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92-403: The word flux comes from Latin : fluxus means "flow", and fluere is "to flow". As fluxion , this term was introduced into differential calculus by Isaac Newton . The concept of heat flux was a key contribution of Joseph Fourier , in the analysis of heat transfer phenomena. His seminal treatise Théorie analytique de la chaleur ( The Analytical Theory of Heat ), defines fluxion as

184-474: A r g m a x n ^ n ^ p d q d t ( A , p , n ^ ) . {\displaystyle \mathbf {I} (A,\mathbf {p} )={\underset {\mathbf {\hat {n}} }{\operatorname {arg\,max} }}\mathbf {\hat {n}} _{\mathbf {p} }{\frac {\mathrm {d} q}{\mathrm {d} t}}(A,\mathbf {p} ,\mathbf {\hat {n}} ).} In this case, there

276-415: A region U is a real or complex-valued function or distribution on U . The region U may be a set in some Euclidean space , Minkowski space , or more generally a subset of a manifold , and it is typical in mathematics to impose further conditions on the field, such that it be continuous or often continuously differentiable to some order. A scalar field is a tensor field of order zero, and

368-418: A scalar field is a function associating a single number to each point in a region of space – possibly physical space . The scalar may either be a pure mathematical number ( dimensionless ) or a scalar physical quantity (with units ). In a physical context, scalar fields are required to be independent of the choice of reference frame. That is, any two observers using the same units will agree on

460-423: A (single) scalar: j = I A , {\displaystyle j={\frac {I}{A}},} where I = lim Δ t → 0 Δ q Δ t = d q d t . {\displaystyle I=\lim _{\Delta t\to 0}{\frac {\Delta q}{\Delta t}}={\frac {\mathrm {d} q}{\mathrm {d} t}}.} In this case

552-439: A central quantity and proceeds to derive the now well-known expressions of flux in terms of temperature differences across a slab, and then more generally in terms of temperature gradients or differentials of temperature, across other geometries. One could argue, based on the work of James Clerk Maxwell , that the transport definition precedes the definition of flux used in electromagnetism . The specific quote from Maxwell is: In

644-402: A closed surface, in other words the net outflux from a 3D region, is found by adding the local net outflow from each point in the region (which is expressed by the divergence ). If the surface is not closed, it has an oriented curve as boundary. Stokes' theorem states that the flux of the curl of a vector field is the line integral of the vector field over this boundary. This path integral

736-526: A faster pace. It is characterised by greater use of prepositions, and word order that is closer to modern Romance languages, for example, while grammatically retaining more or less the same formal rules as Classical Latin. Ultimately, Latin diverged into a distinct written form, where the commonly spoken form was perceived as a separate language, for instance early French or Italian dialects, that could be transcribed differently. It took some time for these to be viewed as wholly different from Latin however. After

828-718: A few in German , Dutch , Norwegian , Danish and Swedish . Latin is still spoken in Vatican City, a city-state situated in Rome that is the seat of the Catholic Church . The works of several hundred ancient authors who wrote in Latin have survived in whole or in part, in substantial works or in fragments to be analyzed in philology . They are in part the subject matter of the field of classics . Their works were published in manuscript form before

920-404: A few. Famous and well regarded writers included Petrarch, Erasmus, Salutati , Celtis , George Buchanan and Thomas More . Non fiction works were long produced in many subjects, including the sciences, law, philosophy, historiography and theology. Famous examples include Isaac Newton 's Principia . Latin was also used as a convenient medium for translations of important works first written in

1012-461: A flux according to the electromagnetism definition, the corresponding flux density , if that term is used, refers to its derivative along the surface that was integrated. By the Fundamental theorem of calculus , the corresponding flux density is a flux according to the transport definition. Given a current such as electric current—charge per time, current density would also be a flux according to

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1104-488: A magnetic field opposite to the change. This is the basis for inductors and many electric generators . Using this definition, the flux of the Poynting vector S over a specified surface is the rate at which electromagnetic energy flows through that surface, defined like before: The flux of the Poynting vector through a surface is the electromagnetic power , or energy per unit time , passing through that surface. This

