in January. This is the only time these topics contribute to your mark in MATH1055. They will not be part of the final exam in May. 2
PREREQUISITES: Modules 1 and 2 on Algebra and Trigonometry are included for revision. It is assumed that you know this material. If you find that you have forgotten some of the topics in these modules, you are strongly advised to work through them! If you find yourself having to do a lot of catching up with these prerequisites, get some help, see above, under HELP. PACE YOURSELF: The course is self-paced and you are not expected to follow a strict timetable. However, the mathematics is used in the other engineering courses you are taking in your department. You are expected to cover approximately one module per week, in the order listed above. Module 5 on complex numbers is needed early in ELEC1200, and modules 16 and 17 on matrices are needed elsewhere, early in semester two. Your other lecturers will expect you to have learned this by then. And, of course, you need to have advanced to Module 11 by the end of teaching in December, for the multiple-choice test in January, which covers everything you did in semester one. Similarly, in the weeks after the Easter break, only the last X modules will be tested, details in the Organizational Data Sheet. The separate Organizational Data sheet summarises the main facts. Use it for reference. 3. People involved (i) Self-Study-Course Organisers: Ruben Sanchez Garcia and Carsten Gundlach (both Mathematical Sciences) are responsible for the course as a whole. (ii) Academic Supervisors, Mathematics/FEE staff, each run one weekly self-paced session. (iii) Course Administrators are responsible for the distribution of tests, supervising the taking of the tests and keeping an accurate record of progress. (iv) Markers, engineering PhD students plus the academic supervisors: (a) mark and discuss tests and (b) answer all your other mathematical questions. 4. Modules Here is the complete list of self-paced modules, in numerical order. You will study most of them in a slightly different order, but feel free to dip into any of them: 1. Algebra (revision of basic rules; equations; inequalities; partial fractions) 2. Trigonometry (revision of standard trigonometric functions and formulae) 3. Differentiation I (basic rules; standard differentials; Newton s method for finding roots; simple partial differentiation) 4. Integration I (standard integrals; simple substitutions; integration by parts; numerical integration) 5. Complex numbers I (Argand diagram; polar form; exponential form and Euler s formula) 6. Differential equations I (classification; separable first order ODEs; homogeneous second order ODEs with constant coefficients) 7. Functions (functions and inverse functions; trigonometric and inverse trigonometric functions; exponential and logarithmic functions; hyperbolic and inverse hyperbolic functions; differentiation of inverse trigonometric and hyperbolic functions) 3
8. Differentiation II (maxima, minima and points of inflection; curve sketching; parametric, implicit and logarithmic differentiation; Maclaurin s series; Taylor s series) 9. Integration II (more advanced substitutions; applications including volumes of revolution, centroids, centres of gravity, mean values, arc length) 10. Integration III (integration of rational functions; improper integrals) 11. Integration IV (double integrals; polar integrals; triple integrals) 12. Differential equations II (dx/dt = f(x/t); linear and exact first order ODEs) 13. Differential equations III (inhomogeneous second order ODEs with constant coefficients; free and forced oscillations) 14. Vectors I (basic properties; Cartesian components; scalar and vector products) 15. Vectors II (triple products; differentiation and integration of vectors; vector equations of lines and planes) 16. Matrices I (terminology; basic properties; determinants) 17. Matrices II (solving sets of linear equations; calculation of inverse using cofactor and elimination methods) 18. Matrices III (rank of a matrix; eigenvalues and eigenvectors) 19. Further calculus I (chain rule for partial derivatives; higher partial derivatives; total differentials and errors) 20. Further calculus II (sequences and series; Rolle s theorem; Taylor s and Maclaurin s theorems; l Hôpital s rule) 21. Laplace transforms (definition; simple transforms and properties; solution of first and second order linear ODEs with constant coefficients) 22. Complex numbers II (complex trigonometric and hyperbolic functions; logarithm of a complex number; de Moivre s theorem; nth roots; simple loci) 23. Fourier series (periodic signals; whole-range Fourier series; even and odd functions) 24. Statistics I (probability; conditional probability; combinations and permutations; discrete and continuous random variables) 25. Statistics II (mean and standard error of sample data; normal distribution; sampling; confidence intervals; hypothesis testing) 26. Applications to electrical circuits (complex numbers and alternating currents; complex impedance; differential equations for RLC circuits; forced oscillations and resonance; complex solutions of differential equations; phasors) 27. Further applications to electrical circuits (inverse of a matrix using elimination; mesh analysis of circuits; node analysis of circuits) 5. What is in a Module? A Module consists of a batch of printed material which is to be used in conjunction with either of the course textbooks, and from Module 3 onwards it essential to have easy access to a copy of a book. The main book, which will be referred to as J. throughout the Modules, is Modern Engineering Mathematics by Glyn James, Prentice Hall 2007 (4th edition), ISBN 978-0132391443, Library Code TA 150 JAM (15 copies, 3 in short loan). Those of you with a less mathematical background may prefer to use the alternative text (referred to as S.), 4
