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The opah is the only fish known to exhibit whole body endothermy where all the internal organs are kept at a higher temperature than the surrounding water. This feature allows opahs to maintain an active lifestyle in the cold waters they inhabit. Unlike birds and mammals, the opah is not a homeotherm despite being an e... | Opah | Wikipedia | 437 | 1490711 | https://en.wikipedia.org/wiki/Opah | Biology and health sciences | Acanthomorpha | Animals | vital_articles |
In addition to the rete mirabile in its gills, the opah also has a rete in the blood supply to its brain and eyes. This helps to trap heat in the cranium and further raise its temperature above the rest of the body. While the rete mirabile in the gills is unique to the opah, the cranial rete mirabile has also evolved i... | Opah | Wikipedia | 430 | 1490711 | https://en.wikipedia.org/wiki/Opah | Biology and health sciences | Acanthomorpha | Animals | vital_articles |
The planktonic opah larvae initially resemble those of certain ribbonfishes (Trachipteridae), but are distinguished by the former's lack of dorsal and pelvic fin ornamentation. The slender hatchlings later undergo a marked and rapid transformation from a slender to deep-bodied form; this transformation is complete by 1... | Opah | Wikipedia | 94 | 1490711 | https://en.wikipedia.org/wiki/Opah | Biology and health sciences | Acanthomorpha | Animals | vital_articles |
Tridacna gigas, the giant clam, is the best-known species of the giant clam genus Tridacna. Giant clams are the largest living bivalve mollusks. Several other species of "giant clam" in the genus Tridacna are often misidentified as Tridacna gigas.
These clams were known to indigenous peoples of East Asia for thousands... | Giant clam | Wikipedia | 407 | 1491594 | https://en.wikipedia.org/wiki/Giant%20clam | Biology and health sciences | Bivalvia | Animals | vital_articles |
Anatomy
Young T. gigas are difficult to distinguish from other species of Tridacninae. Adult T. gigas are the only giant clams unable to close their shells completely, allowing part of the brownish-yellow mantle to remain visible. Tridacna gigas has four or five vertical folds in its shell, which serves as the main ch... | Giant clam | Wikipedia | 492 | 1491594 | https://en.wikipedia.org/wiki/Giant%20clam | Biology and health sciences | Bivalvia | Animals | vital_articles |
In very small clams— dry tissue weight—filter feeding provides approximately 65% of total carbon needed for respiration and growth; comparatively larger clams () acquire only 34% of carbon from this source. A single species of zooxenthellae may be symbionts of both giant clams and nearby reef–building (hermatypic) cora... | Giant clam | Wikipedia | 499 | 1491594 | https://en.wikipedia.org/wiki/Giant%20clam | Biology and health sciences | Bivalvia | Animals | vital_articles |
The fertilized egg floats in the sea for approximately 12 hours until eventually a larva (trochophore) hatches. It then starts to produce a calcium carbonate shell. Two days after fertilization it measures . Soon it develops a "foot," which is used to move on the ground. Larvae also can swim to search for appropriate h... | Giant clam | Wikipedia | 495 | 1491594 | https://en.wikipedia.org/wiki/Giant%20clam | Biology and health sciences | Bivalvia | Animals | vital_articles |
Even in countries where giant clams are easily seen, stories incorrectly depict giant clams as aggressive beings. For instance, although the clams are unable to close their shells completely, a Polynesian folk tale relates that a monkey's hand was bitten off by one, and even though once past larval stage, the clams are... | Giant clam | Wikipedia | 467 | 1491594 | https://en.wikipedia.org/wiki/Giant%20clam | Biology and health sciences | Bivalvia | Animals | vital_articles |
T. gigas has been reported as locally extinct in peninsular Malaysia, while T. derasa and Hippopus porcellanus are restricted to Eastern Malaysia. These recent local extinctions have motivated the introduction of giant clams to Hawaii and Micronesia following maricultural advancements. Restocked individuals in the Phil... | Giant clam | Wikipedia | 83 | 1491594 | https://en.wikipedia.org/wiki/Giant%20clam | Biology and health sciences | Bivalvia | Animals | vital_articles |
An electric stove, electric cooker or electric range is a stove with an integrated electrical heating device to cook and bake. Electric stoves became popular as replacements for solid-fuel (wood or coal) stoves which required more labor to operate and maintain. Some modern stoves come in a unit with built-in extractor ... | Electric stove | Wikipedia | 454 | 1491814 | https://en.wikipedia.org/wiki/Electric%20stove | Technology | Household appliances | null | vital_articles |
After the patent was granted in 1906, manufacturing of Curle Smith's design commenced in October that year. The entire production run was acquired by the electricity supply department of Kalgoorlie Municipality, which hired out the stoves to residents. About 50 appliances were produced before cost overruns became a fac... | Electric stove | Wikipedia | 464 | 1491814 | https://en.wikipedia.org/wiki/Electric%20stove | Technology | Household appliances | null | vital_articles |
In the 1970s, glass-ceramic cooktops started to appear. Glass-ceramic has very low thermal conductivity and a near-zero coefficient of thermal expansion, but lets infrared radiation pass very well. Electrical heating coils or halogen lamps are used as heating elements. Because of its physical characteristics,
A third... | Electric stove | Wikipedia | 119 | 1491814 | https://en.wikipedia.org/wiki/Electric%20stove | Technology | Household appliances | null | vital_articles |
The eastern mole or common mole (Scalopus aquaticus) is a medium-sized North American mole. It is the only species in the genus Scalopus. It is found in forested and open areas with moist sandy soils in northern Mexico, the eastern United States and the southwestern corner of Ontario in Canada.