1196-548: A native language, Medieval Latin was used across Western and Catholic Europe during the Middle Ages as a working and literary language from the 9th century to the Renaissance , which then developed a classicizing form, called Renaissance Latin . This was the basis for Neo-Latin which evolved during the early modern period . In these periods Latin was used productively and generally taught to be written and spoken, at least until

1288-528: A positive point charge can be visualized as a dot radiating electric field lines (sometimes also called "lines of force"). Conceptually, electric flux can be thought of as "the number of field lines" passing through a given area. Mathematically, electric flux is the integral of the normal component of the electric field over a given area. Hence, units of electric flux are, in the MKS system , newtons per coulomb times meters squared, or N m/C. (Electric flux density

1380-551: A result, the list has variants, as well as alternative names. In addition to the historical phases, Ecclesiastical Latin refers to the styles used by the writers of the Roman Catholic Church from late antiquity onward, as well as by Protestant scholars. The earliest known form of Latin is Old Latin, also called Archaic or Early Latin, which was spoken from the Roman Kingdom , traditionally founded in 753 BC, through

1472-407: A separate language, existing more or less in parallel with the literary or educated Latin, but this is now widely dismissed. The term 'Vulgar Latin' remains difficult to define, referring both to informal speech at any time within the history of Latin, and the kind of informal Latin that had begun to move away from the written language significantly in the post-Imperial period, that led ultimately to

1564-695: A small number of Latin services held in the Anglican church. These include an annual service in Oxford, delivered with a Latin sermon; a relic from the period when Latin was the normal spoken language of the university. In the Western world, many organizations, governments and schools use Latin for their mottos due to its association with formality, tradition, and the roots of Western culture . Canada's motto A mari usque ad mare ("from sea to sea") and most provincial mottos are also in Latin. The Canadian Victoria Cross

1656-429: A sort of informal language academy dedicated to maintaining and perpetuating educated speech. Philological analysis of Archaic Latin works, such as those of Plautus , which contain fragments of everyday speech, gives evidence of an informal register of the language, Vulgar Latin (termed sermo vulgi , "the speech of the masses", by Cicero ). Some linguists, particularly in the nineteenth century, believed this to be

1748-565: A spoken and written language by the scholarship by the Renaissance humanists . Petrarch and others began to change their usage of Latin as they explored the texts of the Classical Latin world. Skills of textual criticism evolved to create much more accurate versions of extant texts through the fifteenth and sixteenth centuries, and some important texts were rediscovered. Comprehensive versions of authors' works were published by Isaac Casaubon , Joseph Scaliger and others. Nevertheless, despite

1840-432: A strictly left-to-right script. During the late republic and into the first years of the empire, from about 75 BC to AD 200, a new Classical Latin arose, a conscious creation of the orators, poets, historians and other literate men, who wrote the great works of classical literature , which were taught in grammar and rhetoric schools. Today's instructional grammars trace their roots to such schools , which served as

1932-410: A surface. By contrast, according to the electromagnetism definition, flux is the integral over a surface; it makes no sense to integrate a second-definition flux for one would be integrating over a surface twice. Thus, Maxwell's quote only makes sense if "flux" is being used according to the transport definition (and furthermore is a vector field rather than single vector). This is ironic because Maxwell

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2024-689: A vernacular, such as those of Descartes . Latin education underwent a process of reform to classicise written and spoken Latin. Schooling remained largely Latin medium until approximately 1700. Until the end of the 17th century, the majority of books and almost all diplomatic documents were written in Latin. Afterwards, most diplomatic documents were written in French (a Romance language ) and later native or other languages. Education methods gradually shifted towards written Latin, and eventually concentrating solely on reading skills. The decline of Latin education took several centuries and proceeded much more slowly than

2116-411: Is Veritas ("truth"). Veritas was the goddess of truth, a daughter of Saturn, and the mother of Virtue. Switzerland has adopted the country's Latin short name Helvetia on coins and stamps, since there is no room to use all of the nation's four official languages . For a similar reason, it adopted the international vehicle and internet code CH , which stands for Confoederatio Helvetica ,

