Engineering in the context of Numerical linear algebra

Technology is the application of conceptual knowledge to achieve practical goals, especially in a reproducible way. The word technology can also mean the products resulting from such efforts, including both tangible tools such as utensils or machines, and intangible ones such as software. Technology plays a critical role in science, engineering, and everyday life.

Technological advancements have led to significant changes in society. The earliest known technology is the stone tool, used during prehistory, followed by the control of fire—which in turn contributed to the growth of the human brain and the development of language during the Ice Age, according to the cooking hypothesis. The invention of the wheel in the Bronze Age allowed greater travel and the creation of more complex machines. More recent technological inventions, including the printing press, telephone, and the Internet, have lowered barriers to communication and ushered in the knowledge economy.

Ancient Chinese scientists and engineers made significant scientific innovations, findings and technological advances across various scientific disciplines including the natural sciences, engineering, medicine, military technology, mathematics, geology and astronomy.

Among the earliest inventions were the abacus, the sundial, and the Kongming lantern. The Four Great Inventions – the compass, gunpowder, papermaking, and printing – were among the most important technological advances, only known to Europe by the end of the Middle Ages 1000 years later. The Tang dynasty (AD 618–906) in particular was a time of great innovation. A good deal of exchange occurred between Western and Chinese discoveries up to the Qing dynasty.

In microeconomics, economic efficiency, depending on the context, is usually one of the following two related concepts:

• Allocative or Pareto efficiency: any changes made to assist one person would harm another.
• Productive efficiency: no additional output of one good can be obtained without decreasing the output of another good, and production proceeds at the lowest possible average total cost.

These definitions are not equivalent: a market or other economic system may be allocatively but not productively efficient, or productively but not allocatively efficient. There are also other definitions and measures. All characterizations of economic efficiency are encompassed by the more general engineering concept that a system is efficient or optimal when it maximizes desired outputs (such as utility) given available inputs.

A mathematical model is an abstract description of a concrete system using mathematical concepts and language. The process of developing a mathematical model is termed mathematical modeling. Mathematical models are used in many fields, including applied mathematics, natural sciences, social sciences and engineering. In particular, the field of operations research studies the use of mathematical modelling and related tools to solve problems in business or military operations. A model may help to characterize a system by studying the effects of different components, which may be used to make predictions about behavior or solve specific problems.

During the Renaissance, great advances occurred in geography, astronomy, chemistry, physics, mathematics, manufacturing, anatomy and engineering. The collection of ancient scientific texts began in earnest at the start of the 15th century and continued up to the Fall of Constantinople in 1453, and the invention of printing allowed a faster propagation of new ideas. Nevertheless, some have seen the Renaissance, at least in its initial period, as one of scientific backwardness. Historians like George Sarton and Lynn Thorndike criticized how the Renaissance affected science, arguing that progress was slowed for some amount of time. Humanists favored human-centered subjects like politics and history over study of natural philosophy or applied mathematics. More recently, however, scholars have acknowledged the positive influence of the Renaissance on mathematics and science, pointing to factors like the rediscovery of lost or obscure texts and the increased emphasis on the study of language and the correct reading of texts.

Marie Boas Hall coined the term Scientific Renaissance to designate the early phase of the Scientific Revolution, 1450–1630. More recently, Peter Dear has argued for a two-phase model of early modern science: a Scientific Renaissance of the 15th and 16th centuries, focused on the restoration of the natural knowledge of the ancients; and a Scientific Revolution of the 17th century, when scientists shifted from recovery to innovation.

Manufacturing engineering or production engineering is a branch of professional engineering that shares many common concepts and ideas with other fields of engineering such as mechanical, chemical, electrical, and industrial engineering. Manufacturing engineering requires the ability to plan the practices of manufacturing; to research and to develop tools, processes, machines, and equipment; and to integrate the facilities and systems for producing quality products with the optimum expenditure of capital.

The manufacturing or production engineer's primary focus is to turn raw material into an updated or new product in the most effective, efficient & economic way possible. An example would be a company uses computer integrated technology in order for them to produce their product so that it is faster and uses less human labor.

