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108-411: The university's teaching and research are focused on engineering , computer science , and natural sciences . It currently has about 28,100 students (29% women), eight faculties, and about 5,000 staff members (3,800 academics). The institution was founded in 1815 by Emperor Francis I of Austria as the k.k. Polytechnisches Institut (English: Imperial-Royal Polytechnic Institute ). The first rector

216-591: A set whose elements are unspecified, of operations acting on the elements of the set, and rules that these operations must follow. The scope of algebra thus grew to include the study of algebraic structures. This object of algebra was called modern algebra or abstract algebra , as established by the influence and works of Emmy Noether . Some types of algebraic structures have useful and often fundamental properties, in many areas of mathematics. Their study became autonomous parts of algebra, and include: The study of types of algebraic structures as mathematical objects

324-694: A broad range of more specialized fields of engineering , each with a more specific emphasis on particular areas of applied mathematics , applied science , and types of application. See glossary of engineering . The term engineering is derived from the Latin ingenium , meaning "cleverness". The American Engineers' Council for Professional Development (ECPD, the predecessor of ABET ) has defined "engineering" as: The creative application of scientific principles to design or develop structures, machines, apparatus, or manufacturing processes, or works utilizing them singly or in combination; or to construct or operate

432-407: A commercial scale, such as the manufacture of commodity chemicals , specialty chemicals , petroleum refining , microfabrication , fermentation , and biomolecule production . Civil engineering is the design and construction of public and private works, such as infrastructure (airports, roads, railways, water supply, and treatment etc.), bridges, tunnels, dams, and buildings. Civil engineering

540-533: A count of 2,000. There were fewer than 50 engineering graduates in the U.S. before 1865. In 1870 there were a dozen U.S. mechanical engineering graduates, with that number increasing to 43 per year in 1875. In 1890, there were 6,000 engineers in civil, mining , mechanical and electrical. There was no chair of applied mechanism and applied mechanics at Cambridge until 1875, and no chair of engineering at Oxford until 1907. Germany established technical universities earlier. The foundations of electrical engineering in

648-420: A demand for machinery with metal parts, which led to the development of several machine tools . Boring cast iron cylinders with precision was not possible until John Wilkinson invented his boring machine , which is considered the first machine tool . Other machine tools included the screw cutting lathe , milling machine , turret lathe and the metal planer . Precision machining techniques were developed in

756-614: A foundation for all mathematics). Mathematics involves the description and manipulation of abstract objects that consist of either abstractions from nature or—in modern mathematics—purely abstract entities that are stipulated to have certain properties, called axioms . Mathematics uses pure reason to prove properties of objects, a proof consisting of a succession of applications of deductive rules to already established results. These results include previously proved theorems , axioms, and—in case of abstraction from nature—some basic properties that are considered true starting points of

864-637: A fruitful interaction between mathematics and science , to the benefit of both. Mathematical discoveries continue to be made to this very day. According to Mikhail B. Sevryuk, in the January ;2006 issue of the Bulletin of the American Mathematical Society , "The number of papers and books included in the Mathematical Reviews (MR) database since 1940 (the first year of operation of MR)

972-404: A mathematical problem. In turn, the axiomatic method allows for the study of various geometries obtained either by changing the axioms or by considering properties that do not change under specific transformations of the space . Today's subareas of geometry include: Algebra is the art of manipulating equations and formulas. Diophantus (3rd century) and al-Khwarizmi (9th century) were

1080-422: A mathematical statement that is taken to be true without need of proof. If a mathematical statement has yet to be proven (or disproven), it is termed a conjecture . Through a series of rigorous arguments employing deductive reasoning , a statement that is proven to be true becomes a theorem. A specialized theorem that is mainly used to prove another theorem is called a lemma . A proven instance that forms part of

1188-402: A more general finding is termed a corollary . Numerous technical terms used in mathematics are neologisms , such as polynomial and homeomorphism . Other technical terms are words of the common language that are used in an accurate meaning that may differ slightly from their common meaning. For example, in mathematics, " or " means "one, the other or both", while, in common language, it

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1296-535: A population mean with a given level of confidence. Because of its use of optimization , the mathematical theory of statistics overlaps with other decision sciences , such as operations research , control theory , and mathematical economics . Computational mathematics is the study of mathematical problems that are typically too large for human, numerical capacity. Numerical analysis studies methods for problems in analysis using functional analysis and approximation theory ; numerical analysis broadly includes

1404-400: A result, many engineers continue to learn new material throughout their careers. If multiple solutions exist, engineers weigh each design choice based on their merit and choose the solution that best matches the requirements. The task of the engineer is to identify, understand, and interpret the constraints on a design in order to yield a successful result. It is generally insufficient to build

1512-411: A separate branch of mathematics until the seventeenth century. At the end of the 19th century, the foundational crisis in mathematics and the resulting systematization of the axiomatic method led to an explosion of new areas of mathematics. The 2020 Mathematics Subject Classification contains no less than sixty-three first-level areas. Some of these areas correspond to the older division, as