The eastern mole has gr... | Eastern mole | Wikipedia | 226 | 1493798 | https://en.wikipedia.org/wiki/Eastern%20mole | Biology and health sciences | Eulipotyphla | Animals | vital_articles |
Glacial landforms are landforms created by the action of glaciers. Most of today's glacial landforms were created by the movement of large ice sheets during the Quaternary glaciations. Some areas, like Fennoscandia and the southern Andes, have extensive occurrences of glacial landforms; other areas, such as the Sahara,... | Glacial landform | Wikipedia | 387 | 1494235 | https://en.wikipedia.org/wiki/Glacial%20landform | Physical sciences | Glacial landforms | null | vital_articles |
Later, when the glaciers retreated leaving behind their freight of crushed rock and sand (glacial drift), they created characteristic depositional landforms. Depositional landforms are often made of glacial till, which is composed of unsorted sediments (some quite large, others small) that were eroded, carried, and dep... | Glacial landform | Wikipedia | 449 | 1494235 | https://en.wikipedia.org/wiki/Glacial%20landform | Physical sciences | Glacial landforms | null | vital_articles |
Kettle lake: Depression, formed by a block of ice separated from the main glacier, in which the lake forms
Tarn: A lake formed in a cirque by overdeepening
Paternoster lake: A series of lakes in a glacial valley, formed when a stream is dammed by successive recessional moraines left by an advancing or retreating glac... | Glacial landform | Wikipedia | 469 | 1494235 | https://en.wikipedia.org/wiki/Glacial%20landform | Physical sciences | Glacial landforms | null | vital_articles |
Google Maps is a web mapping platform and consumer application offered by Google. It offers satellite imagery, aerial photography, street maps, 360° interactive panoramic views of streets (Street View), real-time traffic conditions, and route planning for traveling by foot, car, bike, air (in beta) and public transport... | Google Maps | Wikipedia | 493 | 1494648 | https://en.wikipedia.org/wiki/Google%20Maps | Technology | Utility | null | vital_articles |
Acquisitions
Google Maps first started as a C++ program designed by two Danish brothers, Lars and Jens Eilstrup Rasmussen, and Noel Gordon and Stephen Ma, at the Sydney-based company Where 2 Technologies, which was founded in early 2003. The program was initially designed to be separately downloaded by users, but the ... | Google Maps | Wikipedia | 497 | 1494648 | https://en.wikipedia.org/wiki/Google%20Maps | Technology | Utility | null | vital_articles |
2011–2015
On April 19, 2011, Map Maker was added to the American version of Google Maps, allowing any viewer to edit and add changes to Google Maps. This provides Google with local map updates almost in real-time instead of waiting for digital map data companies to release more infrequent updates.
On January 31, 2012... | Google Maps | Wikipedia | 493 | 1494648 | https://en.wikipedia.org/wiki/Google%20Maps | Technology | Utility | null | vital_articles |
On June 27, 2016, Google rolled out new satellite imagery worldwide sourced from Landsat 8, comprising over 700 trillion pixels of new data. In September 2016, Google Maps acquired mapping analytics startup Urban Engines.
In 2016, the Government of South Korea offered Google conditional access to the country's geograp... | Google Maps | Wikipedia | 478 | 1494648 | https://en.wikipedia.org/wiki/Google%20Maps | Technology | Utility | null | vital_articles |
In January 2021, Google announced updates to the route planner that would accommodate drivers of electric vehicles. Routing would take into account the type of vehicle, vehicle status including current charge, and the locations of charging stations.
In June 2022, Google Maps added a layer displaying air quality for ce... | Google Maps | Wikipedia | 485 | 1494648 | https://en.wikipedia.org/wiki/Google%20Maps | Technology | Utility | null | vital_articles |
In late 2014, Google launched Google Underwater Street View, including of the Australian Great Barrier Reef in 3D. The images are taken by special cameras which turn 360 degrees and take shots every 3 seconds.
In 2017, in both Google Maps and Google Earth, Street View navigation of the International Space Station int... | Google Maps | Wikipedia | 493 | 1494648 | https://en.wikipedia.org/wiki/Google%20Maps | Technology | Utility | null | vital_articles |
Lens in Maps
Previously called Search with Live View, Lens In Maps identifies shops, restaurants, transit stations and other street features with a phone's camera and places relevant information and a category pin on top, like closing/opening times, current busyness, pricing and reviews using AI and augmented reality.... | Google Maps | Wikipedia | 481 | 1494648 | https://en.wikipedia.org/wiki/Google%20Maps | Technology | Utility | null | vital_articles |
Genuine business owners can also optimize their business listings to gain greater visibility in Google Maps, through a type of search engine marketing called local search engine optimization.
Indoor maps
In March 2011, indoor maps were added to Google Maps, giving users the ability to navigate themselves within build... | Google Maps | Wikipedia | 512 | 1494648 | https://en.wikipedia.org/wiki/Google%20Maps | Technology | Utility | null | vital_articles |
Implementation
As the user drags the map, the grid squares are downloaded from the server and inserted into the page. When a user searches for a business, the results are downloaded in the background for insertion into the side panel and map; the page is not reloaded. A hidden iframe with form submission is used beca... | Google Maps | Wikipedia | 507 | 1494648 | https://en.wikipedia.org/wiki/Google%20Maps | Technology | Utility | null | vital_articles |
Google Maps API
Google Maps API, now called Google Maps Platform, hosts about 17 different APIs, which are themed under the following categories: Maps, Places and Routes.