Engineering Mathematics by K A Stroud, Palgrave 2007 (6th edition), ISBN 978-403942463, Library Code QA 100 STR (8 copies, 2 in short loan). These two textbooks, J. and S. were chosen by a group of engineers and mathematicians to be the most appropriate books for this course. However, no book (or set of lecture notes) is perfect for everyone and the intention of the Modules is to guide you through the book, sometimes supplementing its contents. Each Module begins with Module Topics a list of the main points covered in it. In some Modules there may also be an introductory paragraph. This is followed by the Work Scheme based on James (4th edition), which is split into numbered sections. Most of the sections will refer to particular parts of J., which you will be asked to read (usually for background) or study (essential for the understanding of the Module). Some sections of the work scheme, however, will contain material not in J. At various places in the work scheme you will be asked to do Exercises, most of which will be taken from J. The handwritten Worked Solutions to these Exercises are included in the solutions booklet. Some of you with a weaker mathematical background may find the book by Stroud, a programmed learning text, more appropriate. This covers less of the syllabus, but can be very useful for certain Modules. Therefore, a Work Scheme based on Stroud (6th edition) is also included in each Module. After completing the work scheme you should attempt the Specimen Test, which is included in the Module and is similar to the test you will take in the testing session. Solutions to the Specimen Test are included at the end of each Module. These tests show whether you know the basic material in the Modules, although the examination Section B questions (see Section 10) can be longer and more difficult. Lecture notes which summarise the content of the Modules 1 25 are on the web (see URL at end). Using these is optional but they may be a good introduction to each module. But you still want to work through the modules and sample problems. You learn by doing! A new edition of S. has been published in 2012, but the modules refer to the old 6 th edition, published in 2007. If you feel you need the gentler approach of Stroud, you can get it from the Library or buy a second-hand copy of the sixth edition. Do not buy the new seventh edition, as the references in the modules may not be accurate All the references to these textbooks in Modules 1-27 are up to date for the editions published in 2007. On the course web-page (see Section 11) you will also find versions of the typeset parts of Modules 1-26 which refer to the previous editions of J. and S, also available in the Library. These are equally well suitable for the course. For those that buy a new copy of J., they will find that they get an access card, at the back of the book, to the MyMathLab online programme run by Pearson, the publisher. This is for those of you that want extra practice, see next page. If you bought a second-hand copy of the text, it may not have the access card or the code may have become outdated. In that case, you can still get access from Pearson, for a price. 5
How to Register for MyMathLab Global and Enrol in Your Course Your instructor has set up a MyMathLab Global course for you. The Course name is: MyMathLab Global with James, Modern Engineering Maths 4th Edition It is based on the following book: James, Modern Engineering Mathematics, 4e Your Course ID is: XL00-G1W0-401Z-63N4 In order to join this course, you will need to: 1. Register for MyMathLab Global 2. Enrol in your instructor's course To register, you will need a student access code and a course ID. If you don't have a kit with a student access code, you can purchase access online at http://www.mymathlab.com/global - you will have the choice to purchase access with or without a full etext. 1. Registering for MyMathLab Global 1. Go to the MyMathLab Global home page at http://www.mymathlab.com/global and click the Student button, in the Register section. 2. Enter your Course ID XL00-G1W0-401Z-63N4 and click Next. 3. Choose to register an access code (came with your new book) or purchase access if you don t have an access kit or code. 4. Click the button to proceed to registration. 5. Follow the instructions to create your account. 6. Click the link to login. You ll be directed to http://www.mymathlab.com/global, where you can enter the user name and password you just created. Click the login button. The first time you enter the site you ll be asked to enter your Course ID. Be sure to click on the Browser Check link in the upper right of the screen the first time you login and anytime you use a new computer. You will be alerted to any new software required for your course, such as Flash. You will need administrative rights to install any new software on your computer. 2. Enrolling in your Instructor's course After registering, log in to MyMathLab Global with your username and password. To enrol in the appropriate course, use the Course ID for your instructor's course. IMPORTANT: The Course ID for your course is: XL00-G1W0-401Z-63N4 To view a complete set of instructions on registering and enrolling, visit the Tours page of the MyMathLab Global web site. 6