A point source of pollution is a single identifiable source of air, water, thermal, noise or light pollution. A point source has negligible extent, distinguishing it from other pollution source geometrics (such as nonpoint source or area source). The sources are called point sources because in mathematical modeling, they can be approximated as a mathematical point to simplify analysis. Pollution point sources are identical to other physics, engineering, optics, and chemistry point sources and include:

• Air pollution from an industrial source (rather than an airport or a road, considered a line source, or a forest fire, which is considered an area source, or volume source)
• Water pollution from factories, power plants, municipal sewage treatment plants and some farms (see concentrated animal feeding operation). The U.S. Clean Water Act also defines municipal separate storm sewer systems and industrial stormwater discharges (such as construction sites) as point sources.
  • Man-made, natural, and groundwater reservoirs can all be contaminated by point source pollution which can threaten human health and safety.
• Noise pollution from a jet engine
• Disruptive seismic vibration from a localized seismic study
• Light pollution from an intrusive street light
• Radio emissions from an interference-producing electrical device

A vacuum (pl.: vacuums or vacua) is space devoid of matter. The word is derived from the Latin adjective vacuus (neuter vacuum) meaning "vacant" or "void". An approximation to such vacuum is a region with a gaseous pressure much less than atmospheric pressure. Physicists often discuss ideal test results that would occur in a perfect vacuum, which they sometimes simply call "vacuum" or free space, and use the term partial vacuum to refer to an actual imperfect vacuum as one might have in a laboratory or in space. In engineering and applied physics on the other hand, vacuum refers to any space in which the pressure is considerably lower than atmospheric pressure. The Latin term in vacuo is used to describe an object that is surrounded by a vacuum.

The quality of a partial vacuum refers to how closely it approaches a perfect vacuum. Other things equal, lower gas pressure means higher-quality vacuum. For example, a typical vacuum cleaner produces enough suction to reduce air pressure by around 20%. But higher-quality vacuums are possible. Ultra-high vacuum chambers, common in chemistry, physics, and engineering, operate below one trillionth (10) of atmospheric pressure (100 nPa), and can reach around 100 particles/cm. Outer space is an even higher-quality vacuum, with the equivalent of just a few hydrogen atoms per cubic meter on average in intergalactic space.

China has been the source of many innovations, scientific discoveries and inventions. This includes the Four Great Inventions: papermaking, the compass, gunpowder, and early printing (both woodblock and movable type). The list below contains these and other inventions in ancient and modern China attested by archaeological or historical evidence, including prehistoric inventions of Neolithic and early Bronze Age China.

The historical region now known as China experienced a history involving mechanics, hydraulics and mathematics applied to horology, metallurgy, astronomy, agriculture, engineering, music theory, craftsmanship, naval architecture and warfare. Use of the plow during the Neolithic period Longshan culture (c. 3000 – c. 2000 BC) allowed for high agricultural production yields and rise of Chinese civilization during the Shang dynasty (c. 1600 – c. 1050 BC). Later inventions such as the multiple-tube seed drill and the heavy moldboard iron plow enabled China to sustain a much larger population through improvements in agricultural output.

Applied mathematics is the application of mathematical methods by different fields such as physics, engineering, medicine, biology, finance, business, computer science, and industry. Thus, applied mathematics is a combination of mathematical science and specialized knowledge. The term "applied mathematics" also describes the professional specialty in which mathematicians work on practical problems by formulating and studying mathematical models.

In the past, practical applications have motivated the development of mathematical theories, which then became the subject of study in pure mathematics where abstract concepts are studied for their own sake. The activity of applied mathematics is thus intimately connected with research in pure mathematics.

Hydrostatics is the branch of fluid mechanics that studies fluids at hydrostatic equilibrium and "the pressure in a fluid or exerted by a fluid on an immersed body". The word "hydrostatics" is sometimes used to refer specifically to water and other liquids, but more often it includes both gases and liquids, whether compressible or incompressible. It encompasses the study of the conditions under which fluids are at rest in stable equilibrium. It is opposed to fluid dynamics, the study of fluids in motion.

Hydrostatics is fundamental to hydraulics, the engineering of equipment for storing, transporting and using fluids. It is also relevant to geophysics and astrophysics (for example, in understanding plate tectonics and the anomalies of the Earth's gravitational field), to meteorology, to medicine (in the context of blood pressure), and many other fields.

Basic research, also called pure research, fundamental research, basic science, or pure science, is a type of scientific research with the aim of improving scientific theories for better understanding and prediction of natural or other phenomena. In contrast, applied research uses scientific theories to develop technology or techniques, which can be used to intervene and alter natural or other phenomena. Though often driven simply by curiosity, basic research often fuels the technological innovations of applied science. The two aims are often practiced simultaneously in coordinated research and development.

In addition to innovations, basic research serves to provide insights and public support of nature, possibly improving conservation efforts. Technological innovations may influence engineering concepts, such as the beak of a kingfisher influencing the design of a high-speed bullet train.

Applied science is the application of the scientific method and scientific knowledge to attain practical goals. It includes a broad range of disciplines, such as engineering and medicine. Applied science is often contrasted with basic science, which is focused on advancing scientific theories and laws that explain and predict natural or other phenomena.

There are applied natural sciences, as well as applied formal and social sciences. Applied science examples include genetic epidemiology which applies statistics and probability theory, and applied psychology, including criminology.

