سئوالات متداول در مورد ریاضیات و مهندسی
ترجـمه: ابوالفضل گروئی
از سایت دانشگاه « بریستول» انگلستان
ریاضیات مهندسی چیست؟
موضوع آن کاربرد ریاضیات و حساب کردن برای مسائل مهندسی نوین است.
با چند مثال چطورید؟
در اینجا نقل قولی از کتاب «بازی، مجموعه و ریاضی»* نوشته ریاضیدان برجسته انگلیسی و ترویج کننده علم « یان استوارت» آورده می شود:
« ریاضیات اساس شیوه زندگی ما هستند. چه تعداد از مردمی که یک برنامه تلویزیونی را تماشا می کنند، درک می کنند که بدون ریاضیات چیزی برای تماشا وجود نداشت؟ ریاضیات یکی از اجزای مهم در کشف امواج رادیوئی بود. ریاضیات، طرح مدارهای الکترونیکی را که سیگنالها را پردازش می کند، کنـترل می نماید. وقتی تصویر روی صفحه در درون لامپ برچیده می شود و می چرخد تا تصویر دیگری را ظاهر کند، حجم ریاضیاتی که مانند گرافیک رایانه ای وارد زندگی شده است، به درستی معلوم نیست... چند سال پیش، گواهی است بر یکی شدن دوباره ریاضیات محض و کاربسته.
توپولوژی به تمام حوزه های جدید دینامیک گشوده شد، هندسه بیضوی چند بعدی برای AT&T پول ساز شد؛ موارد گمنام مانند گروههای p-adic در طراحی شبکه های کارآمد تلفن ظاهر شدند و مجموعه کانتور (Cantor) چگونگی کار قلب را شرح داد...»
* "Game, Set and Math" by: Ian Stewart
فرآیند هندسی استفاده شده برای ساخت «مجموعه کانتور».
برای آشنائی با پروفسور «یان استوارت» و مقالات و کتابهای ایشان (اینجا) را کلیک کنید.
چرا ریاضیات مهندسی بخوانیم؟
برنامه های ریاضیات مهندسی توازنی بین نظریه و عمل برقرار می کنند که هم برانگیزاننده قوای عقلانی و هم موضوعی هستند. آموختن هنر اعمال ریاضیات به مسائل دنیای حقیقی نه تنها شما را به مهارتهای فنی مجهز خواهد کرد، بلکه توانائی شما را برای قضاوت در مورد نقش مهم و فزاینده ای که علم و فناوری در دنیای نوین ایفا می کنند، افزایش خواهد داد. صنعت به خوبی از کمبود مهارتهای لازم در حوزه های مختلف شامل ریاضیات کاربردی و حساب آگاه است. این کمبودها در آینده نزدیک به دلیل تقاضای فزاینده از سوی کارفرمایانی که در رشد صنعتی اروپا نقش دارند، بیشتر خود را نشان خواهد داد.
تفاوت میان ریاضیات کاربردی و ریاضیات مهندسی چیست؟
واضح است که همپوشانی میان ریاضیات کاربردی و ریاضیات مهندسی وجود دارد؛ اما:
ریاضیات مهندسی می تواند به عنوان ریاضیات قابل کاربرد یعنی ریاضیات برای حل مسئله، مدل سازی سیستم ها و کاربردهای صنعتی عمومی در نظر گرفته شود. از سوی دیگر، ریاضیات کاربردی روی مطالب فیزیک ریاضی تمرکز می کند: ریاضیات به کار رفته برای سیستم های فیزیکی؛ برای مثال مکانیک، دینامیک سیال و مدل سازی تصادفی در زیست شناسی.
مدل سازی تصادفی یک نورون (سلول عصبی).
تفاوت میان مهندسی و ریاضیات مهندسی چیست؟
دوره های مهندسی روی اصول، روشها و کاربردها در شاخه اختصاصی شان از مهندسی (هوانوردی، مکانیک، برق/الکترونیک، عمران یا حساب) تمرکز می کنند. ریاضیات مهندسی پایه های ریاضی را برای همه اینها و هم چنین حوزه های کاربرد در پزشکی و علوم اجتماعی فراهم می کند.