2208-404: Is j  cos  θ , while the component of flux passing tangential to the area is j  sin  θ , but there is no flux actually passing through the area in the tangential direction. The only component of flux passing normal to the area is the cosine component. For vector flux, the surface integral of j over a surface S , gives the proper flowing per unit of time through

2300-857: Is a classical language belonging to the Italic branch of the Indo-European languages . Latin was originally spoken by the Latins in Latium (now known as Lazio ), the lower Tiber area around Rome , Italy. Through the expansion of the Roman Republic it became the dominant language in the Italian Peninsula and subsequently throughout the Roman Empire . By the late Roman Republic , Old Latin had evolved into standardized Classical Latin . Vulgar Latin refers to

2392-626: Is a reversal of the original phrase Non terrae plus ultra ("No land further beyond", "No further!"). According to legend , this phrase was inscribed as a warning on the Pillars of Hercules , the rocks on both sides of the Strait of Gibraltar and the western end of the known, Mediterranean world. Charles adopted the motto following the discovery of the New World by Columbus, and it also has metaphorical suggestions of taking risks and striving for excellence. In

2484-417: Is also called circulation , especially in fluid dynamics. Thus the curl is the circulation density. We can apply the flux and these theorems to many disciplines in which we see currents, forces, etc., applied through areas. An electric "charge," such as a single proton in space, has a magnitude defined in coulombs. Such a charge has an electric field surrounding it. In pictorial form, the electric field from

2576-534: Is commonly used in analysis of electromagnetic radiation , but has application to other electromagnetic systems as well. Confusingly, the Poynting vector is sometimes called the power flux , which is an example of the first usage of flux, above. It has units of watts per square metre (W/m). Latin Latin ( lingua Latina , pronounced [ˈlɪŋɡʷa ɫaˈtiːna] , or Latinum [ɫaˈtiːnʊ̃] )

2668-408: Is drawn by curves (field lines) following the "flow"; the magnitude of the vector field is then the line density, and the flux through a surface is the number of lines. Lines originate from areas of positive divergence (sources) and end at areas of negative divergence (sinks). See also the image at right: the number of red arrows passing through a unit area is the flux density, the curve encircling

2760-548: Is found in any widespread language, the languages of Spain, France, Portugal, and Italy have retained a remarkable unity in phonological forms and developments, bolstered by the stabilising influence of their common Christian (Roman Catholic) culture. It was not until the Muslim conquest of Spain in 711, cutting off communications between the major Romance regions, that the languages began to diverge seriously. The spoken Latin that would later become Romanian diverged somewhat more from

2852-408: Is maximized across the disk that is perpendicular to it. The unit vector thus uniquely maximizes the function when it points in the "true direction" of the flow. (Strictly speaking, this is an abuse of notation because the "arg max" cannot directly compare vectors; we take the vector with the biggest norm instead.) These direct definitions, especially the last, are rather unwieldy. For example,

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2944-661: Is modelled after the British Victoria Cross which has the inscription "For Valour". Because Canada is officially bilingual, the Canadian medal has replaced the English inscription with the Latin Pro Valore . Spain's motto Plus ultra , meaning "even further", or figuratively "Further!", is also Latin in origin. It is taken from the personal motto of Charles V , Holy Roman Emperor and King of Spain (as Charles I), and

3036-399: Is no fixed surface we are measuring over. q is a function of a point, an area, and a direction (given by a unit vector n ^ {\displaystyle \mathbf {\hat {n}} } ), and measures the flow through the disk of area A perpendicular to that unit vector. I is defined picking the unit vector that maximizes the flow around the point, because the true flow

3128-466: Is sometimes referred to as the probability current or current density, or probability flux density. As a mathematical concept, flux is represented by the surface integral of a vector field , where F is a vector field , and d A is the vector area of the surface A , directed as the surface normal . For the second, n is the outward pointed unit normal vector to the surface. The surface has to be orientable , i.e. two sides can be distinguished:

3220-958: Is taught at many high schools, especially in Europe and the Americas. It is most common in British public schools and grammar schools, the Italian liceo classico and liceo scientifico , the German Humanistisches Gymnasium and the Dutch gymnasium . Occasionally, some media outlets, targeting enthusiasts, broadcast in Latin. Notable examples include Radio Bremen in Germany, YLE radio in Finland (the Nuntii Latini broadcast from 1989 until it

