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Cobourg

Cobourg is a town in the Canadian province of Ontario, located in Southern Ontario 95 km east of Toronto and 62 km east of Oshawa. It is the largest town in and seat of Northumberland County. Its nearest neighbour is Port Hope, 7 km to the west. It is located along Highway 401 (exits 472 and 474) and the former Highway 2 (now Northumberland County Road 2). To the south, Cobourg borders Lake Ontario. To the north, east and west, it is surrounded by Hamilton Township.

History
The land which present-day Cobourg occupies was previously inhabited by Mississauga (Anishinaabe-speaking) peoples. The settlements that make up today's Cobourg were founded by United Empire Loyalists in 1798 within Northumberland County, Home District, Province of Upper Canada. Some of the founding fathers and early settlers were Eliud Nickerson, Joseph Ash, Zacheus Burnham and Asa Allworth Burnham. The Town was originally a group of smaller villages such as Amherst and Hardscrabble, which were later named Hamilton. In 1808 it became the district town for the Newcastle District. It was renamed Cobourg in 1819, in recognition of the marriage of Princess Charlotte Augusta of Wales to Prince Leopold of Saxe-Coburg-Saalfeld (later Saxe-Coburg and Gotha, who would later become King of Belgium).

By the 1830s, Cobourg had become a regional centre, mostly due to its fine harbour on Lake Ontario. In 1835 the Upper Canada Academy was established in Cobourg by Egerton Ryerson and the Wesleyan Conference of Bishops. On 1 July 1837, Cobourg was officially incorporated as a town. In 1841 the Upper Canada Academy's name was changed to Victoria College. In 1842 Victoria College was granted powers to confer degrees. Victoria College remained in Cobourg until 1892, when it was moved to Toronto and federated with the University of Toronto. In 1842, John Strachan founded the Diocesan Theological Institute in Cobourg, an Anglican seminary that became integrated into the University of Trinity College in Toronto in 1852.

The Railway to Rice Lake
The timber and other resources of Cobourg's large hinterland were identified as the key to its prosperity, and if they could be brought to the harbour, Lake Ontario opened up a large and prosperous market. Peterborough to the north, founded in 1825 by Peter Robinson, had become the principal source area, and in the 1830s the waterways were still the prime method of bulk transport. Rice Lake and the Otonabee River were brought into use when James Gray Bethune established a steamer running across the lake and up the Otonabee which was navigable through to Peterborough. This meant goods and passengers could be brought at least to the south shore of Rice Lake. The remaining 8 miles of rough tracks was viable for passengers and light goods, but no use for the valuable timber and mine products. By 1835, only 10 years after the first steam railway in the world, there was active discussion about building a railway up to what later became Harwood. However, the townspeople invested instead in a plank road, using 300,000 feet of 3-inch wooden planks, allowing horse-drawn vehicles to haul heavy goods. By 1850 the plank road was breaking up, and was impassible in wet conditions, so the railway scheme was revived.

By 1852, there was considerable enthusiasm for the railway project within the town. River traffic had become seen as yesterday's solution by this time, so the plans were expanded to include a 4 km long bridge across Rice Lake, to take the railway right up to Peterborough. By 1854 the rails reached the shore of the lake, and it found good work transporting passengers and nearly 2 million feet of lumber from the Rice Lake down to Cobourg that summer. However, revenue from the project had to be diverted into building an ill-fated bridge, using hundreds of wooden trestles, 31 Burr Truss spans, and a centre-pivot swing bridge to allow boats to pass. The prime mover locally for getting the Railway company off the ground was D'Arcy E. Boulton, a lawyer based in Cobourg, who enthused the town with the plan. They agreed to begin funding the scheme that was initially expected to cost £150,000, but ended leaving many people with worthless railway bonds and the town council with a debt that was only finally repaid in the 1930s. The man appointed to manage the project was Samuel Zimmerman, who had previously been instrumental in building the Great Western Railway (Ontario). The bridge was constructed over the summer of 1854 and was officially opened on 29 December that year. Three days later, it collapsed when ice movements shifted the trestles out of line, splintering the Burr Truss sections. The proposed solution was to stabilise the trestles (or 'stilts' as their critics dubbed them) by an infill of soil, which did happen on the southern side, still visible as a strip of land still remaining running into the lake near Harwood. But funds were not forthcoming for the northern side, and winter ice and shifting lake mud meant that it was frequently unusable.