1620-424: A single unknown , which were called algebraic equations (a term still in use, although it may be ambiguous). During the 19th century, mathematicians began to use variables to represent things other than numbers (such as matrices , modular integers , and geometric transformations ), on which generalizations of arithmetic operations are often valid. The concept of algebraic structure addresses this, consisting of

1728-418: A statistical action, such as using a procedure in, for example, parameter estimation , hypothesis testing , and selecting the best . In these traditional areas of mathematical statistics , a statistical-decision problem is formulated by minimizing an objective function , like expected loss or cost , under specific constraints. For example, designing a survey often involves minimizing the cost of estimating

1836-411: A technically successful product, rather, it must also meet further requirements. Constraints may include available resources, physical, imaginative or technical limitations, flexibility for future modifications and additions, and other factors, such as requirements for cost, safety , marketability, productivity, and serviceability . By understanding the constraints, engineers derive specifications for

1944-618: A testament to the ingenuity and skill of ancient civil and military engineers. Other monuments, no longer standing, such as the Hanging Gardens of Babylon and the Pharos of Alexandria , were important engineering achievements of their time and were considered among the Seven Wonders of the Ancient World . The six classic simple machines were known in the ancient Near East . The wedge and

2052-909: A useful purpose. Examples of bioengineering research include bacteria engineered to produce chemicals, new medical imaging technology, portable and rapid disease diagnostic devices, prosthetics, biopharmaceuticals, and tissue-engineered organs. Interdisciplinary engineering draws from more than one of the principle branches of the practice. Historically, naval engineering and mining engineering were major branches. Other engineering fields are manufacturing engineering , acoustical engineering , corrosion engineering , instrumentation and control , aerospace , automotive , computer , electronic , information engineering , petroleum , environmental , systems , audio , software , architectural , agricultural , biosystems , biomedical , geological , textile , industrial , materials , and nuclear engineering . These and other branches of engineering are represented in

2160-622: A way to distinguish between those specializing in the construction of such non-military projects and those involved in the discipline of military engineering . The pyramids in ancient Egypt , ziggurats of Mesopotamia , the Acropolis and Parthenon in Greece, the Roman aqueducts , Via Appia and Colosseum, Teotihuacán , and the Brihadeeswarar Temple of Thanjavur , among many others, stand as

2268-477: A wide expansion of mathematical logic, with subareas such as model theory (modeling some logical theories inside other theories), proof theory , type theory , computability theory and computational complexity theory . Although these aspects of mathematical logic were introduced before the rise of computers , their use in compiler design, formal verification , program analysis , proof assistants and other aspects of computer science , contributed in turn to

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2376-703: Is Fermat's Last Theorem . This conjecture was stated in 1637 by Pierre de Fermat, but it was proved only in 1994 by Andrew Wiles , who used tools including scheme theory from algebraic geometry , category theory , and homological algebra . Another example is Goldbach's conjecture , which asserts that every even integer greater than 2 is the sum of two prime numbers . Stated in 1742 by Christian Goldbach , it remains unproven despite considerable effort. Number theory includes several subareas, including analytic number theory , algebraic number theory , geometry of numbers (method oriented), diophantine equations , and transcendence theory (problem oriented). Geometry

2484-513: Is a broad discipline that is often broken down into several sub-disciplines. Although an engineer will usually be trained in a specific discipline, he or she may become multi-disciplined through experience. Engineering is often characterized as having four main branches: chemical engineering, civil engineering, electrical engineering, and mechanical engineering. Chemical engineering is the application of physics, chemistry, biology, and engineering principles in order to carry out chemical processes on

2592-730: Is associated with anything constructed on or within the Earth. This discipline applies geological sciences and engineering principles to direct or support the work of other disciplines such as civil engineering , environmental engineering , and mining engineering . Geological engineers are involved with impact studies for facilities and operations that affect surface and subsurface environments, such as rock excavations (e.g. tunnels ), building foundation consolidation, slope and fill stabilization, landslide risk assessment, groundwater monitoring, groundwater remediation , mining excavations, and natural resource exploration. One who practices engineering

2700-497: Is called an engineer , and those licensed to do so may have more formal designations such as Professional Engineer , Chartered Engineer , Incorporated Engineer , Ingenieur , European Engineer , or Designated Engineering Representative . In the engineering design process, engineers apply mathematics and sciences such as physics to find novel solutions to problems or to improve existing solutions. Engineers need proficient knowledge of relevant sciences for their design projects. As

2808-403: Is commonly used for advanced parts. Analysis is further subdivided into real analysis , where variables represent real numbers , and complex analysis , where variables represent complex numbers . Analysis includes many subareas shared by other areas of mathematics which include: Discrete mathematics, broadly speaking, is the study of individual, countable mathematical objects. An example

2916-509: Is defined by the set of all similar objects and the properties that these objects must have. For example, in Peano arithmetic , the natural numbers are defined by "zero is a number", "each number has a unique successor", "each number but zero has a unique predecessor", and some rules of reasoning. This mathematical abstraction from reality is embodied in the modern philosophy of formalism , as founded by David Hilbert around 1910. The "nature" of