6. How to study You may go about the Self-study course in whatever manner you choose, but these notes may be of some guidance. First of all, understanding is not an all or nothing process; one understands at various levels. Thus, when studying a section of a book it is suggested that you might adopt the following approach. First read the section quickly and try to get a feeling for the scope of the material and its level. Then go back over it carefully, more than once if necessary, each time trying to get a deeper understanding of what you are reading. If you come across something on which you are stuck leave it for the time being and carry on. The penny may well drop later. Even leaving it for a day or two can sometimes help. If it does not, see any Marker before going for a test. You will get more out of your reading if you become actively involved in the sense of constantly asking yourself questions such as What is this all about?, Why is it done this way? and Is this a significant or a trivial point?. Simply underlining key words in the text or jotting down thoughts and queries in the margin can be valuable ways of increasing your concentration. Another golden rule in studying is that a little and often is far better than a lot in one go. So do not try to cram everything in just before taking a test. It is far better to spread your learning over the week. The cramming approach may just enable you to pass the test but you will find that the knowledge gained will not stick and your ignorance will catch up with you later. Remember that if you pass the test without being fully prepared for it you are cheating nobody but yourself the tests are primarily for your benefit and to tell you whether you have properly understood material on which you will subsequently be examined and which is essential for the proper understanding of your engineering work. It is very important to go through the worked examples in J - one often first understands a piece of bookwork by seeing it in action in a particular instance. Try the Module exercises - unless you can do the exercises you haven t fully understood the bookwork! This is so important that it is worth repeating: try the exercises before you look at the worked solutions. Even if you do not get very far with some of them it will be of much greater benefit to you if you try them on your own first. This way you can often isolate your difficulty and then, when you do see how something is done, it is much more likely to stick. It is far too easy to read through a solution thinking you understand what is going on but then to find that you are completely unable to do anything like it yourself subsequently. Finally, do the specimen test and check your answers! The specimen test is of the same length, difficulty and choice of material as the module test and will give you a very good idea of how well you have absorbed the material. 7. Testing and Marking For each mathematics period on your timetable your group has been allocated two rooms; a testing room and a marking room details are given on the sheet Organizational Data. However, for the smooth running of the system (i.e. minimising queues) we suggest you don t all come at once, see the Organizational Data sheet for detail, although, of course, if you need help or are behind you can attend any of the available 4 periods. 7
When you feel competent in a Module, go to the testing room and ask the Administrator for the appropriate test. Please show the Administrator your ID card so s/he can easily recognize your name in his/her list of all students. When you have collected the test, sit down, write your name and School on the test sheet and answer the questions by writing on the test sheet. Note that during the testing sessions you MAY consult the Formula Sheet but nothing else. The object of the exercise is for you and the Marker to see what YOU know! The test should take approximately 25 minutes to complete. If you get stuck with something in the test do not spend too long over it. In any case do not spend more than 35 minutes on a test unless you are very near completion. After you have finished the test bring it to the marking room. Choose any Marker in the marking room and ask him/her to mark your test. If you prefer you may choose the same Marker every week. In order to allow adequate time for discussion you should take your test sheet to the marking room at least 15 minutes before the end of the session. There may not be sufficient time to mark all tests on that day. In this case you should return the unmarked test to the Administrator and come back at the beginning of the next session. The test will be marked and discussed with you and you will be given a mark of 0, 1, 