A further problem emerged when Port Hope, not far along the coast, pursued its own plans for a Railway to Peterborough. In 1857, the Port Hope and Lindsay line was constructed, and the following year opened a branch to Peterborough, going round the western end of the lake, in direct competition with the struggling Cobourg route. The response of the Cobourg directors was to oust D.E. Boulton, who then invested in the Port Hope line. Conflicts of interest among various personnel resulted in the deliberate removal of the bolts on sections of the bridge in early 1861, ensuring that when the ice moved again the bridge was destroyed, and this time it was left unrepaired. The railway reverted to linking Cobourg harbour with Harwood and the Rice Lake water traffic.

In 1865, the railway was bought by a consortium of Pittsburgh steel manufacturers, who had already bought the Marmora Iron quarries north-east of Rice Lake. They established an iron-ore supply route using barges up the Trent River and across Rice Lake to the railway at Harwood. From there it was brought along the Railway to Cobourg Harbour for shipment across Lake Ontario to feed the steel mills of America. This provided a steady income for the railway and the town until the ore ran out in 1878. It also had two longer term spin-offs in the form of a rail car company and the beginnings of a tourist industry.

Crossen Car Works
When the iron ore scheme was getting underway, a small iron foundry based in Cobourg, was approached to cast the wheels and frames of wagons to move the Marmora iron ore down to the harbour. James Crossen saw an opportunity to combine his cast-iron products with the abundant local timber to produce railway rolling stock. With the foundry located near both the Cobourg line and train station of the Grand Trunk Railway, which had been built along the shore of Lake Ontario in 1856, it was well placed to expand as Canada's railway network grew. It was later named the Crossen Car Manufacturing Company and went on to become the largest builder of timber-framed rail cars in Canada, making everything from coal and freight wagons through to dining cars and first-class carriages. By 1910, wood was going out of date, when all-steel cars took over, and in 1915 the company, unable to adapt, went into liquidation, and parts of the site reverted to being an iron foundry. A replica of one of the Crossen ore cars was built in 2016 and is on display near Cobourg waterfront.

Wealthy Americans
The connections and trade links which developed through the iron shipments brought many American industrialists to Cobourg, which became a popular summer destination. High class hotels were established, followed in the late 19th century and early 20th century by enormous summer homes for wealthy Americans, a few of which still stand today. One notable home, on King Street East, later became the Brookside School, a youth detention centre. (It has now closed, and the property stands idle.) A major ferry service connected Cobourg and Rochester, New York from 1907 to 1952, transporting passengers and cargo across Lake Ontario, allowing Americans to reach the town more readily.

Victoria Hall
For a brief moment in 1856, the town, with both its new railway link to the interior and an east–west rail connection along the Grand Trunk Railway, was feeling secure in its future prosperity, and thought a new Town Hall would encourage further investment and be an asset to the area. Victoria Hall stands at the heart of the downtown, a building that now serves as the town hall, as well as home of the Art Gallery of Northumberland, the Cobourg Concert Hall, and an Old Bailey-style courtroom that is now used as the Council chamber. Victoria Hall was designed by architect Kivas Tully. The landmark is known for its impressive stone work. Charles Thomas Thomas (1820-1867), an English-born master stonecarver and building contractor, executed the fine stone carvings, including the bearded faced keystone over the main entrance into the building. Victoria Hall was officially opened on September 6, 1860 by the Prince of Wales, later to become King Edward VII. At that time, Cobourg was a significant town in the Province of Canada, and some townspeople felt that Cobourg would be a suitable capital for the newly united provinces; this privilege went to Ottawa, Ontario, however.

Heritage Centre
One of the oldest buildings in the town was for many years known as The Barracks, suggesting military connections. However it is equally likely that it was built for industrial uses, either in the very early 1800s or as a malting house and brewery by James Calcutt in the early 1830s. It probably served that purpose until a larger brewery was built by the McKechnies in 1863. The old stone-built building had a variety of industrial and storage purposes, and twenty different owners. In 2000 it was acquired in a run-down state by the Cobourg Museum Foundation, who have restored it and it is now open as the Sifton-Cook Heritage Centre.

Cobourg Jail
In 1906 on 77 Albert Street the Cobourg Jail, originally referred to as The County Jail, began to house inmates. Over time the Cobourg jail continued to expand to house the increasing population of inmates. In 1998 the jail closed and the year after in 1999 was purchased by private buyers. The jail has since been converted to a tourist attraction called the King George Inn with the jail still intact to see or stay in. Additionally, a family restaurant was implemented for guests and non-guests of the hotel.