3024-407: Is either ambiguous or means "one or the other but not both" (in mathematics, the latter is called " exclusive or "). Finally, many mathematical terms are common words that are used with a completely different meaning. This may lead to sentences that are correct and true mathematical assertions, but appear to be nonsense to people who do not have the required background. For example, "every free module

3132-487: Is in Babylonian mathematics that elementary arithmetic ( addition , subtraction , multiplication , and division ) first appear in the archaeological record. The Babylonians also possessed a place-value system and used a sexagesimal numeral system which is still in use today for measuring angles and time. In the 6th century BC, Greek mathematics began to emerge as a distinct discipline and some Ancient Greeks such as

3240-493: Is led by the Rector and four Vice Rectors (responsible for Research, Academic Affairs, Finance as well as Human Resources and Gender). The Senate has 26 members. The University Council, consisting of seven members, acts as a supervisory board. Development work in almost all areas of technology is encouraged by the interaction between basic research and the different fields of engineering sciences at Vienna University of Technology. Also,

3348-586: Is mostly used for numerical calculations . Number theory dates back to ancient Babylon and probably China . Two prominent early number theorists were Euclid of ancient Greece and Diophantus of Alexandria. The modern study of number theory in its abstract form is largely attributed to Pierre de Fermat and Leonhard Euler . The field came to full fruition with the contributions of Adrien-Marie Legendre and Carl Friedrich Gauss . Many easily stated number problems have solutions that require sophisticated methods, often from across mathematics. A prominent example

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3456-404: Is not sufficient to verify by measurement that, say, two lengths are equal; their equality must be proven via reasoning from previously accepted results ( theorems ) and a few basic statements. The basic statements are not subject to proof because they are self-evident ( postulates ), or are part of the definition of the subject of study ( axioms ). This principle, foundational for all mathematics,

3564-1192: Is now more than 1.9 million, and more than 75 thousand items are added to the database each year. The overwhelming majority of works in this ocean contain new mathematical theorems and their proofs." Mathematical notation is widely used in science and engineering for representing complex concepts and properties in a concise, unambiguous, and accurate way. This notation consists of symbols used for representing operations , unspecified numbers, relations and any other mathematical objects, and then assembling them into expressions and formulas. More precisely, numbers and other mathematical objects are represented by symbols called variables, which are generally Latin or Greek letters, and often include subscripts . Operation and relations are generally represented by specific symbols or glyphs , such as + ( plus ), × ( multiplication ), ∫ {\textstyle \int } ( integral ), = ( equal ), and < ( less than ). All these symbols are generally grouped according to specific rules to form expressions and formulas. Normally, expressions and formulas do not appear alone, but are included in sentences of

3672-547: Is often held to be Archimedes ( c.  287  – c.  212 BC ) of Syracuse . He developed formulas for calculating the surface area and volume of solids of revolution and used the method of exhaustion to calculate the area under the arc of a parabola with the summation of an infinite series , in a manner not too dissimilar from modern calculus. Other notable achievements of Greek mathematics are conic sections ( Apollonius of Perga , 3rd century BC), trigonometry ( Hipparchus of Nicaea , 2nd century BC), and

3780-433: Is one of the oldest branches of mathematics. It started with empirical recipes concerning shapes, such as lines , angles and circles , which were developed mainly for the needs of surveying and architecture , but has since blossomed out into many other subfields. A fundamental innovation was the ancient Greeks' introduction of the concept of proofs , which require that every assertion must be proved . For example, it

3888-1019: Is ranked #190 by the QS World University Ranking , #406 by the Center of World University Rankings, and it is positioned among the best 251-300 higher education institutions globally by the Times Higher Education World University Rankings . The computer science department has been consistently ranked among the top 100 in the world by the QS World University Ranking and The Times Higher Education World University Rankings respectively. The Vienna University of Technology has eight faculties led by deans: Architecture and Planning, Chemistry, Civil Engineering, Computer Sciences, Electrical Engineering and Information Technology, Mathematics and Geoinformation, Mechanical and Industrial Engineering, and Physics. The university

3996-554: Is sometimes mistranslated as a condemnation of mathematicians. The apparent plural form in English goes back to the Latin neuter plural mathematica ( Cicero ), based on the Greek plural ta mathēmatiká ( τὰ μαθηματικά ) and means roughly "all things mathematical", although it is plausible that English borrowed only the adjective mathematic(al) and formed the noun mathematics anew, after

4104-428: Is the design and manufacture of physical or mechanical systems, such as power and energy systems, aerospace / aircraft products, weapon systems , transportation products, engines , compressors , powertrains , kinematic chains , vacuum technology, vibration isolation equipment, manufacturing , robotics, turbines, audio equipments, and mechatronics . Bioengineering is the engineering of biological systems for

4212-418: Is the purpose of universal algebra and category theory . The latter applies to every mathematical structure (not only algebraic ones). At its origin, it was introduced, together with homological algebra for allowing the algebraic study of non-algebraic objects such as topological spaces ; this particular area of application is called algebraic topology . Calculus, formerly called infinitesimal calculus,

4320-405: Is the set of all integers. Because the objects of study here are discrete, the methods of calculus and mathematical analysis do not directly apply. Algorithms —especially their implementation and computational complexity —play a major role in discrete mathematics. The four color theorem and optimal sphere packing were two major problems of discrete mathematics solved in the second half of