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در زیر می توانید متـن کامل و اصلی را به زبان انگلیسی مطالعه فرمائید:
What is Engineering Mathematics?
The subject is the application of mathematics and computing to problems of modern engineering. At Bristol, we are committed to using mathematics to study real world problems of direct engineering, scientific or industrial relevance. Research interests cover mathematics for tomorrow's technology, ranging from artificial intelligence (including fuzzy logic and computational intelligence) to applied nonlinear mathematics and chaos. How about some examples?
Here is a quotation from the book "Game, Set and Math" by the eminent British mathematician and populariser of science Ian Stewart:
"...Mathematics is fundamental to our lifestyle. How many people, watching a television program, realise that without mathematics there would be nothing to watch? Mathematics was a crucial ingredient in the discovery of radio waves. It controls the design of electronic circuits that process the signals. When the picture on the screen rolls up into a tube and spins off to reveal another picture, the quantity of mathematics that has come to life as computer graphics is staggering. ...The last few years have witnessed a remarkable re-unification of pure and applied mathematics. Topology is opening up entire new areas of dynamics; the geometry of multi-dimensional ellipsoids is minting money for AT&T; obscure items such as p-adic groups turn up in the design of efficient telephone networks; and the Cantor set describes how your heart works..."
Why study Engineering Mathematics?
The Engineering Mathematics programmes offer a balance of theory and practice, which is both intellectually stimulating and topical. Learning the craft of applying mathematics to real world problems will not only equip you with technical skills, but will also enhance your ability to make sound judgements on the increasingly important role played by science and technology in the modern world. Industry is well aware of acute skills shortage in various fields including applied mathematics and computing. These shortages are bound to persist in the near future, due to increasing demand from employers in line with European industrial growth.
Why are there two `Maths' Departments at Bristol?
Short Answer: It is a historical accident. The School of Mathematics is part of the Science Faculty, whereas the Engineering Mathematics Department, original a department of `Theoretical Mechanics' grew out of the Engineering Faculty. But serendipity dictates that historical accidents often lead to strength and new opportunities! Long Answer: The two departments fulfill different roles. Both are research led departments, with complementary research expertise. Both run their own degree programmes and both provide service teaching to other departments; the School of Mathematics in the Science Faculty and the Engineering Mathematics Department in the Engineering Faculty. Physically the departments are distinct. The School of Mathematics maintains its own identity by having its own building on University Walk, whereas Engineering Mathematics is fully integrated with other Engineering Departments in the Queens Building and the Merchant Venturers Building thus enhancing its close links with engineering application. There is strong co-operation between the departments, e.g. in the area of nonlinear systems . The teaching of the two departments was assessed jointly in the most recent Teaching Quality Assessment. Some of the strengths and specialisms of the two departments are given below.
School of Mathematics
Engineering Mathematics Department
The department focusses on development and application of the latest mathematics of relevance to current and future technologies. It has two main research groups in artificial intelligence and applied nonlinear mathematics, and a range of other research activities. It has about 25 staff. It runs its own undergraduate degree programmes (MEng) in Engineering Mathematics. In addition it jointly runs degrees in Computer Systems Engineering and Avionics Engineering. Admissions information is available. Research interests :
What is the difference between Applied Mathematics and Engineering Mathematics?
There is clearly an overlap between Applied Mathematics and Engineering Mathematics, BUT: Engineering Mathematics can be thought of as Applicable Mathematics - mathematics for problem solving, modelling of systems, and general industrial applications. Applied Mathematics, on the other hand, focuses on the material of Mathematical Physics: the mathematics applied to physical systems - for example: mechanics, fluid dynamics and stochastic modelling in biology.
What is the difference between Engineering and Engineering Mathematics?
Engineering courses focus on the principles, methods and applications in their particular branch of Engineering (Aeronautical, Mechanical, Electrical/Electronic, Civil or Computing); Engineering Mathematics provides the mathematical foundations for all of these as well as areas of application in Medicine and the Social Sciences.