3312-599: Is the concentration ( mol /m) of component A. This flux has units of mol·m·s, and fits Maxwell's original definition of flux. For dilute gases, kinetic molecular theory relates the diffusion coefficient D to the particle density n = N / V , the molecular mass m , the collision cross section σ {\displaystyle \sigma } , and the absolute temperature T by D = 2 3 n σ k T π m {\displaystyle D={\frac {2}{3n\sigma }}{\sqrt {\frac {kT}{\pi m}}}} where

3404-407: Is the vector area  – combination A = A n ^ {\displaystyle \mathbf {A} =A\mathbf {\hat {n}} } of the magnitude of the area A through which the property passes and a unit vector n ^ {\displaystyle \mathbf {\hat {n}} } normal to the area. Unlike in the second set of equations,

3496-500: Is the electric flux per unit area, and is a measure of strength of the normal component of the electric field averaged over the area of integration. Its units are N/C, the same as the electric field in MKS units.) Two forms of electric flux are used, one for the E -field: and one for the D -field (called the electric displacement ): This quantity arises in Gauss's law – which states that

3588-524: Is zero. As mentioned above, chemical molar flux of a component A in an isothermal , isobaric system is defined in Fick's law of diffusion as: J A = − D A B ∇ c A {\displaystyle \mathbf {J} _{A}=-D_{AB}\nabla c_{A}} where the nabla symbol ∇ denotes the gradient operator, D AB is the diffusion coefficient (m·s) of component A diffusing through component B, c A

3680-531: The Corpus Inscriptionum Latinarum (CIL). Authors and publishers vary, but the format is about the same: volumes detailing inscriptions with a critical apparatus stating the provenance and relevant information. The reading and interpretation of these inscriptions is the subject matter of the field of epigraphy . About 270,000 inscriptions are known. The Latin influence in English has been significant at all stages of its insular development. In

3772-521: The D -field flux equals the charge Q A within it. Here the expression "flux of" indicates a mathematical operation and, as can be seen, the result is not necessarily a "flow", since nothing actually flows along electric field lines. The magnetic flux density ( magnetic field ) having the unit Wb/m ( Tesla ) is denoted by B , and magnetic flux is defined analogously: with the same notation above. The quantity arises in Faraday's law of induction , where

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3864-562: The Holy See , the primary language of its public journal , the Acta Apostolicae Sedis , and the working language of the Roman Rota . Vatican City is also home to the world's only automatic teller machine that gives instructions in Latin. In the pontifical universities postgraduate courses of Canon law are taught in Latin, and papers are written in the same language. There are

3956-561: The Middle Ages , borrowing from Latin occurred from ecclesiastical usage established by Saint Augustine of Canterbury in the 6th century or indirectly after the Norman Conquest , through the Anglo-Norman language . From the 16th to the 18th centuries, English writers cobbled together huge numbers of new words from Latin and Greek words, dubbed " inkhorn terms ", as if they had spilled from a pot of ink. Many of these words were used once by

4048-559: The Roman Rite of the Catholic Church at the Vatican City . The church continues to adapt concepts from modern languages to Ecclesiastical Latin of the Latin language. Contemporary Latin is more often studied to be read rather than spoken or actively used. Latin has greatly influenced the English language , along with a large number of others, and historically contributed many words to

4140-553: The Romance languages . During the Classical period, informal language was rarely written, so philologists have been left with only individual words and phrases cited by classical authors, inscriptions such as Curse tablets and those found as graffiti . In the Late Latin period, language changes reflecting spoken (non-classical) norms tend to be found in greater quantities in texts. As it

4232-622: The Western Roman Empire fell in 476 and Germanic kingdoms took its place, the Germanic people adopted Latin as a language more suitable for legal and other, more formal uses. While the written form of Latin was increasingly standardized into a fixed form, the spoken forms began to diverge more greatly. Currently, the five most widely spoken Romance languages by number of native speakers are Spanish , Portuguese , French , Italian , and Romanian . Despite dialectal variation, which