Crash landing
On 20 December 1951, Cobourg experienced media attention as a chartered Curtiss C-46 Commando airplane, bound for Newark, New Jersey, made an emergency landing in local farmer Charles Wilson's field, alongside Highway 2 and Roger's Road. The pilot had lost his way after losing radio contact, and unwittingly drifted north. The 44 passengers and three crew escaped unhurt, but extremely cold in the sub-zero temperatures. The plane, having crash landed on deep snow, was able to be repaired and the field smoothed out enough for it to get airborne again.

Cobourg was the site of No. 26 Ordnance Depot, later Canadian Forces Station Cobourg, from 1953 to 1971.

Regeneration
Throughout the late 1980s and early 1990s, the town invested heavily in purchasing property along the waterfront and beautifying the area. A boardwalk was developed to connect the harbour and large sandy beach while further pathways were created to encompass Victoria Park and the historic downtown. Because of this renewal and revitalization, many community activities now revolve in and around these spaces.

Climate
Cobourg has a humid continental climate (Dfb) with warm summers and cold winters.

Transportation
Highway 401 and Northumberland County Road 2 (formerly Highway 2) pass through town. County Road 45 (formerly Highway 45) begins in Cobourg and heads north-east to Norwood.

The Canadian Pacific and Canadian National main railway lines (Toronto – Ottawa/Montreal) traverse the town. Via Rail offers passenger service to Cobourg railway station with multiple daily departures on the Toronto – Ottawa/Montreal route.

Cobourg Transit is a fully accessible community transit system that provides daily bus service to the town. The community transit system ends its runs by 7pm, which can hamper travel for those without other means of transportation. It formerly ran until 9:30pm however the hours were reduced.

The Ontario Waterfront Trail passes through Cobourg and cyclists can be found moving through daily during the warmer months.

Demographics
In the 2021 Census of Population conducted by Statistics Canada, Cobourg had a population of $20,519$ living in $9,134$ of its $9,520$ total private dwellings, a change of NaN% from its 2016 population of $19,440$. With a land area of 22.41 km2, it had a population density of in 2021.

Age Profile
 * Age 0-14: 18.1%
 * Age 15-24: 11.9%
 * Age 25-44: 24.7%
 * Age 45-64: 24.2%
 * Age 65-74: 10.8%
 * Age 75+: 10.3%

Religious Make Up
 * 56.1% Protestant
 * 23.9% Catholic
 * 2.2% other Christian
 * 0.7% other religions
 * 17.1% non-professing

Racial Profile
 * 91.4% White
 * 4.4% Aboriginal
 * 3.8% Korean
 * 0.7% Black

Government
The Town of Cobourg is governed by a seven-member town council. The Town is responsible for delivering a variety of local services, including planning, transit, fire and police protection, and the maintenance of local roads and parks. Certain responsibilities, such as waste management or paramedic services, are the responsibility of Northumberland County.

There are six advisory committees appointed by the city council. These bodies are made up of city councillors.

 * Accessibility Advisory Committee - effective 2023
 * Civic Awards Advisory Committee - to re-convene in 2024
 * Cobourg Heritage Advisory Committee - effective 2023
 * Cobourg Police Service Board - effective 2023
 * Cobourg Public Library Board - effective 2023
 * Committee of Adjustment - effective 2023

Previous bodies include:

 * Equity, Diversity and Inclusion Committee - no longer active
 * Parks and Recreation Advisory Committee - no longer active
 * Planning and Development Advisory Committee - no longer active
 * Sustainability and Climate Change Emergency Advisory Committee - no longer active
 * Transportation Advisory Committee, Zero Emission Ad Hoc Committee (on hold)

The Town maintains an independent municipal police force, the Cobourg Police Service, which provides policing services to all areas of the municipality as well as seasonal by-law enforcement services in the neighbouring Township of Hamilton.

The Town of Cobourg had an approved operating budget of CA$28.1 million in 2023 and a capital budget plan of CA$33 million. The town's revenues include subsidies from the Government of Canada and the Government of Ontario (for programs mandated by those governments). The town's largest operating expenditure is the Harbour Rehabilitation project costing nearly CA$23 million in 2023.

Education
Public education in Cobourg is under the management of the Kawartha Pine Ridge District School Board, and Catholic education is by the Peterborough Victoria Northumberland and Clarington Catholic District School Board.