2, 3 or 4. The main purpose of the test is to discover whether you have sufficient knowledge to proceed to the next Module. The discussion with the Marker is the key feedback in this module. It provides you with an opportunity to obtain help with any difficulties you may have. If you are given a 3 or 4, you have passed. Then take the marked test back to the Administrator who will record your mark. The marked test will be returned to you; please keep it for the whole semester; it will be useful when you revise for the exam and, in some rare cases, you may be asked to show it once more for administrative reasons. If you are given a mark of 0, 1 or 2 you have not passed, the Administrator will record your mark and you will be asked to return on a different day to take a different test. The Marker may also suggest that you should attend the Mathematics Workshop, see Section 8. A maximum of 2 attempts is allowed for each Module, but only if you get a 0, 1 or 2 the first time around: you don t get a second go if you achieved a 3. The marks awarded for each test (the higher of the 2 marks if you retake a test) count towards your final coursework mark for the course. You may want to check the correct entry of your marks in the Grade Center of the respective Blackboard page, see http://blackboard.soton.ac.uk. If you find any error, please show the corresponding marked test to the Administrator at your next visit to the testing room. 8. Available help (i) Quick queries Except when they are engaged in marking tests the Markers will be available during the timetable periods in the marking room to help you with any points that cause you difficulty. The most appropriate time is usually in the first 30 minutes of the session whilst most people are still taking their tests, or during the last half-hour when most people have finished! Don t be shy: come and ask for help, we are always pleased when people keep us busy. Remember: there are no stupid questions! 8
(ii) All queries If your problems require lengthy discussions then you should attend the Mathematics Workshop (MATH3075). This runs throughout the teaching year, in Eustice J (5/2017), from 15.00 to 18.00 each Monday and Wednesday. It is staffed each afternoon by two tutors and a few copies of the course texts are available for consultation. The Workshop is there principally to support this course but it can also be used by any other student in the University with mathematical queries. The Workshop has proved an extremely useful facility for first and second year engineering students. If you experience difficulties during the year, or your mathematical background is weak or rusty, then you are strongly advised to make use of the Workshop. You can drop in any time it is open, for five minutes with a quick query, or go along for the full three hours each session and work through the week s Module with help readily available when you get stuck. It is there to help you. Use it. 9. Progress through the Modules On the sheet Organizational Data you will find a schedule which shows the order of the Modules and the deadlines by which you should have passed the test on each Module. Some of you will be able to proceed at a faster rate. If you fall behind you will be sent a reminder e-mail, with a copy to your personal Engineering Tutor. 10. Assessment At the end the year, in May, you sit a two-hour written final examination. The final exam covers the entire year s material. It counts for 70%, the self-paced tests for 20%, and the one-hour multiple-choice test in January for 10%. MATH1055 is core to your programme, so you must pass the course. If you fail, you will normally be required to take a referral/resit examination paper in August/September (or at the normal examining time in the following year). The format of the examination paper will be as follows: Section A with short multiplechoice questions, similar to the multiple-choice questions in the January test, and Section B containing longer questions. All questions are compulsory. You must be able to do the longer questions in the Modules to be able to cope with the longer questions in the examination. 11. Web A web site for the self-study course can be found at http://www.soton.ac.uk/~cjg/eng1/ This site is also linked through http://blackboard.soton.ac.uk. Click on External Links and Course webpage on the Blackboard page of your self-study course. At the course web-site you will find the Formula Sheet, this Course Description (= Module 0), Organizational Data sheets, Modules 1-27, lecture notes for Modules 1-25 and examination papers and solutions from recent years. Note that MATH1055 ran as a full-year module in 2012/13 for the first time, but the syllabus of its predecessors MATH1013 and MATH1017 were very similar, so you will find their old exams for practice, too. 9
12. Feedback Please send any comments you have about the course (e.g typographical errors in the paperwork, topics which you feel could be better explained) to the respective Academic Supervisors or to the Self-Study-Course Organisers: Prof Carsten Gundlach, Mathematical Sciences, office 54/2017, telephone 023 8059 5116, email cjg@soton.ac.uk Dr Ruben Sanchez Garcia, Mathematical Sciences, office 54/8023, telephone 023 8059 3655, email R.Sanchez-Garcia@soton.ac.uk 10