What are the Career prospects for Engineering Mathematics graduates?
The payoff of a degree in one of the Department's programmes can be an interesting and rewarding career with good travel opportunities. Recent Engineering Mathematics graduates have found employment in academic research, software engineering, electronic engineering, computer manufacture, chemical industries, as well as in management, accountancy and the voluntary sector. The employers include:
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Arthur Anderson and Co.
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BDO Binder Hamlyn, Accountants
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Bovis Construction Company
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Bowaters
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British Aerospace
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British Petroleum
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British Oxygen Company (BOC)
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British Telecom
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CEGB (Berkeley Research Station)
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Civil Service
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Credit Suisse Financial Products
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Ford Motor Company
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GEC Avionics
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GEC Turbine Generators
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Hunting Engineering
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ICI
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Inlingua, Bangkok
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Inmos
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SGS Thompson
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Logica
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Metal Box Company
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Midland Bank
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Mobil Oil
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Motor Industries Research Association
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Oracle Corporation UK
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Peat Marwick Mitchell
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RACAL
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Rank Hovis McDougall
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Renishaw
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Rolls Royce Aero Engines
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Royal Air Force
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Royal Marines
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Royal Navy
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Reuters
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Shell
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STC (Components & Telecommunications)
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SD-Scicon
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Sun Life
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Thames Water Utilities
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Transco
What happened to our Graduates of the last 5 years in the first 6 months since leaving?
32%
Aerospace, Oil and other Industry
22%
Research and Higher Degrees
10%
Consultancy and Information Technology
6%
Self employed
6%
Foreign students returned home
6%
Accountancy, Management consultants
6%
Unemployed
4%
Health Authorities, Charities
2%
Teaching
What about Sponsorship?
Some of the companies listed above have sponsored one or more of our students in the past. However, instead of direct sponsorship, many employers are now moving over to using "A Year in Industry" as a route to finding future employees. They select some of those who have done a year with them for sponsorship and offer others vacation employment. For more information go to the Year in Industry web-site at: http://www.yini.org.uk/ What is the role of Computing in the degree programmes?
Computer based tools are a principal stock in trade of all mathematicians working in science, technology, industry and commerce. Engineering Mathematics students must, therefore, learn to get the best out of these tools. Engineering Mathematics is not Computer Science so you will not spend all your time programming and learning about the inner workings of computers. But you will acquire a deep knowledge of the capabilities of computers and how they can support mathematical work.
Do we cater for students who are not experienced in computer work?
The short answer is a big YES. Students come on the Engineering Mathematics programmes from a very wide variety of different backgrounds. The computing laboratory is the principal laboratory of the Engineering Mathematician. The emphasis is on computer use.
Some students come with no experience at all of computers or computer use. Other students may have as much as ten years experience of computer use, and they may also own a Personal Computer. Computing instruction does not assume any prior knowledge, and the students rapidly converge to a common standard by gaining confidence and experience in supervised laboratories.
How different are the programmes from other Engineering Mathematics type programmes?
Traditional Engineering Mathematics courses focus on Continuum Mechanics - the mathematics needed to model the behaviour of solids, liquids and gases. The Engineering Mathematics programmes at Bristol also specialise in Continuum Mechanics with particular emphasis on Non-Linear Dynamics and Chaos; BUT in addition the programmes specialise in Information Engineering topics such as Artificial Intelligence, Knowledge Engineering and Operations Research. Continuum Mechanics is the mathematical discipline rooted in the ENERGY oriented resources of the Industrial Society. Artificial Intelligence is a key technology for the INFORMATION oriented resources of our Post-Industrial Society. What kind of projects can we do?
In the final year of the Engineering Mathematics programmes, students work in pairs or by themselves on a technical research project of their own. The topic is either chosen from a list of suggestions made by members of staff, or is proposed by the student(s). The project is the largest and most important piece of work they do. It is a creative exercise and is usually a very rewarding experience. Listed below are the titles of a selection of recent projects together with short descriptions.