After the success of reverse-engineered mashups such as chicagocrime.org and housingmaps.com, Google launched the Google Maps API in June 2005 to a... | Google Maps | Wikipedia | 413 | 1494648 | https://en.wikipedia.org/wiki/Google%20Maps | Technology | Utility | null | vital_articles |
Within China, the State Council mandates that all maps of China use the GCJ-02 coordinate system, which is offset from the WGS-84 system used in most of the world. google.cn/maps (formerly Google Ditu) uses the GCJ-02 system for both its street maps and satellite imagery. google.com/maps also uses GCJ-02 data for the s... | Google Maps | Wikipedia | 481 | 1494648 | https://en.wikipedia.org/wiki/Google%20Maps | Technology | Utility | null | vital_articles |
Google has said it created its maps from third-party data, public sources, satellites, and users, but many names used have not been connected to any official record. According to a former Google Maps employee (who was not authorized to speak publicly), users can submit changes to Google Maps, but some submissions are r... | Google Maps | Wikipedia | 504 | 1494648 | https://en.wikipedia.org/wiki/Google%20Maps | Technology | Utility | null | vital_articles |
In February 2020, the artist Simon Weckert used 99 cell phones to fake a Google Maps traffic jam.
In September 2024, several schools in Taiwan and Hong Kong were altered to incorrect labels, such as "psychiatric hospitals" or "prisons". Initially, it was believed to be the result of hacker attacks. However, police lat... | Google Maps | Wikipedia | 478 | 1494648 | https://en.wikipedia.org/wiki/Google%20Maps | Technology | Utility | null | vital_articles |
In November 2023, a hiker was rescued by helicopter on the backside of Mount Fromme in Vancouver. North Shore Rescue stated on its Facebook page that the hiker had followed a non-existent hiking trail on Google Maps. This was also the second hiker in two months to require rescuing after following the same trail. The fa... | Google Maps | Wikipedia | 492 | 1494648 | https://en.wikipedia.org/wiki/Google%20Maps | Technology | Utility | null | vital_articles |
If imagery was missing, outdated, misaligned, or generally incorrect, one could notify Google through their contact request form.
In November 2016, Google announced the discontinuation of Google Map Maker as of March 2017.
Mobile app
Google Maps is available as a mobile app for the Android and iOS mobile operating ... | Google Maps | Wikipedia | 459 | 1494648 | https://en.wikipedia.org/wiki/Google%20Maps | Technology | Utility | null | vital_articles |
At the end of 2015 Google Maps announced its new offline functionality, but with various limitations – downloaded area cannot exceed 120,000 square kilometers and require a considerable amount of storage space. In January 2017, Google added a feature exclusively to Android that will, in some U.S. cities, indicate the l... | Google Maps | Wikipedia | 292 | 1494648 | https://en.wikipedia.org/wiki/Google%20Maps | Technology | Utility | null | vital_articles |
Reception
USA Today welcomed the application back to iOS, saying: "The reemergence in the middle of the night of a Google Maps app for the iPhone is like the return of an old friend. Only your friend, who'd gone missing for three months, comes back looking better than ever." Jason Parker of CNET, calling it "the king ... | Google Maps | Wikipedia | 509 | 1494648 | https://en.wikipedia.org/wiki/Google%20Maps | Technology | Utility | null | vital_articles |
Artistic and literary uses
The German "geo-novel" Senghor on the Rocks (2008) presents its story as a series of spreads showing a Google Maps location on the left and the story's text on the right. Annika Richterich explains that the "satellite pictures in Senghor on the Rocks illustrate the main character's travel th... | Google Maps | Wikipedia | 164 | 1494648 | https://en.wikipedia.org/wiki/Google%20Maps | Technology | Utility | null | vital_articles |
A Faraday cage or Faraday shield is an enclosure used to block some electromagnetic fields. A Faraday shield may be formed by a continuous covering of conductive material, or in the case of a Faraday cage, by a mesh of such materials. Faraday cages are named after scientist Michael Faraday, who first constructed one in... | Faraday cage | Wikipedia | 415 | 151590 | https://en.wikipedia.org/wiki/Faraday%20cage | Technology | Signal transmission | null | vital_articles |
The reception or transmission of radio waves, a form of electromagnetic radiation, to or from an antenna within a Faraday cage is heavily attenuated or blocked by the cage; however, a Faraday cage has varied attenuation depending on wave form, frequency, or the distance from receiver or transmitter, and receiver or tra... | Faraday cage | Wikipedia | 339 | 151590 | https://en.wikipedia.org/wiki/Faraday%20cage | Technology | Signal transmission | null | vital_articles |
Interior charges
If charge +Q is placed inside an ungrounded Faraday shield without touching the walls, the internal face of the shield becomes charged with −Q, leading to field lines originating at the charge and extending to charges inside the inner surface of the metal. The field line paths in this inside space (to... | Faraday cage | Wikipedia | 447 | 151590 | https://en.wikipedia.org/wiki/Faraday%20cage | Technology | Signal transmission | null | vital_articles |
Faraday cage
Faraday cages are Faraday shields that have holes in them and are therefore more complex to analyze. Whereas continuous shields essentially attenuate all wavelengths whose skin depth in the hull material is less than the thickness of the hull, the holes in a cage may permit shorter wavelengths to pass thr... | Faraday cage | Wikipedia | 141 | 151590 | https://en.wikipedia.org/wiki/Faraday%20cage | Technology | Signal transmission | null | vital_articles |
Faraday cages are routinely used in analytical chemistry to reduce noise while making sensitive measurements.
Faraday cages, more specifically dual paired seam Faraday bags, are often used in digital forensics to prevent remote wiping and alteration of criminal digital evidence.