Interdisciplinarity or interdisciplinary studies involves the combination of multiple academic disciplines into one activity (e.g., a research project). It draws knowledge from several fields such as sociology, anthropology, psychology, economics, etc. It is related to an interdiscipline or an interdisciplinary field, which is an organizational unit that crosses traditional boundaries between academic disciplines or schools of thought, as new needs and professions emerge. Large engineering teams are usually interdisciplinary, as a power station or mobile phone or other project requires the melding of several specialties. However, the term "interdisciplinary" is sometimes confined to academic settings.

The term interdisciplinary is applied within education and training pedagogies to describe studies that use methods and insights of several established disciplines or traditional fields of study. Interdisciplinarity involves researchers, students, and teachers in the goals of connecting and integrating several academic schools of thought, professions, or technologies—along with their specific perspectives—in the pursuit of a common task. The epidemiology of HIV/AIDS or global warming requires understanding of diverse disciplines to solve complex problems. Interdisciplinary may be applied where the subject is felt to have been neglected or even misrepresented in the traditional disciplinary structure of research institutions, for example, women's studies or ethnic area studies. Interdisciplinarity can likewise be applied to complex subjects that can only be understood by combining the perspectives of two or more fields.

In mathematics, analytic geometry, also known as coordinate geometry or Cartesian geometry, is the study of geometry using a coordinate system. This contrasts with synthetic geometry.

Analytic geometry is used in physics and engineering, and also in aviation, rocketry, space science, and spaceflight. It is the foundation of most modern fields of geometry, including algebraic, differential, discrete and computational geometry.

In physics and engineering, in particular fluid dynamics, the volumetric flow rate (also known as volume flow rate, or volume velocity) is the volume of fluid which passes per unit time; usually it is represented by the symbol Q (sometimes ${\displaystyle {\dot {V}}}$). Its SI unit is cubic metres per second (m/s).

It contrasts with mass flow rate, which is the other main type of fluid flow rate. In most contexts a mention of "rate of fluid flow" is likely to refer to the volumetric rate. In hydrometry, the volumetric flow rate is known as discharge.

Science, technology, engineering, and mathematics (STEM) is an umbrella term used to group together the related technical disciplines of science, technology, engineering, and mathematics. It represents a broad and interconnected set of fields that are crucial for innovation and technological advancement. These disciplines are often grouped together because they share a common emphasis on critical thinking, problem-solving, and analytical skills. The term is typically used in the context of education policy or curriculum choices in schools. It has implications for workforce development, national security concerns (as a shortage of STEM-educated citizens can reduce effectiveness in this area), and immigration policy, with regard to admitting foreign students and tech workers.

There is no universal agreement on which disciplines are included in STEM; in particular, whether or not the science in STEM includes social sciences, such as psychology, sociology, economics, and political science. In the United States, these are typically included by the National Science Foundation (NSF), the Department of Labor's O*Net online database for job seekers, and the Department of Homeland Security. In the United Kingdom, the social sciences are categorized separately and are instead grouped with humanities and arts to form another counterpart acronym HASS (humanities, arts, and social sciences), rebranded in 2020 as SHAPE (social sciences, humanities and the arts for people and the economy). Some sources also use HEAL (health, education, administration, and literacy) as the counterpart of STEM.

Mathematical optimization (alternatively spelled optimisation) or mathematical programming is the selection of a best element, with regard to some criteria, from some set of available alternatives. It is generally divided into two subfields: discrete optimization and continuous optimization. Optimization problems arise in all quantitative disciplines from computer science and engineering to operations research and economics, and the development of solution methods has been of interest in mathematics for centuries.

In the more general approach, an optimization problem consists of maximizing or minimizing a real function by systematically choosing input values from within an allowed set and computing the value of the function. The generalization of optimization theory and techniques to other formulations constitutes a large area of applied mathematics.

Cybernetics is the transdisciplinary study of circular causal processes such as feedback and recursion, where the effects of a system's actions (its outputs) return as inputs to that system, influencing subsequent action. It is concerned with general principles that are relevant across multiple contexts, including in engineering, ecological, economic, biological, cognitive and social systems and also in practical activities such as designing, learning, and managing. Cybernetics' transdisciplinary character has meant that it intersects with a number of other fields, leading to it having both a wide influence and diverse interpretations.

The field is named after an example of circular causal feedback—that of steering a ship (the ancient Greek κυβερνήτης (kybernḗtēs) refers to the person who steers a ship). In steering a ship, the position of the rudder is adjusted in continual response to the effect it is observed as having, forming a feedback loop through which a steady course can be maintained in a changing environment, responding to disturbances from cross winds and tide.