4324-617: The British Crown. The motto is featured on all presently minted coinage and has been featured in most coinage throughout the nation's history. Several states of the United States have Latin mottos , such as: Many military organizations today have Latin mottos, such as: Some law governing bodies in the Philippines have Latin mottos, such as: Some colleges and universities have adopted Latin mottos, for example Harvard University 's motto

4416-599: The English lexicon , particularly after the Christianization of the Anglo-Saxons and the Norman Conquest . Latin and Ancient Greek roots are heavily used in English vocabulary in theology , the sciences , medicine , and law . A number of phases of the language have been recognized, each distinguished by subtle differences in vocabulary, usage, spelling, and syntax. There are no hard and fast rules of classification; different scholars emphasize different features. As

4508-580: The Grinch Stole Christmas! , The Cat in the Hat , and a book of fairy tales, " fabulae mirabiles ", are intended to garner popular interest in the language. Additional resources include phrasebooks and resources for rendering everyday phrases and concepts into Latin, such as Meissner's Latin Phrasebook . Some inscriptions have been published in an internationally agreed, monumental, multivolume series,

4600-461: The United States the unofficial national motto until 1956 was E pluribus unum meaning "Out of many, one". The motto continues to be featured on the Great Seal . It also appears on the flags and seals of both houses of congress and the flags of the states of Michigan, North Dakota, New York, and Wisconsin. The motto's 13 letters symbolically represent the original Thirteen Colonies which revolted from

4692-552: The University of Kentucky, the University of Oxford and also Princeton University. There are many websites and forums maintained in Latin by enthusiasts. The Latin Misplaced Pages has more than 130,000 articles. Italian , French , Portuguese , Spanish , Romanian , Catalan , Romansh , Sardinian and other Romance languages are direct descendants of Latin. There are also many Latin borrowings in English and Albanian , as well as

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4784-424: The area at an angle θ to the area normal n ^ {\displaystyle \mathbf {\hat {n}} } , then the dot product j ⋅ n ^ = j cos ⁡ θ . {\displaystyle \mathbf {j} \cdot \mathbf {\hat {n}} =j\cos \theta .} That is, the component of flux passing through the surface (i.e. normal to it)

4876-499: The area of that cross section, or the amount of sunlight energy that lands on a patch of ground each second divided by the area of the patch, are kinds of flux. Here are 3 definitions in increasing order of complexity. Each is a special case of the following. In all cases the frequent symbol j , (or J ) is used for flux, q for the physical quantity that flows, t for time, and A for area. These identifiers will be written in bold when and only when they are vectors. First, flux as

4968-406: The arg max construction is artificial from the perspective of empirical measurements, when with a weathervane or similar one can easily deduce the direction of flux at a point. Rather than defining the vector flux directly, it is often more intuitive to state some properties about it. Furthermore, from these properties the flux can uniquely be determined anyway. If the flux j passes through

5060-455: The author and then forgotten, but some useful ones survived, such as 'imbibe' and 'extrapolate'. Many of the most common polysyllabic English words are of Latin origin through the medium of Old French . Romance words make respectively 59%, 20% and 14% of English, German and Dutch vocabularies. Those figures can rise dramatically when only non-compound and non-derived words are included. Scalar field In mathematics and physics ,

5152-425: The benefit of those who do not understand Latin. There are also songs written with Latin lyrics . The libretto for the opera-oratorio Oedipus rex by Igor Stravinsky is in Latin. Parts of Carl Orff 's Carmina Burana are written in Latin. Enya has recorded several tracks with Latin lyrics. The continued instruction of Latin is seen by some as a highly valuable component of a liberal arts education. Latin

5244-409: The careful work of Petrarch, Politian and others, first the demand for manuscripts, and then the rush to bring works into print, led to the circulation of inaccurate copies for several centuries following. Neo-Latin literature was extensive and prolific, but less well known or understood today. Works covered poetry, prose stories and early novels, occasional pieces and collections of letters, to name

5336-428: The case of fluxes, we have to take the integral, over a surface, of the flux through every element of the surface. The result of this operation is called the surface integral of the flux. It represents the quantity which passes through the surface. According to the transport definition, flux may be a single vector, or it may be a vector field / function of position. In the latter case flux can readily be integrated over