4428-664: Is traditionally broken into a number of sub-disciplines, including structural engineering , environmental engineering , and surveying . It is traditionally considered to be separate from military engineering . Electrical engineering is the design, study, and manufacture of various electrical and electronic systems, such as broadcast engineering , electrical circuits , generators , motors , electromagnetic / electromechanical devices, electronic devices , electronic circuits , optical fibers , optoelectronic devices , computer systems, telecommunications , instrumentation , control systems , and electronics . Mechanical engineering

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4536-508: Is true regarding number theory (the modern name for higher arithmetic ) and geometry. Several other first-level areas have "geometry" in their names or are otherwise commonly considered part of geometry. Algebra and calculus do not appear as first-level areas but are respectively split into several first-level areas. Other first-level areas emerged during the 20th century or had not previously been considered as mathematics, such as mathematical logic and foundations . Number theory began with

4644-574: The Babylonians and Egyptians began using arithmetic, algebra, and geometry for taxation and other financial calculations, for building and construction, and for astronomy. The oldest mathematical texts from Mesopotamia and Egypt are from 2000 to 1800 BC. Many early texts mention Pythagorean triples and so, by inference, the Pythagorean theorem seems to be the most ancient and widespread mathematical concept after basic arithmetic and geometry. It

4752-753: The Golden Age of Islam , especially during the 9th and 10th centuries, mathematics saw many important innovations building on Greek mathematics. The most notable achievement of Islamic mathematics was the development of algebra . Other achievements of the Islamic period include advances in spherical trigonometry and the addition of the decimal point to the Arabic numeral system. Many notable mathematicians from this period were Persian, such as Al-Khwarizmi , Omar Khayyam and Sharaf al-Dīn al-Ṭūsī . The Greek and Arabic mathematical texts were in turn translated to Latin during

4860-742: The IFIUS World Interuniversity Games in October 2007. 48°11′56″N 16°22′12″E  /  48.19889°N 16.37000°E  / 48.19889; 16.37000 Engineering Engineering is the practice of using natural science , mathematics , and the engineering design process to solve technical problems, increase efficiency and productivity, and improve systems. Modern engineering comprises many subfields which include designing and improving infrastructure , machinery , vehicles , electronics , materials , and energy systems. The discipline of engineering encompasses

4968-831: The Neo-Assyrian period (911–609) BC. The Egyptian pyramids were built using three of the six simple machines, the inclined plane, the wedge, and the lever, to create structures like the Great Pyramid of Giza . The earliest civil engineer known by name is Imhotep . As one of the officials of the Pharaoh , Djosèr , he probably designed and supervised the construction of the Pyramid of Djoser (the Step Pyramid ) at Saqqara in Egypt around 2630–2611 BC. The earliest practical water-powered machines,

5076-542: The Newcomen steam engine . Smeaton designed the third Eddystone Lighthouse (1755–59) where he pioneered the use of ' hydraulic lime ' (a form of mortar which will set under water) and developed a technique involving dovetailed blocks of granite in the building of the lighthouse. He is important in the history, rediscovery of, and development of modern cement , because he identified the compositional requirements needed to obtain "hydraulicity" in lime; work which led ultimately to

5184-505: The Pythagoreans appeared to have considered it a subject in its own right. Around 300 BC, Euclid organized mathematical knowledge by way of postulates and first principles, which evolved into the axiomatic method that is used in mathematics today, consisting of definition, axiom, theorem, and proof. His book, Elements , is widely considered the most successful and influential textbook of all time. The greatest mathematician of antiquity

5292-524: The Renaissance , mathematics was divided into two main areas: arithmetic , regarding the manipulation of numbers, and geometry , regarding the study of shapes. Some types of pseudoscience , such as numerology and astrology , were not then clearly distinguished from mathematics. During the Renaissance, two more areas appeared. Mathematical notation led to algebra which, roughly speaking, consists of

5400-403: The U.S. Army Corps of Engineers . The word "engine" itself is of even older origin, ultimately deriving from the Latin ingenium ( c.  1250 ), meaning "innate quality, especially mental power, hence a clever invention." Later, as the design of civilian structures, such as bridges and buildings, matured as a technical discipline, the term civil engineering entered the lexicon as

5508-446: The controversy over Cantor's set theory . In the same period, various areas of mathematics concluded the former intuitive definitions of the basic mathematical objects were insufficient for ensuring mathematical rigour . This became the foundational crisis of mathematics. It was eventually solved in mainstream mathematics by systematizing the axiomatic method inside a formalized set theory . Roughly speaking, each mathematical object

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5616-517: The inclined plane (ramp) were known since prehistoric times. The wheel , along with the wheel and axle mechanism, was invented in Mesopotamia (modern Iraq) during the 5th millennium BC. The lever mechanism first appeared around 5,000 years ago in the Near East , where it was used in a simple balance scale , and to move large objects in ancient Egyptian technology . The lever was also used in

