MATH1055 MATHEMATICS FOR ELECTRONIC AND ELECTRICAL ENGINEERING 1. Introduction: Aims and Objectives MODULE 0: COURSE DESCRIPTION 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. The aims and objectives of this course include the knowledge and understanding of the material, but also other additional skills you are expected to develop through this self-paced course. Having successfully completed this module, you will be able to: Demonstrate knowledge and understanding of basic differential and integral calculus, differential equations, complex numbers, vectors and matrices, and be familiar with partial differentiation and some more advanced techniques of calculus; Show logical thinking in problem solving; Work more effectively with self-study material; Demonstrate organisational and time-management skills; Critically analyse and solve some mathematical problems; Perform calculations in simple situations and work through some longer examples. 2. Outline 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 a similar 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.) The following are the main points to describing how this self-study course works. All yearspecific information (marking/testing room, timetable, people, etc.) can be found in the Organisational Data document given to you separately. You have a list of modules (topics) to study and a recommended list of deadlines for the topics. For each module (topic) you get a set of instructions detailing which section of the textbook to read, which exercises to do, and a specimen test. All the modules are in in this booklet, in the order they must be studied; all solutions are in a separate solutions booklet, in the same order. Occasionally a module will be self-contained (i.e. all the material is in the module itself with no reference to the textbook). After working through a module, and when you think you are ready, come to the testing room and get a test from the administrator. The test should take you about 20-30 minutes. After finishing the test, take it to the marking room, and sit down with any free marker. They will mark your work 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. 1
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 opportunity at passing the test. In January there is a one-hour multiple-choice class test worth 10% of your final mark. This test covers first half of the syllabus (the first 10 modules). In May/June there is a final exam, worth 70% of the mark. This covers the material of the entire year. 2. Details TEXTBOOK: To be able to follow this self-study course, you need a copy of the textbook. The main book is Modern Engineering Mathematics by Glyn James, Pearson, 2015 (5th edition). However there are alternatives to this textbook: see section 5. 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 you come at the beginning of a session you will probably find a free marker. For longer queries, come to the Mathematics Workshop. See also section 8 below for more details about available help. 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 final exam, so get familiar with what is in there. 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 (this should rarely be necessary). Note that no tests begin in the last half hour of a session, so you cannot start a test after 12:15pm on Mondays or 11.15am on Wednesdays. So please arrive early to avoid disappointment, as there may be queues. 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 class test in January, and especially, for the final exam in May/June, that s when it counts most. The marks are whole numbers from 0 to 4. Marks 0, 1 or 2 means fail and you have one more chance at taking a different test on the same topic. If you fail the same module a second time, the best mark will be recorded and you move on to the next topic. You can only retake a module is you fail, that is, you cannot retake a test to improve a mark of 3 to a 4, as the test are not about scoring a high mark but about checking your understanding, learning, and preparing yourself for the final exam. See also section 7 below. 2
ARRIVE EARLY: Make sure you turn up on time for the testing/marking sessions, especially as you cannot take a test in the last half an hour, and queues may build up at busy times. RECORDING YOUR MARK: Don t walk off without having your test marked, and the mark recorded by the administrator. If you take an unmarked or unrecorded test home, you will get a zero mark for that attempt. Please look at Grade Centre on Blackboard regularly to check that your marks have been correctly recorded. It is your responsibility to make sure that the correct mark has been recorded on Blackboard. Please let the administrator know if there is a problem, but allow at least 2 weeks since you did the test for the marks to be uploaded. MULTIPLE TESTS: You may do more than one test on the same session, time permitting: if you pass a test, you may take a test for the next module if you wish to. However, if you fail a test (for the first time), you must go home, revise, and take another test on the same module on a different day. Also, you cannot take more than one test per session after the Easter break, see below. ORDER OF MODULES: The modules (topics) that you will study are shown below, in the order chosen by your Faculty. You must study the modules, and take the tests, in this order (no skipping allowed). This is the material you are expected to learn this year, to a greater or lesser degree, without picking and choosing the material you want to learn, and the material you want to discard: they are all equally important. Moreover, some modules are prerequisites for later modules, or are needed for other courses in your first year (see below), and all this have been reflected in the order below. Semester 1: 5, 3, 4, 14, 15, 7, 8, 9, 10, 11, class test. Semester 2: 16, 17, 23, 6, 22, 12, 13, 21, 18, 19, final exam. Note: You may start Semester 2 modules in Semester 1, if you have already finished all the previous modules. In fact, it is not impossible to finish all 20 modules by January! 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, or section 8 below). 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 class test in January. Similarly, in the weeks after the Easter break, you can only take one module per session, to avoid overcrowding. Note that the University is closed on May Bank Holiday Monday, that you may miss test/marking sessions due to field trips, illness, etc. and that you must come another day if you fail a module test for the first time. Please plan ahead and make sure you do not run out of time for taking tests. Untaken 3
tests get a mark of zero. Finally, note that we send personal tutors a weekly progress report, and that if you do fall behind, we may contact you and your tutor. REMEMBER: Taking and passing the module tests is not a goal but the means to a goal: It is very important to plan your pace throughout the year so that you do each module in a way that prepares you for the final exam. In particular, avoid shortterm memorisation of topics/exercises and make sure you are confident with the material, including all types of exercises, before moving on. 3. People involved (i) Co-ordinators: Ruben Sanchez Garcia and Carsten Gundlach (both Mathematical Sciences) are responsible for the course as a whole. (ii) Academic Supervisors: Mathematics or FEE staff, each run one weekly self-paced session. (iii) Administrators: responsible for the distribution and invigilation of tests, and keeping an accurate record of your progress. (iv) Markers: Engineering or Mathematics PhD students plus the academic supervisors; they (a) mark and discuss tests, and (b) answer all your short mathematical questions. (v) Mathematics helpers: Mathematics PhD students who run the Mathematics Workshop (Mondays & Wednesdays 3-6pm); they will answer all your (short or long) questions and provide any mathematical help you may need. 4. List of modules (topics) Here is the complete list of self-paced modules, in numerical order. You will study most of them in a slightly different order (see above, or in the Organisational Data sheet), but feel free to dip into any of them. You will only be tested and examined on the modules in your syllabus, but all the modules are available on Blackboard and on the course website, and you are welcome to try them out for yourself. If you want help on any of the other modules, you can ask a marker, or use the Mathematics Workshop, see section 8 below. 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) 8. Differentiation II (maxima, minima and points of inflection; curve sketching; parametric, implicit and logarithmic differentiation; Maclaurin s series; Taylor s series) 4