5428-415: The classicised Latin that followed through to the present are often grouped together as Neo-Latin , or New Latin, which have in recent decades become a focus of renewed study , given their importance for the development of European culture, religion and science. The vast majority of written Latin belongs to this period, but its full extent is unknown. The Renaissance reinforced the position of Latin as

5520-461: The country's full Latin name. Some film and television in ancient settings, such as Sebastiane , The Passion of the Christ and Barbarians (2020 TV series) , have been made with dialogue in Latin. Occasionally, Latin dialogue is used because of its association with religion or philosophy, in such film/television series as The Exorcist and Lost (" Jughead "). Subtitles are usually shown for

5612-445: The curve ∂ A {\displaystyle \partial A} , with the sign determined by the integration direction. The time-rate of change of the magnetic flux through a loop of wire is minus the electromotive force created in that wire. The direction is such that if current is allowed to pass through the wire, the electromotive force will cause a current which "opposes" the change in magnetic field by itself producing

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5704-493: The decline in written Latin output. Despite having no native speakers, Latin is still used for a variety of purposes in the contemporary world. The largest organisation that retains Latin in official and quasi-official contexts is the Catholic Church . The Catholic Church required that Mass be carried out in Latin until the Second Vatican Council of 1962–1965 , which permitted the use of the vernacular . Latin remains

5796-570: The educated and official world, Latin continued without its natural spoken base. Moreover, this Latin spread into lands that had never spoken Latin, such as the Germanic and Slavic nations. It became useful for international communication between the member states of the Holy Roman Empire and its allies. Without the institutions of the Roman Empire that had supported its uniformity, Medieval Latin

5888-456: The field, the flux for the sides is zero and there is an equal and opposite flux at both ends of the tube. This is a consequence of Gauss's Law applied to an inverse square field. The flux for any cross-sectional surface of the tube will be the same. The total flux for any surface surrounding a charge q is q / ε 0 . In free space the electric displacement is given by the constitutive relation D = ε 0 E , so for any bounding surface

5980-442: The flux of the electric field E out of a closed surface is proportional to the electric charge Q A enclosed in the surface (independent of how that charge is distributed), the integral form is: where ε 0 is the permittivity of free space . If one considers the flux of the electric field vector, E , for a tube near a point charge in the field of the charge but not containing it with sides formed by lines tangent to

6072-679: The invention of printing and are now published in carefully annotated printed editions, such as the Loeb Classical Library , published by Harvard University Press , or the Oxford Classical Texts , published by Oxford University Press . Latin translations of modern literature such as: The Hobbit , Treasure Island , Robinson Crusoe , Paddington Bear , Winnie the Pooh , The Adventures of Tintin , Asterix , Harry Potter , Le Petit Prince , Max and Moritz , How

6164-675: The language of the Roman Rite . The Tridentine Mass (also known as the Extraordinary Form or Traditional Latin Mass) is celebrated in Latin. Although the Mass of Paul VI (also known as the Ordinary Form or the Novus Ordo) is usually celebrated in the local vernacular language, it can be and often is said in Latin, in part or in whole, especially at multilingual gatherings. It is the official language of

6256-440: The large areas where it had come to be natively spoken. However, even after the fall of Western Rome , Latin remained the common language of international communication , science, scholarship and academia in Europe until well into the early 19th century, by which time modern languages had supplanted it in common academic and political usage. Late Latin is the literary language from the 3rd century AD onward. No longer spoken as

6348-462: The late seventeenth century, when spoken skills began to erode. It then became increasingly taught only to be read. Latin grammar is highly fusional , with classes of inflections for case , number , person , gender , tense , mood , voice , and aspect . The Latin alphabet is directly derived from the Etruscan and Greek alphabets . Latin remains the official language of the Holy See and

6440-431: The later part of the Roman Republic , up to 75 BC, i.e. before the age of Classical Latin . It is attested both in inscriptions and in some of the earliest extant Latin literary works, such as the comedies of Plautus and Terence . The Latin alphabet was devised from the Etruscan alphabet . The writing later changed from what was initially either a right-to-left or a boustrophedon script to what ultimately became