5724-553: The shadoof water-lifting device, the first crane machine, which appeared in Mesopotamia c.  3000 BC , and then in ancient Egyptian technology c.  2000 BC . The earliest evidence of pulleys date back to Mesopotamia in the early 2nd millennium BC, and ancient Egypt during the Twelfth Dynasty (1991–1802 BC). The screw , the last of the simple machines to be invented, first appeared in Mesopotamia during

5832-624: The water wheel and watermill , first appeared in the Persian Empire , in what are now Iraq and Iran, by the early 4th century BC. Kush developed the Sakia during the 4th century BC, which relied on animal power instead of human energy. Hafirs were developed as a type of reservoir in Kush to store and contain water as well as boost irrigation. Sappers were employed to build causeways during military campaigns. Kushite ancestors built speos during

5940-412: The 14th century when an engine'er (literally, one who builds or operates a siege engine ) referred to "a constructor of military engines". In this context, now obsolete, an "engine" referred to a military machine, i.e. , a mechanical contraption used in war (for example, a catapult ). Notable examples of the obsolete usage which have survived to the present day are military engineering corps, e.g. ,

6048-400: The 17th century, when René Descartes introduced what is now called Cartesian coordinates . This constituted a major change of paradigm : Instead of defining real numbers as lengths of line segments (see number line ), it allowed the representation of points using their coordinates , which are numbers. Algebra (and later, calculus) can thus be used to solve geometrical problems. Geometry

6156-426: The 1800s included the experiments of Alessandro Volta , Michael Faraday , Georg Ohm and others and the invention of the electric telegraph in 1816 and the electric motor in 1872. The theoretical work of James Maxwell (see: Maxwell's equations ) and Heinrich Hertz in the late 19th century gave rise to the field of electronics . The later inventions of the vacuum tube and the transistor further accelerated

6264-405: The 19th century, mathematicians discovered non-Euclidean geometries , which do not follow the parallel postulate . By questioning that postulate's truth, this discovery has been viewed as joining Russell's paradox in revealing the foundational crisis of mathematics . This aspect of the crisis was solved by systematizing the axiomatic method, and adopting that the truth of the chosen axioms is not

6372-532: The 20th century. The P versus NP problem , which remains open to this day, is also important for discrete mathematics, since its solution would potentially impact a large number of computationally difficult problems. Discrete mathematics includes: The two subjects of mathematical logic and set theory have belonged to mathematics since the end of the 19th century. Before this period, sets were not considered to be mathematical objects, and logic , although used for mathematical proofs, belonged to philosophy and

6480-593: The 36 licensed member institutions of the UK Engineering Council . New specialties sometimes combine with the traditional fields and form new branches – for example, Earth systems engineering and management involves a wide range of subject areas including engineering studies , environmental science , engineering ethics and philosophy of engineering . Aerospace engineering covers the design, development, manufacture and operational behaviour of aircraft , satellites and rockets . Marine engineering covers

6588-576: The 9th century. In 1206, Al-Jazari invented programmable automata / robots . He described four automaton musicians, including drummers operated by a programmable drum machine , where they could be made to play different rhythms and different drum patterns. Before the development of modern engineering, mathematics was used by artisans and craftsmen, such as millwrights , clockmakers , instrument makers and surveyors. Aside from these professions, universities were not believed to have had much practical significance to technology. A standard reference for

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6696-531: The Antikythera mechanism, required sophisticated knowledge of differential gearing or epicyclic gearing , two key principles in machine theory that helped design the gear trains of the Industrial Revolution, and are widely used in fields such as robotics and automotive engineering . Ancient Chinese, Greek, Roman and Hunnic armies employed military machines and inventions such as artillery which

6804-478: The Bronze Age between 3700 and 3250 BC. Bloomeries and blast furnaces were also created during the 7th centuries BC in Kush. Ancient Greece developed machines in both civilian and military domains. The Antikythera mechanism , an early known mechanical analog computer , and the mechanical inventions of Archimedes , are examples of Greek mechanical engineering. Some of Archimedes' inventions, as well as

6912-620: The Middle Ages and made available in Europe. During the early modern period , mathematics began to develop at an accelerating pace in Western Europe , with innovations that revolutionized mathematics, such as the introduction of variables and symbolic notation by François Viète (1540–1603), the introduction of logarithms by John Napier in 1614, which greatly simplified numerical calculations, especially for astronomy and marine navigation ,

7020-638: The United States went to Josiah Willard Gibbs at Yale University in 1863; it was also the second PhD awarded in science in the U.S. Only a decade after the successful flights by the Wright brothers , there was extensive development of aeronautical engineering through development of military aircraft that were used in World War I . Meanwhile, research to provide fundamental background science continued by combining theoretical physics with experiments. Engineering

7128-409: The aviation pioneers around the start of the 20th century although the work of Sir George Cayley has recently been dated as being from the last decade of the 18th century. Early knowledge of aeronautical engineering was largely empirical with some concepts and skills imported from other branches of engineering. The first PhD in engineering (technically, applied science and engineering ) awarded in

7236-574: The beginnings of algebra (Diophantus, 3rd century AD). The Hindu–Arabic numeral system and the rules for the use of its operations, in use throughout the world today, evolved over the course of the first millennium AD in India and were transmitted to the Western world via Islamic mathematics . Other notable developments of Indian mathematics include the modern definition and approximation of sine and cosine , and an early form of infinite series . During