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 (topic) 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, Pearson 2015 (5th edition), ISBN 978-1-292-08073-4 (print) You will be given a copy of the 5th edition, and copies of the 3rd and 4th edition are also available in the library. Those of you with a less mathematical background may prefer to use the alternative text, referred to as S. throughout the modules, 5
Engineering Mathematics by K A Stroud, Palgrave 2007 (6th edition), ISBN 978-403942463, Library Code QA 100 STR (10 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 (FIFTH 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 in the solutions booklet too. 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. Optional lecture notes which summarise the content of the Modules 1 25 are on Blackboard and on the course website (see section 11). Using these is optional, but they may be a good introduction to each module, although you still need to work through the modules and sample problems in the self-study guide. You learn mathematics 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 J. in Modules 1-27 are up to date for the 5th edition published in 2015. However, on Blackboard and on the course website (see section 11) you will also find versions of the self-study guide for Modules 1-25 which refer to the previous editions of J. and S., also available in the Library. These are equally well suitable for the course. Pearson, the publisher of J., runs an online platform called MyMathLab. This can be helpful for students who need extra practice, as they provide many additional exercises with interactive step-by-step solutions. However, you will need to buy an access card from Pearson, at a cost. You can find more information about MyMathLab on Blackboard. 6
6. How to study You may work through the self-study guide in whatever manner you choose, but the following notes and tips 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. Even leaving it for a day or two before going back to it can sometimes help. If it does not, get help (see section 8 ) 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 will pay the consequences later. Remember that if you pass the test without being fully prepared for it you are only cheating 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 remember it. 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 later on. 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 understood the material. 7. Testing and Marking When you feel competent in a module (topic), go to the testing room and ask the administrator for the appropriate test. Please show the administrator your student ID card. If you do not have your student ID card you will not be allowed to take a test. No other document will be accepted. When you have collected the test, sit down, write your name and department/faculty on the test sheet and answer the questions by writing on the test sheet. During the testing 7
sessions you MAY use a university approved calculator (please bring your own), and consult the Formula Sheet (please bring a copy with you) but nothing else. The test should take approximately 25 minutes to complete. Remember that the object of the test is not to get the best possible score, but for you and the marker to check what you already know, identify those points where you are struggling, and learning through the process. Hence if you get stuck with something in the test do not spend too long over it, but rather ask the marker later. 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 your test. In this case you should return the unmarked test to the administrator and come back at the beginning of the next session. Please do not take unmarked tests home. 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 the unlikely case that we need to double-check your mark. 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 on the same topic. The marker may also suggest that you should attend the Mathematics Workshop, see Section 8. A maximum of two attempts is allowed for each module, but only if you fail (0, 1 or 2 marks); you don t get a second attempt if you already achieved a 3, since the tests are about learning and preparing you for the final exam, not about achieving perfect scores. The marks awarded for each test (the higher of the two marks if you retake a test) count towards your 20% coursework mark for the course. You can check the correct entry of your marks in Blackboard Grade Center (please allow at least 2 weeks for the administrator to upload the marks). 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! (ii) All queries If your problems require lengthy discussions then you should attend the Mathematics Workshop (MATH3075). This runs Mondays and Wednesdays 3-8
6pm throughout the teaching weeks of both semesters, and continues in the May exam period until the final exam. There are two or three mathematics helpers to answer your questions. There are also a few copies of the course texts 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. Assessment At the end the year, in May/June, you sit a two-hour written final examination. The final exam covers the entire year s material. It counts for 70% of your final mark, the self-paced tests for 20%, and the one-hour multiple-choice class test in January for 10%. MATH1055 is core to your programme, so you must pass the course (a final mark of 40% or more). If you fail, you will normally be required to take a referral/resit examination paper in August/September and your final mark will be 100% the referral exam mark (normally capped at 40%). Please ask your personal tutor or the student office if in doubt about referral rules. The format of the examination paper will be as follows: Part A with short multiplechoice questions, similar to the multiple-choice questions in the January test, and Part 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. The rubric of the exam (the first page, containing the exam instructions) will be published on Blackboard in advance. 10. Blackboard and Website All the material for this course is available on Blackboard, and on the following website http://www.soton.ac.uk/~cjg/eng1/ Here 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 MATH1054 ran as a full-year module in 2012/13 for the first time, but the syllabus of its predecessors MATH1010 and MATH1015 were very similar, so you will find their old exams for practice, too. 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 Coordinators: Prof Carsten Gundlach, Mathematical Sciences, office 54/2017, telephone 023 8059 5116, email cjg@soton.ac.uk 9
Dr Ruben Sanchez Garcia, Mathematical Sciences, office 54/8023, telephone 023 8059 3655, email R.Sanchez-Garcia@soton.ac.uk 10