6532-421: The less prestigious colloquial registers , attested in inscriptions and some literary works such as those of the comic playwrights Plautus and Terence and the author Petronius . While often called a "dead language", Latin did not undergo language death . By the 6th to 9th centuries, natural language change eventually resulted in Latin as a vernacular language evolving into distinct Romance languages in

6624-405: The magnetic flux is time-dependent either because the boundary is time-dependent or magnetic field is time-dependent. In integral form: where d ℓ is an infinitesimal vector line element of the closed curve ∂ A {\displaystyle \partial A} , with magnitude equal to the length of the infinitesimal line element, and direction given by the tangent to

6716-412: The most common forms of flux from the transport phenomena literature are defined as follows: These fluxes are vectors at each point in space, and have a definite magnitude and direction. Also, one can take the divergence of any of these fluxes to determine the accumulation rate of the quantity in a control volume around a given point in space. For incompressible flow , the divergence of the volume flux

6808-462: The other varieties, as it was largely separated from the unifying influences in the western part of the Empire. Spoken Latin began to diverge into distinct languages by the 9th century at the latest, when the earliest extant Romance writings begin to appear. They were, throughout the period, confined to everyday speech, as Medieval Latin was used for writing. For many Italians using Latin, though, there

6900-726: The probability of finding a particle in a differential volume element d r is d P = | ψ | 2 d 3 r . {\displaystyle dP=|\psi |^{2}\,d^{3}\mathbf {r} .} Then the number of particles passing perpendicularly through unit area of a cross-section per unit time is the probability flux; J = i ℏ 2 m ( ψ ∇ ψ ∗ − ψ ∗ ∇ ψ ) . {\displaystyle \mathbf {J} ={\frac {i\hbar }{2m}}\left(\psi \nabla \psi ^{*}-\psi ^{*}\nabla \psi \right).} This

6992-408: The red arrows denotes the boundary of the surface, and the orientation of the arrows with respect to the surface denotes the sign of the inner product of the vector field with the surface normals. If the surface encloses a 3D region, usually the surface is oriented such that the influx is counted positive; the opposite is the outflux . The divergence theorem states that the net outflux through

7084-412: The same units must agree on the numerical value of a scalar field at any given point of physical space. Scalar fields are contrasted with other physical quantities such as vector fields , which associate a vector to every point of a region, as well as tensor fields and spinor fields . More subtly, scalar fields are often contrasted with pseudoscalar fields. In physics, scalar fields often describe

7176-622: The second factor is the mean free path and the square root (with the Boltzmann constant k ) is the mean velocity of the particles. In turbulent flows, the transport by eddy motion can be expressed as a grossly increased diffusion coefficient. In quantum mechanics , particles of mass m in the quantum state ψ ( r , t ) have a probability density defined as ρ = ψ ∗ ψ = | ψ | 2 . {\displaystyle \rho =\psi ^{*}\psi =|\psi |^{2}.} So

7268-407: The surface does not fold back onto itself. Also, the surface has to be actually oriented, i.e. we use a convention as to flowing which way is counted positive; flowing backward is then counted negative. The surface normal is usually directed by the right-hand rule . Conversely, one can consider the flux the more fundamental quantity and call the vector field the flux density. Often a vector field

7360-497: The surface here need not be flat. Finally, we can integrate again over the time duration t 1 to t 2 , getting the total amount of the property flowing through the surface in that time ( t 2  −  t 1 ): q = ∫ t 1 t 2 ∬ S j ⋅ d A d t . {\displaystyle q=\int _{t_{1}}^{t_{2}}\iint _{S}\mathbf {j} \cdot d\mathbf {A} \,dt.} Eight of

7452-763: The surface in which flux is being measured is fixed and has area A . The surface is assumed to be flat, and the flow is assumed to be everywhere constant with respect to position and perpendicular to the surface. Second, flux as a scalar field defined along a surface, i.e. a function of points on the surface: j ( p ) = ∂ I ∂ A ( p ) , {\displaystyle j(\mathbf {p} )={\frac {\partial I}{\partial A}}(\mathbf {p} ),} I ( A , p ) = d q d t ( A , p ) . {\displaystyle I(A,\mathbf {p} )={\frac {\mathrm {d} q}{\mathrm {d} t}}(A,\mathbf {p} ).} As before,