7344-503: The concept of a proof and its associated mathematical rigour first appeared in Greek mathematics , most notably in Euclid 's Elements . Since its beginning, mathematics was primarily divided into geometry and arithmetic (the manipulation of natural numbers and fractions ), until the 16th and 17th centuries, when algebra and infinitesimal calculus were introduced as new fields. Since then,

7452-399: The current language, where expressions play the role of noun phrases and formulas play the role of clauses . Mathematics has developed a rich terminology covering a broad range of fields that study the properties of various abstract, idealized objects and how they interact. It is based on rigorous definitions that provide a standard foundation for communication. An axiom or postulate is

7560-553: The derived expression mathēmatikḗ tékhnē ( μαθηματικὴ τέχνη ), meaning ' mathematical science ' . It entered the English language during the Late Middle English period through French and Latin. Similarly, one of the two main schools of thought in Pythagoreanism was known as the mathēmatikoi (μαθηματικοί)—which at the time meant "learners" rather than "mathematicians" in the modern sense. The Pythagoreans were likely

7668-571: The design, development, manufacture and operational behaviour of watercraft and stationary structures like oil platforms and ports . Computer engineering (CE) is a branch of engineering that integrates several fields of computer science and electronic engineering required to develop computer hardware and software . Computer engineers usually have training in electronic engineering (or electrical engineering ), software design , and hardware-software integration instead of only software engineering or electronic engineering. Geological engineering

7776-407: The development and large scale manufacturing of chemicals in new industrial plants. The role of the chemical engineer was the design of these chemical plants and processes. Aeronautical engineering deals with aircraft design process design while aerospace engineering is a more modern term that expands the reach of the discipline by including spacecraft design. Its origins can be traced back to

7884-418: The development of electronics to such an extent that electrical and electronics engineers currently outnumber their colleagues of any other engineering specialty. Chemical engineering developed in the late nineteenth century. Industrial scale manufacturing demanded new materials and new processes and by 1880 the need for large scale production of chemicals was such that a new industry was created, dedicated to

7992-428: The expansion of these logical theories. The field of statistics is a mathematical application that is employed for the collection and processing of data samples, using procedures based on mathematical methods especially probability theory . Statisticians generate data with random sampling or randomized experiments . Statistical theory studies decision problems such as minimizing the risk ( expected loss ) of

8100-421: The first commercial piston steam engine in 1712, was not known to have any scientific training. The application of steam-powered cast iron blowing cylinders for providing pressurized air for blast furnaces lead to a large increase in iron production in the late 18th century. The higher furnace temperatures made possible with steam-powered blast allowed for the use of more lime in blast furnaces , which enabled

8208-409: The first half of the 19th century. These included the use of gigs to guide the machining tool over the work and fixtures to hold the work in the proper position. Machine tools and machining techniques capable of producing interchangeable parts lead to large scale factory production by the late 19th century. The United States Census of 1850 listed the occupation of "engineer" for the first time with

8316-567: The first to constrain the use of the word to just the study of arithmetic and geometry. By the time of Aristotle (384–322 BC) this meaning was fully established. In Latin and English, until around 1700, the term mathematics more commonly meant " astrology " (or sometimes " astronomy ") rather than "mathematics"; the meaning gradually changed to its present one from about 1500 to 1800. This change has resulted in several mistranslations: For example, Saint Augustine 's warning that Christians should beware of mathematici , meaning "astrologers",

8424-824: The framework of cooperative projects with other universities, research institutes and business sector partners is established by the research section of Vienna University of Technology. TU Wien has sharpened its research profile by defining competence fields and setting up interdisciplinary collaboration centres, and clearer outlines will be developed. Research focus points of Vienna University of Technology are introduced as computational science and engineering, quantum physics and quantum technologies, materials and matter, information and communication technology and energy and environment. The EU Research Support (EURS) provides services at Vienna University of Technology and informs both researchers and administrative staff in preparing and carrying out EU research projects. The TU Vienna Bibliothek,

8532-491: The interaction between mathematical innovations and scientific discoveries has led to a correlated increase in the development of both. At the end of the 19th century, the foundational crisis of mathematics led to the systematization of the axiomatic method , which heralded a dramatic increase in the number of mathematical areas and their fields of application. The contemporary Mathematics Subject Classification lists more than sixty first-level areas of mathematics. Before

8640-400: The introduction of coordinates by René Descartes (1596–1650) for reducing geometry to algebra, and the development of calculus by Isaac Newton (1643–1727) and Gottfried Leibniz (1646–1716). Leonhard Euler (1707–1783), the most notable mathematician of the 18th century, unified these innovations into a single corpus with a standardized terminology, and completed them with the discovery and

8748-514: The invention of Portland cement . Applied science led to the development of the steam engine. The sequence of events began with the invention of the barometer and the measurement of atmospheric pressure by Evangelista Torricelli in 1643, demonstration of the force of atmospheric pressure by Otto von Guericke using the Magdeburg hemispheres in 1656, laboratory experiments by Denis Papin , who built experimental model steam engines and demonstrated