Cobourg also has a satellite campus of Fleming College.

In 1871, Cobourg's first Collegiate Institute was erected on King Street East. In 1899, the student population at the original Cobourg Collegiate Institute had outgrown the school. New plans for another school on the west end of King Street were put in place and the new school building on 135 King Street was built. This new building was called the West high school and the older school building was then named the East high school. The West high school had many additions in the 1920s as the school continued to grow. The two schools co-existed for 116 years until the closing of both schools in 2015. The purpose of closing the two schools was to bring them back together with the fabrication of a brand new Cobourg Collegiate Institute in 2015. The new school was built on the property of the old east high school. The old west high school was not demolished, but renamed William Academy Cobourg West and is a functioning co-ed private school for grades 7-12.

Attractions
Cobourg retains its small-town atmosphere, in part due to the downtown and surrounding residential area's status as a Heritage Conservation District. The downtown is a well-preserved example of a traditional small-town main street. Victoria Hall, the town hall completed in 1860, is a National Historic Site of Canada. The oldest building in the town is now open as the Sifton Cook Heritage Centre and operated by the Cobourg Museum Foundation.

Cobourg has several parks. The Cobourg Waterfront Festival, held in Victoria Park and the nearby beach and harbour, is an annual arts and crafts event occurring on Canada Day. It began in 1987 as part of the town's sesquicentennial celebrations, and was conceived by the Art Gallery of Northumberland's former Director/Curator Peter Tulumello and former Concert Hall Manager Mark Finnan. Cobourg's oldest annual event, the Cobourg Highland Games, was started in 1963 in Donegan Park by Dave Carr to celebrate the Scottish culture in the area. The event is now held in Victoria Park in June.

Cobourg's beach, Victoria Park Beach, is used as a location for volleyball tournaments, events, beach days, family picnics and other events. The beach is equipped with a splash park, playground set, park area for eating, local restaurants and a pier. Additionally, there is a small trailer park located directly next to the beach. There are lifeguards on duty some days and it is safe to swim and play in. Close to the downtown, tourists can shop in local shops and eat out at many of Cobourg's restaurants.

Cobourg is home to the Port Hope Drive-In, Canada's oldest drive-in.

Print

 * Northumberland Today, owned by Osprey Media, is a daily newspaper with content contributed from the Cobourg Star and Port Hope Guide.

Radio
Five radio stations are licensed to Cobourg, providing music, news and local programming to Northumberland County and surrounding areas.
 * FM 89.7 - CFWN, not-for-profit community-based, volunteer driven, programming focused on the community
 * FM 90.7 - CHJJ, Christian music
 * FM 93.3 - CKSG, adult contemporary
 * FM 103.1 - CFMZ, classical
 * FM 107.9 - CHUC, classic hits

TV
Cobourg uses cable 10 to broadcast local events and weather. Apart from that, Cobourg has no local terrestrial television stations; the closest is CHEX-TV in Peterborough, Ontario.

Notable people

 * John Douglas Armour, Puisne judge of the Supreme Court of Canada
 * Ada Blenkhorn, hymn writer who wrote "Keep on the Sunny Side" and "Let the Sunshine In"
 * Charles Arkoll Boulton, militia leader in the Red River Rebellion
 * Fern Blodgett Sunde, first Canadian woman to serve with the merchant marines during the Second World War
 * Alan Bradley, author
 * Jeremiah Brown, Canadian Men's Eight Rowing Team, won silver at the 2012 London Olympic Games
 * Mandy Bujold, Canadian boxer
 * James Cockburn, lawyer and Father of Confederation
 * cleopatrick, rock band
 * Martin Dobkin, first mayor of the City of Mississauga, Ontario and served as Mayor from 1973 to 1976.
 * Marie Dressler, silent film actress, Academy Award winner best actress 1931
 * Francis P. Duffy, Canadian-born military chaplain of the Fighting 69th New York Regiment during WWI
 * Sandford Fleming who helped create and define the idea of time zones
 * Rev. John Weir Foote, politician, chaplain, and Victoria Cross recipient
 * Ed Greenwood, author of the Forgotten Realms setting for Dungeons & Dragons
 * W. A. Hewitt, sports executive and journalist, Hockey Hall of Fame inductee
 * Paul Kane, Irish-born Canadian painter
 * Henry Bowyer Lane, English born Toronto architect
 * John Murray, pioneering oceanographer, marine biologist and limnologist. Considered the father of modern oceanography.
 * Ebenezer Perry (1788–1876), politician and Cobourg's first police commissioner
 * Josh Richards, internet personality, musician and actor
 * William Renwick Riddell, judge and historian
 * Dave "Goldie Rogers" Sherwin, professional wrestler
 * Virna Sheard, poet and novelist
 * Steve Smith, NHL hockey player
 * Justin Williams, NHL hockey player, 3-time Stanley Cup champion, winner of 2014 Conn Smythe Trophy