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Capsize prevention in irregular seas. A number of small ships capsize and sink in rough seas each year, despite advances in design. By making a realistic mathematical model of the rolling ship it is possible to relate the risk of capsize to various design parameters and to make recommendations for reducing the risk.
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An expert system in orthodontics. Orthodontics is the art and science of making people’s teeth look and stay good. It is most important for children, whose teeth often get twisted and crowded in the jaw. This work, done in collaboration with a consultant at the Bristol Dental Hospital, resulted in an expert system, which advises ordinary dentists on the best way to treat children’s teeth. This expert system is now used in dental training and is bought by Dentists.
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Helicopter system identification and control. Fine control of a helicopter is crucial in, for example, maritime rescue. The pilot’s expertise has to be supplemented with a sophisticated control system. There is a need to improve the mathematical models on which the control system is based. This project was undertaken in collaboration with the Royal Aircraft Establishment, Bedford.
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Language modelling for speech recognition. Though much progress has been made in automatic recognition of human speech, the listening computer is not yet a reality. The main limitations are in the ideas used and not in the technology needed for their implementation. Recognition of fluent speech requires sophisticated mathematical models of word and sentence formation. Noise, uncertainty in speech data, variations in intonation and dialect, as well as semantic ambiguity have all to be taken into account. There have been a number of projects under this heading, which is a part of a major research programme.
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Modelling of non-linear car suspension components. Classical methods of analysis of engineering components assume linear behaviour. Some modern car suspension components deliberately include non-linear behaviour in an attempt to achieve improved stability and comfort. A dynamic system simulation program was used to model the car/suspension/road system; it demonstrated the superiority of some types of non-linear components over classical linear ones.
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Automatic generation of state space form of non-linear differential equations. To analyse properties of solutions of high order differential equations, it is convenient to rewrite them as systems of first-order equations. For example, the equation
x’’’ + 2xx’’ + x(x’)2 - sin x = 0
can be written as
x’ = y
y’ = z
z’ = -2xz - xy2 + sin x
The student on this project wrote a computer program capable of rewriting certain higher order differential equations in this fashion.
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Optimal sail design. This project, suggested by a competitive sail-boarding student, had a strong experimental component. A mid-section of a full-sized racing sail was produced by a local sail manufacturer. This was then installed in a variable geometry frame and suspended in an aeronautical wind tunnel to explore lift and drag forces induced by the air stream at various angles of attack and various degrees of sail curvature. Stall characteristics were compared with those more usually found in fixed wing cross-sections.
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Linear instability in electrothermal convection. When a planar liquid layer is subjected to both electrical potential and thermal gradients, it is possible to observe either steady or time varying motion. The nature of this instability was studied. This work was related to an international research project studying electrical augmentation of heat transfer in insulating liquids such as aviation kerosene.
Some Additional Project Titles
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Dynamics of the golf swing
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Level-index computer arithmetic
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Computer assisted timetabling
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Modelling of an aerial for space communication
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Error analysis in Applied Mechanics
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Computer simulation of Epidemics
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Computer Aided diagnosis of Dyslexia
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Computer User’s Advisor - an Expert System
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Electro-hydrodynamic singular perturbations
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Interaction between advection, space-charge diffusion and coulombic repulsion
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Chaos in a magnetic pendulum problem
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Mal-development of Children’s teeth
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Expert System for Sail Boat selection
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Can the Heart beat chaotically?
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Mathematical modelling of Health Care Planning
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Mathematics in the service of industrial forming
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Design of tools using Artificial Reality
Why Four Year Programmes?
To ensure that Bristol Engineering Graduates have professional status throughout Europe (where courses are normally at least four years long).
How hard are the Programmes?
All engineering courses involve a lot of hard work, and these are no exceptions. However, all students who obtain the required entry qualifications should be able to stay the course and graduate, given sufficient motivation.
سئوالات متداول در مورد ریاضیات و مهندسی
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