8856-591: The limits within which a viable object or system may be produced and operated. Mathematics Mathematics is a field of study that discovers and organizes methods, theories and theorems that are developed and proved for the needs of empirical sciences and mathematics itself. There are many areas of mathematics, which include number theory (the study of numbers), algebra (the study of formulas and related structures), geometry (the study of shapes and spaces that contain them), analysis (the study of continuous changes), and set theory (presently used as

8964-409: The manipulation of numbers , that is, natural numbers ( N ) , {\displaystyle (\mathbb {N} ),} and later expanded to integers ( Z ) {\displaystyle (\mathbb {Z} )} and rational numbers ( Q ) . {\displaystyle (\mathbb {Q} ).} Number theory was once called arithmetic, but nowadays this term

9072-400: The natural numbers, there are theorems that are true (that is provable in a stronger system), but not provable inside the system. This approach to the foundations of mathematics was challenged during the first half of the 20th century by mathematicians led by Brouwer , who promoted intuitionistic logic , which explicitly lacks the law of excluded middle . These problems and debates led to

9180-536: The objects defined this way is a philosophical problem that mathematicians leave to philosophers, even if many mathematicians have opinions on this nature, and use their opinion—sometimes called "intuition"—to guide their study and proofs. The approach allows considering "logics" (that is, sets of allowed deducing rules), theorems, proofs, etc. as mathematical objects, and to prove theorems about them. For example, Gödel's incompleteness theorems assert, roughly speaking that, in every consistent formal system that contains

9288-514: The pattern of physics and metaphysics , inherited from Greek. In English, the noun mathematics takes a singular verb. It is often shortened to maths or, in North America, math . In addition to recognizing how to count physical objects, prehistoric peoples may have also known how to count abstract quantities, like time—days, seasons, or years. Evidence for more complex mathematics does not appear until around 3000  BC , when

9396-474: The power to weight ratio of steam engines made practical steamboats and locomotives possible. New steel making processes, such as the Bessemer process and the open hearth furnace, ushered in an area of heavy engineering in the late 19th century. One of the most famous engineers of the mid-19th century was Isambard Kingdom Brunel , who built railroads, dockyards and steamships. The Industrial Revolution created

9504-654: The proof of numerous theorems. Perhaps the foremost mathematician of the 19th century was the German mathematician Carl Gauss , who made numerous contributions to fields such as algebra, analysis, differential geometry , matrix theory , number theory, and statistics . In the early 20th century, Kurt Gödel transformed mathematics by publishing his incompleteness theorems , which show in part that any consistent axiomatic system—if powerful enough to describe arithmetic—will contain true propositions that cannot be proved. Mathematics has since been greatly extended, and there has been

9612-434: The rise of engineering as a profession in the 18th century, the term became more narrowly applied to fields in which mathematics and science were applied to these ends. Similarly, in addition to military and civil engineering, the fields then known as the mechanic arts became incorporated into engineering. Canal building was an important engineering work during the early phases of the Industrial Revolution. John Smeaton

9720-426: The same with full cognizance of their design; or to forecast their behavior under specific operating conditions; all as respects an intended function, economics of operation and safety to life and property. Engineering has existed since ancient times, when humans devised inventions such as the wedge, lever, wheel and pulley, etc. The term engineering is derived from the word engineer , which itself dates back to

9828-410: The state of mechanical arts during the Renaissance is given in the mining engineering treatise De re metallica (1556), which also contains sections on geology, mining, and chemistry. De re metallica was the standard chemistry reference for the next 180 years. The science of classical mechanics , sometimes called Newtonian mechanics, formed the scientific basis of much of modern engineering. With

9936-657: The study and the manipulation of formulas . Calculus , consisting of the two subfields differential calculus and integral calculus , is the study of continuous functions , which model the typically nonlinear relationships between varying quantities, as represented by variables . This division into four main areas—arithmetic, geometry, algebra, and calculus —endured until the end of the 19th century. Areas such as celestial mechanics and solid mechanics were then studied by mathematicians, but now are considered as belonging to physics. The subject of combinatorics has been studied for much of recorded history, yet did not become

10044-561: The study of approximation and discretization with special focus on rounding errors . Numerical analysis and, more broadly, scientific computing also study non-analytic topics of mathematical science, especially algorithmic- matrix -and- graph theory . Other areas of computational mathematics include computer algebra and symbolic computation . The word mathematics comes from the Ancient Greek word máthēma ( μάθημα ), meaning ' something learned, knowledge, mathematics ' , and

10152-672: The theory under consideration. Mathematics is essential in the natural sciences , engineering , medicine , finance , computer science , and the social sciences . Although mathematics is extensively used for modeling phenomena, the fundamental truths of mathematics are independent of any scientific experimentation. Some areas of mathematics, such as statistics and game theory , are developed in close correlation with their applications and are often grouped under applied mathematics . Other areas are developed independently from any application (and are therefore called pure mathematics ) but often later find practical applications. Historically,