7544-675: The surface is assumed to be flat, and the flow is assumed to be everywhere perpendicular to it. However the flow need not be constant. q is now a function of p , a point on the surface, and A , an area. Rather than measure the total flow through the surface, q measures the flow through the disk with area A centered at p along the surface. Finally, flux as a vector field : j ( p ) = ∂ I ∂ A ( p ) , {\displaystyle \mathbf {j} (\mathbf {p} )={\frac {\partial \mathbf {I} }{\partial A}}(\mathbf {p} ),} I ( A , p ) =

7636-429: The surface: d q d t = ∬ S j ⋅ n ^ d A = ∬ S j ⋅ d A , {\displaystyle {\frac {\mathrm {d} q}{\mathrm {d} t}}=\iint _{S}\mathbf {j} \cdot \mathbf {\hat {n}} \,dA=\iint _{S}\mathbf {j} \cdot d\mathbf {A} ,} where A (and its infinitesimal)

7728-411: The term "scalar field" may be used to distinguish a function of this kind with a more general tensor field, density , or differential form . Physically, a scalar field is additionally distinguished by having units of measurement associated with it. In this context, a scalar field should also be independent of the coordinate system used to describe the physical system—that is, any two observers using

7820-404: The term corresponds to. In transport phenomena ( heat transfer , mass transfer and fluid dynamics ), flux is defined as the rate of flow of a property per unit area, which has the dimensions [quantity]·[time]·[area]. The area is of the surface the property is flowing "through" or "across". For example, the amount of water that flows through a cross section of a river each second divided by

7912-428: The transport definition—charge per time per area. Due to the conflicting definitions of flux , and the interchangeability of flux , flow , and current in nontechnical English, all of the terms used in this paragraph are sometimes used interchangeably and ambiguously. Concrete fluxes in the rest of this article will be used in accordance to their broad acceptance in the literature, regardless of which definition of flux

8004-452: The value of the scalar field at the same absolute point in space (or spacetime ) regardless of their respective points of origin. Examples used in physics include the temperature distribution throughout space, the pressure distribution in a fluid, and spin -zero quantum fields, such as the Higgs field . These fields are the subject of scalar field theory . Mathematically, a scalar field on

8096-413: Was free to develop on its own, there is no reason to suppose that the speech was uniform either diachronically or geographically. On the contrary, Romanised European populations developed their own dialects of the language, which eventually led to the differentiation of Romance languages . Late Latin is a kind of written Latin used in the 3rd to 6th centuries. This began to diverge from Classical forms at

8188-496: Was much more liberal in its linguistic cohesion: for example, in classical Latin sum and eram are used as auxiliary verbs in the perfect and pluperfect passive, which are compound tenses. Medieval Latin might use fui and fueram instead. Furthermore, the meanings of many words were changed and new words were introduced, often under influence from the vernacular. Identifiable individual styles of classically incorrect Latin prevail. Renaissance Latin, 1300 to 1500, and

8280-441: Was no complete separation between Italian and Latin, even into the beginning of the Renaissance . Petrarch for example saw Latin as a literary version of the spoken language. Medieval Latin is the written Latin in use during that portion of the post-classical period when no corresponding Latin vernacular existed, that is from around 700 to 1500 AD. The spoken language had developed into the various Romance languages; however, in

8372-502: Was one of the major developers of what we now call "electric flux" and "magnetic flux" according to the electromagnetism definition. Their names in accordance with the quote (and transport definition) would be "surface integral of electric flux" and "surface integral of magnetic flux", in which case "electric flux" would instead be defined as "electric field" and "magnetic flux" defined as "magnetic field". This implies that Maxwell conceived of these fields as flows/fluxes of some sort. Given

8464-478: Was shut down in June 2019), and Vatican Radio & Television, all of which broadcast news segments and other material in Latin. A variety of organisations, as well as informal Latin 'circuli' ('circles'), have been founded in more recent times to support the use of spoken Latin. Moreover, a number of university classics departments have begun incorporating communicative pedagogies in their Latin courses. These include

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