Faraday bags are portable containers f... | Faraday cage | Wikipedia | 478 | 151590 | https://en.wikipedia.org/wiki/Faraday%20cage | Technology | Signal transmission | null | vital_articles |
The scan room of a magnetic resonance imaging (MRI) machine is designed as a Faraday cage. This prevents external RF (radio frequency) signals from being added to data collected from the patient, which would affect the resulting image. Technologists are trained to identify the characteristic artifacts created on images... | Faraday cage | Wikipedia | 320 | 151590 | https://en.wikipedia.org/wiki/Faraday%20cage | Technology | Signal transmission | null | vital_articles |
Fomalhaut (, ) is the brightest star in the southern constellation of Piscis Austrinus, the Southern Fish, and one of the brightest stars in the night sky. It has the Bayer designation Alpha Piscis Austrini, which is an alternative form of α Piscis Austrini, and is abbreviated Alpha PsA or α PsA. This is a class A star... | Fomalhaut | Wikipedia | 458 | 151651 | https://en.wikipedia.org/wiki/Fomalhaut | Physical sciences | Notable stars | Astronomy | vital_articles |
The star's traditional name derives from Fom al-Haut from scientific Arabic "the mouth of the [Southern] Fish" (literally, "mouth of the whale"), a translation of how Ptolemy labeled it.
Fam in Arabic means "mouth", al "the", and ḥūt "fish"
or "whale".
In 2016, the International Astronomical Union organized a Workin... | Fomalhaut | Wikipedia | 505 | 151651 | https://en.wikipedia.org/wiki/Fomalhaut | Physical sciences | Notable stars | Astronomy | vital_articles |
Properties
Fomalhaut is a young star, for many years thought to be only 100 to 300 million years old, with a potential lifespan of a billion years. A 2012 study gave a slightly higher age of . The surface temperature of the star is around . Fomalhaut's mass is about 1.92 times that of the Sun, its luminosity is about 1... | Fomalhaut | Wikipedia | 489 | 151651 | https://en.wikipedia.org/wiki/Fomalhaut | Physical sciences | Notable stars | Astronomy | vital_articles |
The outermost disk is at a radial distance of , in a toroidal shape with a very sharp inner edge, all inclined 24 degrees from edge-on. The dust is distributed in a belt about 25 AU wide. The geometric center of the disk is offset by about from Fomalhaut. The disk is sometimes referred to as "Fomalhaut's Kuiper belt".... | Fomalhaut | Wikipedia | 296 | 151651 | https://en.wikipedia.org/wiki/Fomalhaut | Physical sciences | Notable stars | Astronomy | vital_articles |
On November 13, 2008, astronomers announced an extrasolar planet candidate, orbiting just inside the outer debris ring. This was the first extrasolar orbiting object candidate to be directly imaged in visible light, captured by the Hubble Space Telescope. The mass of the tentative planet, Fomalhaut b, was estimated to ... | Fomalhaut | Wikipedia | 420 | 151651 | https://en.wikipedia.org/wiki/Fomalhaut | Physical sciences | Notable stars | Astronomy | vital_articles |
Fomalhaut forms a binary star with the K4-type star TW Piscis Austrini (TW PsA), which lies away from Fomalhaut, and its space velocity agrees with that of Fomalhaut within , consistent with being a bound companion. A recent age estimate for TW PsA () agrees very well with the isochronal age for Fomalhaut (), further ... | Fomalhaut | Wikipedia | 263 | 151651 | https://en.wikipedia.org/wiki/Fomalhaut | Physical sciences | Notable stars | Astronomy | vital_articles |
LP 876-10 is also associated with the Fomalhaut system, making it a trinary star. In October 2013, Eric Mamajek and collaborators from the RECONS consortium announced that the previously known high-proper-motion star LP 876-10 had a distance, velocity, and color-magnitude position consistent with being another member o... | Fomalhaut | Wikipedia | 375 | 151651 | https://en.wikipedia.org/wiki/Fomalhaut | Physical sciences | Notable stars | Astronomy | vital_articles |
Etymology and cultural significance
Fomalhaut has had various names ascribed to it through time, and has been recognized by many cultures of the northern hemisphere, including the Arabs, Persians, and Chinese. It marked the solstice in 2500 BC. It was also a marker for the worship of Demeter in Eleusis.
It is consider... | Fomalhaut | Wikipedia | 398 | 151651 | https://en.wikipedia.org/wiki/Fomalhaut | Physical sciences | Notable stars | Astronomy | vital_articles |
In mathematics, Stirling's approximation (or Stirling's formula) is an asymptotic approximation for factorials. It is a good approximation, leading to accurate results even for small values of . It is named after James Stirling, though a related but less precise result was first stated by Abraham de Moivre.
One way of... | Stirling's approximation | Wikipedia | 485 | 151783 | https://en.wikipedia.org/wiki/Stirling%27s%20approximation | Mathematics | Specific functions | null | vital_articles |
Higher orders
In fact, further corrections can also be obtained using Laplace's method. From previous result, we know that , so we "peel off" this dominant term, then perform two changes of variables, to obtain:To verify this: .
Now the function is unimodal, with maximum value zero. Locally around zero, it looks lik... | Stirling's approximation | Wikipedia | 428 | 151783 | https://en.wikipedia.org/wiki/Stirling%27s%20approximation | Mathematics | Specific functions | null | vital_articles |
An explicit formula for the coefficients in this series was given by G. Nemes. Further terms are listed in the On-Line Encyclopedia of Integer Sequences as and . The first graph in this section shows the relative error vs. , for 1 through all 5 terms listed above. (Bender and Orszag p. 218) gives the asymptotic formul... | Stirling's approximation | Wikipedia | 502 | 151783 | https://en.wikipedia.org/wiki/Stirling%27s%20approximation | Mathematics | Specific functions | null | vital_articles |
Error bounds
For any positive integer , the following notation is introduced:
and
Then
For further information and other error bounds, see the cited papers.