10260-487: The title of his main treatise . Algebra became an area in its own right only with François Viète (1540–1603), who introduced the use of variables for representing unknown or unspecified numbers. Variables allow mathematicians to describe the operations that have to be done on the numbers represented using mathematical formulas . Until the 19th century, algebra consisted mainly of the study of linear equations (presently linear algebra ), and polynomial equations in

10368-414: The transition from charcoal to coke . These innovations lowered the cost of iron, making horse railways and iron bridges practical. The puddling process , patented by Henry Cort in 1784 produced large scale quantities of wrought iron. Hot blast , patented by James Beaumont Neilson in 1828, greatly lowered the amount of fuel needed to smelt iron. With the development of the high pressure steam engine,

10476-504: The two main precursors of algebra. Diophantus solved some equations involving unknown natural numbers by deducing new relations until he obtained the solution. Al-Khwarizmi introduced systematic methods for transforming equations, such as moving a term from one side of an equation into the other side. The term algebra is derived from the Arabic word al-jabr meaning 'the reunion of broken parts' that he used for naming one of these methods in

10584-517: The university library, was founded in 1815. The main library building was designed by the architects Justus Dahinden , Reinhard Gieselmann  [ de ] , Alexander Marchart  [ de ] , Roland Moebius  [ de ] , and partners. Completed in 1987, it features owl sculptures by the Swiss artist Bruno Weber . The main library has six floors of open access areas and reading rooms, with around 700 study desks. The university hosted

10692-576: The use of a piston, which he published in 1707. Edward Somerset, 2nd Marquess of Worcester published a book of 100 inventions containing a method for raising waters similar to a coffee percolator . Samuel Morland , a mathematician and inventor who worked on pumps, left notes at the Vauxhall Ordinance Office on a steam pump design that Thomas Savery read. In 1698 Savery built a steam pump called "The Miner's Friend". It employed both vacuum and pressure. Iron merchant Thomas Newcomen , who built

10800-564: Was Johann Joseph von Prechtl . It was renamed the Technische Hochschule (English: Technical College ) in 1872. In 1975, it was renamed Technische Universität Wien (English: Vienna University of Technology ). As a university of technology , Vienna University of Technology covers a wide spectrum of scientific concepts from abstract pure research and the fundamental principles of science to applied technological research and partnership with industry. The Vienna University of Technology

10908-623: Was a steam jack driven by a steam turbine , described in 1551 by Taqi al-Din Muhammad ibn Ma'ruf in Ottoman Egypt . The cotton gin was invented in India by the 6th century AD, and the spinning wheel was invented in the Islamic world by the early 11th century, both of which were fundamental to the growth of the cotton industry . The spinning wheel was also a precursor to the spinning jenny , which

11016-513: Was a key development during the early Industrial Revolution in the 18th century. The earliest programmable machines were developed in the Muslim world. A music sequencer , a programmable musical instrument , was the earliest type of programmable machine. The first music sequencer was an automated flute player invented by the Banu Musa brothers, described in their Book of Ingenious Devices , in

11124-588: Was developed by the Greeks around the 4th century BC, the trireme , the ballista and the catapult . In the Middle Ages, the trebuchet was developed. The earliest practical wind-powered machines, the windmill and wind pump , first appeared in the Muslim world during the Islamic Golden Age , in what are now Iran, Afghanistan, and Pakistan, by the 9th century AD. The earliest practical steam-powered machine

11232-457: Was first elaborated for geometry, and was systematized by Euclid around 300 BC in his book Elements . The resulting Euclidean geometry is the study of shapes and their arrangements constructed from lines, planes and circles in the Euclidean plane ( plane geometry ) and the three-dimensional Euclidean space . Euclidean geometry was developed without change of methods or scope until

11340-414: Was introduced independently and simultaneously by 17th-century mathematicians Newton and Leibniz . It is fundamentally the study of the relationship of variables that depend on each other. Calculus was expanded in the 18th century by Euler with the introduction of the concept of a function and many other results. Presently, "calculus" refers mainly to the elementary part of this theory, and "analysis"

11448-437: Was not specifically studied by mathematicians. Before Cantor 's study of infinite sets , mathematicians were reluctant to consider actually infinite collections, and considered infinity to be the result of endless enumeration . Cantor's work offended many mathematicians not only by considering actually infinite sets but by showing that this implies different sizes of infinity, per Cantor's diagonal argument . This led to

11556-571: Was split into two new subfields: synthetic geometry , which uses purely geometrical methods, and analytic geometry , which uses coordinates systemically. Analytic geometry allows the study of curves unrelated to circles and lines. Such curves can be defined as the graph of functions , the study of which led to differential geometry . They can also be defined as implicit equations , often polynomial equations (which spawned algebraic geometry ). Analytic geometry also makes it possible to consider Euclidean spaces of higher than three dimensions. In

11664-505: Was the first self-proclaimed civil engineer and is often regarded as the "father" of civil engineering. He was an English civil engineer responsible for the design of bridges, canals, harbors, and lighthouses. He was also a capable mechanical engineer and an eminent physicist . Using a model water wheel, Smeaton conducted experiments for seven years, determining ways to increase efficiency. Smeaton introduced iron axles and gears to water wheels. Smeaton also made mechanical improvements to

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