A convergent version of Stirling's formula
Thomas Bayes showed, in a letter to John Canton published by the Royal Society in 1763, that Stirling's formula did... | Stirling's approximation | Wikipedia | 386 | 151783 | https://en.wikipedia.org/wiki/Stirling%27s%20approximation | Mathematics | Specific functions | null | vital_articles |
In medicine, a side effect is an effect of the use of a medicinal drug or other treatment, usually adverse but sometimes beneficial, that is unintended. Herbal and traditional medicines also have side effects.
A drug or procedure usually used for a specific effect may be used specifically because of a beneficial side... | Side effect | Wikipedia | 305 | 151828 | https://en.wikipedia.org/wiki/Side%20effect | Biology and health sciences | General concepts_2 | Health | vital_articles |
Most drugs and procedures have a multitude of reported adverse side effects; the information leaflets provided with virtually all drugs list possible side effects. Beneficial side effects are less common; some examples, in many cases of side-effects that ultimately gained regulatory approval as intended effects, are:
... | Side effect | Wikipedia | 512 | 151828 | https://en.wikipedia.org/wiki/Side%20effect | Biology and health sciences | General concepts_2 | Health | vital_articles |
Sildenafil was originally intended for pulmonary hypertension; subsequently, it was discovered that it also produces erections, for which it was later approved.
Terazosin, an α1-adrenergic antagonist approved to treat benign prostatic hyperplasia (enlarged prostate) and hypertension, is (one of several drugs) used off... | Side effect | Wikipedia | 178 | 151828 | https://en.wikipedia.org/wiki/Side%20effect | Biology and health sciences | General concepts_2 | Health | vital_articles |
In vector calculus, the divergence theorem, also known as Gauss's theorem or Ostrogradsky's theorem, is a theorem relating the flux of a vector field through a closed surface to the divergence of the field in the volume enclosed.
More precisely, the divergence theorem states that the surface integral of a vector field... | Divergence theorem | Wikipedia | 432 | 151864 | https://en.wikipedia.org/wiki/Divergence%20theorem | Mathematics | Multivariable and vector calculus | null | vital_articles |
However if a source of liquid is inside the closed surface, such as a pipe through which liquid is introduced, the additional liquid will exert pressure on the surrounding liquid, causing an outward flow in all directions. This will cause a net outward flow through the surface S. The flux outward through S equals the... | Divergence theorem | Wikipedia | 462 | 151864 | https://en.wikipedia.org/wiki/Divergence%20theorem | Mathematics | Multivariable and vector calculus | null | vital_articles |
Informal derivation
The divergence theorem follows from the fact that if a volume is partitioned into separate parts, the flux out of the original volume is equal to the algebraic sum of the flux out of each component volume. This is true despite the fact that the new subvolumes have surfaces that were not part of th... | Divergence theorem | Wikipedia | 512 | 151864 | https://en.wikipedia.org/wiki/Divergence%20theorem | Mathematics | Multivariable and vector calculus | null | vital_articles |
Since this derivation is coordinate free, it shows that the divergence does not depend on the coordinates used.
Proofs
For bounded open subsets of Euclidean space
We are going to prove the following:
Proof of Theorem.
For compact Riemannian manifolds with boundary
We are going to prove the following:
Proof of Theo... | Divergence theorem | Wikipedia | 508 | 151864 | https://en.wikipedia.org/wiki/Divergence%20theorem | Mathematics | Multivariable and vector calculus | null | vital_articles |
because the unit ball has volume .
Applications
Differential and integral forms of physical laws
As a result of the divergence theorem, a host of physical laws can be written in both a differential form (where one quantity is the divergence of another) and an integral form (where the flux of one quantity through a ... | Divergence theorem | Wikipedia | 428 | 151864 | https://en.wikipedia.org/wiki/Divergence%20theorem | Mathematics | Multivariable and vector calculus | null | vital_articles |
Carl Friedrich Gauss was also using surface integrals while working on the gravitational attraction of an elliptical spheroid in 1813, when he proved special cases of the divergence theorem. He proved additional special cases in 1833 and 1839. But it was Mikhail Ostrogradsky, who gave the first proof of the general the... | Divergence theorem | Wikipedia | 470 | 151864 | https://en.wikipedia.org/wiki/Divergence%20theorem | Mathematics | Multivariable and vector calculus | null | vital_articles |
Del, or nabla, is an operator used in mathematics (particularly in vector calculus) as a vector differential operator, usually represented by the nabla symbol ∇. When applied to a function defined on a one-dimensional domain, it denotes the standard derivative of the function as defined in calculus. When applied to a f... | Del | Wikipedia | 473 | 151925 | https://en.wikipedia.org/wiki/Del | Mathematics | Calculus and analysis | null | vital_articles |
Notational uses
Del is used as a shorthand form to simplify many long mathematical expressions. It is most commonly used to simplify expressions for the gradient, divergence, curl, directional derivative, and Laplacian.
Gradient
The vector derivative of a scalar field is called the gradient, and it can be represented... | Del | Wikipedia | 481 | 151925 | https://en.wikipedia.org/wiki/Del | Mathematics | Calculus and analysis | null | vital_articles |
Note that is an operator that takes scalar to a scalar. It can be extended to operate on a vector, by separately operating on each of its components.
Laplacian
The Laplace operator is a scalar operator that can be applied to either vector or scalar fields; for cartesian coordinate systems it is defined as:
and the ... | Del | Wikipedia | 508 | 151925 | https://en.wikipedia.org/wiki/Del | Mathematics | Calculus and analysis | null | vital_articles |
The 3 remaining vector derivatives are related by the equation:
And one of them can even be expressed with the tensor product, if the functions are well-behaved:
Precautions
Most of the above vector properties (except for those that rely explicitly on del's differential properties—for example, the product rule) rely... | Del | Wikipedia | 255 | 151925 | https://en.wikipedia.org/wiki/Del | Mathematics | Calculus and analysis | null | vital_articles |
Vaccinium vitis-idaea is a small evergreen shrub in the heath family, Ericaceae. It is known colloquially as the lingonberry, partridgeberry, foxberry, mountain cranberry, or cowberry. It is native to boreal forest and Arctic tundra throughout the Northern Hemisphere. Commercially cultivated in the United States Pacifi... | Vaccinium vitis-idaea | Wikipedia | 485 | 151942 | https://en.wikipedia.org/wiki/Vaccinium%20vitis-idaea | Biology and health sciences | Berries | Plants | vital_articles |
Etymology
Vaccinium vitis-idaea is most commonly known in English as 'lingonberry' or 'cowberry'. The name 'lingonberry' originates from the Swedish name () for the species deriving from Old Norse lyngr, a cognate (thus also a doublet) to 'ling'.
The genus name Vaccinium is a classical Latin name for a plant, possib... | Vaccinium vitis-idaea | Wikipedia | 503 | 151942 | https://en.wikipedia.org/wiki/Vaccinium%20vitis-idaea | Biology and health sciences | Berries | Plants | vital_articles |
The berries collected in the wild are a popular fruit in northern, central and eastern Europe, notably in the Nordic countries, the Baltic states, central and northern Europe. In some areas, they can be picked legally on both public and private lands in accordance with the freedom to roam.
The berries are quite tart, ... | Vaccinium vitis-idaea | Wikipedia | 481 | 151942 | https://en.wikipedia.org/wiki/Vaccinium%20vitis-idaea | Biology and health sciences | Berries | Plants | vital_articles |
The berries are also popular as a wild picked fruit in Eastern Canada, for example in Newfoundland and Labrador and Cape Breton, where they are locally known as partridgeberries or redberries, and on the mainland of Nova Scotia, where they are known as foxberries. In this region they are incorporated into jams, syrups,... | Vaccinium vitis-idaea | Wikipedia | 455 | 151942 | https://en.wikipedia.org/wiki/Vaccinium%20vitis-idaea | Biology and health sciences | Berries | Plants | vital_articles |
Use of the minus subspecies
The Anticosti people use the fruit to make jams and jellies. The Nihithawak Cree store the berries by freezing them outside during the winter, mix the berries with boiled fish eggs, livers, air bladders and fat and eat them, eat the berries raw as a snack food, or stew them with fish or mea... | Vaccinium vitis-idaea | Wikipedia | 248 | 151942 | https://en.wikipedia.org/wiki/Vaccinium%20vitis-idaea | Biology and health sciences | Berries | Plants | vital_articles |
Vernalization () is the induction of a plant's flowering process by exposure to the prolonged cold of winter, or by an artificial equivalent. After vernalization, plants have acquired the ability to flower, but they may require additional seasonal cues or weeks of growth before they will actually do so. The term is som... | Vernalization | Wikipedia | 456 | 151965 | https://en.wikipedia.org/wiki/Vernalization | Biology and health sciences | Plant reproduction | Biology | vital_articles |
In 1928, the Soviet agronomist Trofim Lysenko published his works on the effects of cold on cereal seeds, and coined the term "яровизация" (yarovizatsiya : "jarovization") to describe a chilling process he used to make the seeds of winter cereals behave like spring cereals (from яровой : yarvoy, Tatar root ярый : yaryi... | Vernalization | Wikipedia | 502 | 151965 | https://en.wikipedia.org/wiki/Vernalization | Biology and health sciences | Plant reproduction | Biology | vital_articles |
Arabidopsis thaliana ("thale cress") is a much-studied model for vernalization. Some ecotypes (varieties), called "winter annuals", have delayed flowering without vernalization; others ("summer annuals") do not. The genes that underlie this difference in plant physiology have been intensively studied.
The reproductive... | Vernalization | Wikipedia | 471 | 151965 | https://en.wikipedia.org/wiki/Vernalization | Biology and health sciences | Plant reproduction | Biology | vital_articles |
Since vernalization also occurs in flc mutants (lacking FLC), vernalization must also activate a non-FLC pathway. A day-length mechanism is also important. Vernalization response works in concert with the photo-periodic genes CO, FT, PHYA, CRY2 to induce flowering.
Devernalization
It is possible to devernalize a plant... | Vernalization | Wikipedia | 139 | 151965 | https://en.wikipedia.org/wiki/Vernalization | Biology and health sciences | Plant reproduction | Biology | vital_articles |
In the mathematical discipline of set theory, forcing is a technique for proving consistency and independence results. Intuitively, forcing can be thought of as a technique to expand the set theoretical universe to a larger universe by introducing a new "generic" object .
Forcing was first used by Paul Cohen in 1963... | Forcing (mathematics) | Wikipedia | 379 | 152205 | https://en.wikipedia.org/wiki/Forcing%20%28mathematics%29 | Mathematics | Set theory | null | vital_articles |
However, in some sense, it may be desirable to "construct the expanded model within ". This would help ensure that "resembles" in certain aspects, such as being the same as (more generally, that cardinal collapse does not occur), and allow fine control over the properties of . More precisely, every member of shou... | Forcing (mathematics) | Wikipedia | 473 | 152205 | https://en.wikipedia.org/wiki/Forcing%20%28mathematics%29 | Mathematics | Set theory | null | vital_articles |
To get around this issue, a standard technique is to let be a standard transitive model of an arbitrary finite subset of (any axiomatization of has at least one axiom schema, and thus an infinite number of axioms), the existence of which is guaranteed by the reflection principle. As the goal of a forcing argument is... | Forcing (mathematics) | Wikipedia | 496 | 152205 | https://en.wikipedia.org/wiki/Forcing%20%28mathematics%29 | Mathematics | Set theory | null | vital_articles |
The forcing poset for Cohen forcing can be formally written as , the finite partial functions from to under reverse inclusion. Cohen forcing satisfies the splitting condition because given any condition , one can always find an element not mentioned in , and add either the sentence or to to get two new forcing co... | Forcing (mathematics) | Wikipedia | 500 | 152205 | https://en.wikipedia.org/wiki/Forcing%20%28mathematics%29 | Mathematics | Set theory | null | vital_articles |
The -names are, in fact, an expansion of the universe. Given , one defines to be the -name
Since , it follows that . In a sense, is a "name for " that does not depend on the specific choice of .
This also allows defining a "name for " without explicitly referring to :
so that .
Rigorous definitions
The concepts... | Forcing (mathematics) | Wikipedia | 511 | 152205 | https://en.wikipedia.org/wiki/Forcing%20%28mathematics%29 | Mathematics | Set theory | null | vital_articles |
The modified forcing relation can be defined recursively as follows:
means
means
means
means
means
Other symbols of the forcing language can be defined in terms of these symbols: For example, means , means , etc. Cases 1 and 2 depend on each other and on case 3, but the recursion always refer to -name... | Forcing (mathematics) | Wikipedia | 445 | 152205 | https://en.wikipedia.org/wiki/Forcing%20%28mathematics%29 | Mathematics | Set theory | null | vital_articles |
Both styles, adjoining to either a countable transitive model or the whole universe , are commonly used. Less commonly seen is the approach using the "internal" definition of forcing, in which no mention of set or class models is made. This was Cohen's original method, and in one elaboration, it becomes the method of... | Forcing (mathematics) | Wikipedia | 455 | 152205 | https://en.wikipedia.org/wiki/Forcing%20%28mathematics%29 | Mathematics | Set theory | null | vital_articles |
This is not yet the falsification of the continuum hypothesis. One must prove that no new maps have been introduced which map onto , or onto . For example, if one considers instead , finite partial functions from to , the first uncountable ordinal, one gets in a bijection from to . In other words, has collapsed, ... | Forcing (mathematics) | Wikipedia | 463 | 152205 | https://en.wikipedia.org/wiki/Forcing%20%28mathematics%29 | Mathematics | Set theory | null | vital_articles |
The importance of antichains in forcing is that for most purposes, dense sets and maximal antichains are equivalent. A maximal antichain is one that cannot be extended to a larger antichain. This means that every element is compatible with some member of . The existence of a maximal antichain follows from Zorn's Lemm... | Forcing (mathematics) | Wikipedia | 454 | 152205 | https://en.wikipedia.org/wiki/Forcing%20%28mathematics%29 | Mathematics | Set theory | null | vital_articles |
Easton's work was notable in that it involved forcing with a proper class of conditions. In general, the method of forcing with a proper class of conditions fails to give a model of . For example, forcing with , where is the proper class of all ordinals, makes the continuum a proper class. On the other hand, forcing w... | Forcing (mathematics) | Wikipedia | 379 | 152205 | https://en.wikipedia.org/wiki/Forcing%20%28mathematics%29 | Mathematics | Set theory | null | vital_articles |
However, a generic filter over a countable transitive model is not in . The real defined by is provably not an element of . The problem is that if , then " is compact", but from the viewpoint of some larger universe , can be non-compact and the intersection of all conditions from the generic filter is actually em... | Forcing (mathematics) | Wikipedia | 446 | 152205 | https://en.wikipedia.org/wiki/Forcing%20%28mathematics%29 | Mathematics | Set theory | null | vital_articles |
A different interpretation of reals in was provided by Dana Scott. Rational numbers in have names that correspond to countably-many distinct rational values assigned to a maximal antichain of Borel sets – in other words, a certain rational-valued function on . Real numbers in then correspond to Dedekind cuts of such... | Forcing (mathematics) | Wikipedia | 496 | 152205 | https://en.wikipedia.org/wiki/Forcing%20%28mathematics%29 | Mathematics | Set theory | null | vital_articles |
The general schema of relative consistency proofs follows. As any proof is finite, it uses only a finite number of axioms:
For any given proof, can verify the validity of this proof. This is provable by induction on the length of the proof.
Then resolve
By proving the following
it can be concluded that
... | Forcing (mathematics) | Wikipedia | 363 | 152205 | https://en.wikipedia.org/wiki/Forcing%20%28mathematics%29 | Mathematics | Set theory | null | vital_articles |
In mathematical logic, the compactness theorem states that a set of first-order sentences has a model if and only if every finite subset of it has a model. This theorem is an important tool in model theory, as it provides a useful (but generally not effective) method for constructing models of any set of sentences that... | Compactness theorem | Wikipedia | 387 | 152207 | https://en.wikipedia.org/wiki/Compactness%20theorem | Mathematics | Model theory | null | vital_articles |
is not satisfiable (because there is no field of characteristic 0 in which holds, and the infinite sequence of sentences ensures any model would be a field of characteristic 0). Therefore, there is a finite subset of these sentences that is not satisfiable. must contain because otherwise it would be satisfiable. Be... | Compactness theorem | Wikipedia | 335 | 152207 | https://en.wikipedia.org/wiki/Compactness%20theorem | Mathematics | Model theory | null | vital_articles |
A second application of the compactness theorem shows that any theory that has arbitrarily large finite models, or a single infinite model, has models of arbitrary large cardinality (this is the Upward Löwenheim–Skolem theorem). So for instance, there are nonstandard models of Peano arithmetic with uncountably many 'na... | Compactness theorem | Wikipedia | 417 | 152207 | https://en.wikipedia.org/wiki/Compactness%20theorem | Mathematics | Model theory | null | vital_articles |
One can prove the compactness theorem using Gödel's completeness theorem, which establishes that a set of sentences is satisfiable if and only if no contradiction can be proven from it. Since proofs are always finite and therefore involve only finitely many of the given sentences, the compactness theorem follows. In ... | Compactness theorem | Wikipedia | 305 | 152207 | https://en.wikipedia.org/wiki/Compactness%20theorem | Mathematics | Model theory | null | vital_articles |
In set theory, Zermelo–Fraenkel set theory, named after mathematicians Ernst Zermelo and Abraham Fraenkel, is an axiomatic system that was proposed in the early twentieth century in order to formulate a theory of sets free of paradoxes such as Russell's paradox. Today, Zermelo–Fraenkel set theory, with the historicall... | Zermelo–Fraenkel set theory | Wikipedia | 465 | 152214 | https://en.wikipedia.org/wiki/Zermelo%E2%80%93Fraenkel%20set%20theory | Mathematics | Axiomatic systems | null | vital_articles |
The metamathematics of Zermelo–Fraenkel set theory has been extensively studied. Landmark results in this area established the logical independence of the axiom of choice from the remaining Zermelo-Fraenkel axioms and of the continuum hypothesis from ZFC. The consistency of a theory such as ZFC cannot be proved within ... | Zermelo–Fraenkel set theory | Wikipedia | 407 | 152214 | https://en.wikipedia.org/wiki/Zermelo%E2%80%93Fraenkel%20set%20theory | Mathematics | Axiomatic systems | null | vital_articles |
Formally, ZFC is a one-sorted theory in first-order logic. The equality symbol can be treated as either a primitive logical symbol or a high-level abbreviation for having exactly the same elements. The former approach is the most common. The signature has a single predicate symbol, usually denoted , which is a predicat... | Zermelo–Fraenkel set theory | Wikipedia | 448 | 152214 | https://en.wikipedia.org/wiki/Zermelo%E2%80%93Fraenkel%20set%20theory | Mathematics | Axiomatic systems | null | vital_articles |
All formulations of ZFC imply that at least one set exists. Kunen includes an axiom that directly asserts the existence of a set, although he notes that he does so only "for emphasis". Its omission here can be justified in two ways. First, in the standard semantics of first-order logic in which ZFC is typically formali... | Zermelo–Fraenkel set theory | Wikipedia | 505 | 152214 | https://en.wikipedia.org/wiki/Zermelo%E2%80%93Fraenkel%20set%20theory | Mathematics | Axiomatic systems | null | vital_articles |
In general, the subset of a set obeying a formula with one free variable may be written as:
The axiom schema of specification states that this subset always exists (it is an axiom schema because there is one axiom for each ). Formally, let be any formula in the language of ZFC with all free variables among ( is n... | Zermelo–Fraenkel set theory | Wikipedia | 465 | 152214 | https://en.wikipedia.org/wiki/Zermelo%E2%80%93Fraenkel%20set%20theory | Mathematics | Axiomatic systems | null | vital_articles |
The axiom schema of specification must be used to reduce this to a set with exactly these two elements. The axiom of pairing is part of Z, but is redundant in ZF because it follows from the axiom schema of replacement if we are given a set with at least two elements. The existence of a set with at least two elements is... | Zermelo–Fraenkel set theory | Wikipedia | 500 | 152214 | https://en.wikipedia.org/wiki/Zermelo%E2%80%93Fraenkel%20set%20theory | Mathematics | Axiomatic systems | null | vital_articles |
By definition, a set is a subset of a set if and only if every element of is also an element of :
The Axiom of power set states that for any set , there is a set that contains every subset of :
The axiom schema of specification is then used to define the power set as the subset of such a containing the subsets ... | Zermelo–Fraenkel set theory | Wikipedia | 399 | 152214 | https://en.wikipedia.org/wiki/Zermelo%E2%80%93Fraenkel%20set%20theory | Mathematics | Axiomatic systems | null | vital_articles |
One motivation for the ZFC axioms is the cumulative hierarchy of sets introduced by John von Neumann. In this viewpoint, the universe of set theory is built up in stages, with one stage for each ordinal number. At stage 0, there are no sets yet. At each following stage, a set is added to the universe if all of its elem... | Zermelo–Fraenkel set theory | Wikipedia | 512 | 152214 | https://en.wikipedia.org/wiki/Zermelo%E2%80%93Fraenkel%20set%20theory | Mathematics | Axiomatic systems | null | vital_articles |
Virtual classes
Proper classes (collections of mathematical objects defined by a property shared by their members which are too big to be sets) can only be treated indirectly in ZF (and thus ZFC).
An alternative to proper classes while staying within ZF and ZFC is the virtual class notational construct introduced by ,... | Zermelo–Fraenkel set theory | Wikipedia | 307 | 152214 | https://en.wikipedia.org/wiki/Zermelo%E2%80%93Fraenkel%20set%20theory | Mathematics | Axiomatic systems | null | vital_articles |
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