Twin towns – sister cities
Cobourg is twinned with:
 * 🇩🇪 Coburg, Germany

title: Cobourg


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Product rule

In calculus, the product rule (or Leibniz rule or Leibniz product rule) is a formula used to find the derivatives of products of two or more functions. For two functions, it may be stated in Lagrange's notation as $$(u \cdot v)' = u ' \cdot v + u \cdot v' $$ or in Leibniz's notation as $$\frac{d}{dx} (u\cdot v) = \frac{du}{dx} \cdot v + u \cdot \frac{dv}{dx}.$$

The rule may be extended or generalized to products of three or more functions, to a rule for higher-order derivatives of a product, and to other contexts.

Discovery
Discovery of this rule is credited to Gottfried Leibniz, who demonstrated it using differentials. (However, J. M. Child, a translator of Leibniz's papers, argues that it is due to Isaac Barrow.) Here is Leibniz's argument: Let u(x) and v(x) be two differentiable functions of x. Then the differential of uv is $$\begin{align} d(u\cdot v) & {} = (u + du)\cdot (v + dv) - u\cdot v \\ & {} = u\cdot dv + v\cdot du + du\cdot dv. \end{align}$$

Since the term du·dv is "negligible" (compared to du and dv), Leibniz concluded that $$d(u\cdot v) = v\cdot du + u\cdot dv $$ and this is indeed the differential form of the product rule. If we divide through by the differential dx, we obtain $$\frac{d}{dx} (u\cdot v) = v \cdot \frac{du}{dx} + u \cdot \frac{dv}{dx} $$ which can also be written in Lagrange's notation as $$(u\cdot v)' = v\cdot u' + u\cdot  v'. $$

Examples

 * Suppose we want to differentiate $$f(x)=x^2\text {sin}(x).$$ By using the product rule, one gets the derivative $$f'(x)=2x\cdot \text {sin}(x)+x^2\text {cos} (x)$$ (since the derivative of $$x^2$$ is $$2x,$$ and the derivative of the sine function is the cosine function).
 * One special case of the product rule is the constant multiple rule, which states: if $c$ is a number, and $$f(x)$$ is a differentiable function, then $$c\cdot f(x)$$ is also differentiable, and its derivative is $$(cf)'(x)=c \cdot f'(x).$$ This follows from the product rule since the derivative of any constant is zero. This, combined with the sum rule for derivatives, shows that differentiation is linear.
 * The rule for integration by parts is derived from the product rule, as is (a weak version of) the quotient rule. (It is a "weak" version in that it does not prove that the quotient is differentiable but only says what its derivative is it is differentiable.)

Limit definition of derivative
Let $h(x) = f(x)g(x)$ and suppose that $f$ and $g$ are each differentiable at $x$. We want to prove that $h$ is differentiable at $x$ and that its derivative, $h'(x)$, is given by $f(x)g(x) + f(x)g(x)$. To do this, $$f(x)g(x+\Delta x)-f(x)g(x+\Delta x)$$ (which is zero, and thus does not change the value) is added to the numerator to permit its factoring, and then properties of limits are used. $$\begin{align} h'(x) &= \lim_{\Delta x\to 0} \frac{h(x+\Delta x)-h(x)}{\Delta x} \\[5pt] &= \lim_{\Delta x\to 0} \frac{f(x+\Delta x)g(x+\Delta x)-f(x)g(x)}{\Delta x} \\[5pt] &= \lim_{\Delta x\to 0} \frac{f(x+\Delta x)g(x+\Delta x)-f(x)g(x+\Delta x)+f(x)g(x+\Delta x)-f(x)g(x)}{\Delta x} \\[5pt] &= \lim_{\Delta x\to 0} \frac{\big[f(x+\Delta x)-f(x)\big] \cdot g(x+\Delta x) + f(x) \cdot \big[g(x+\Delta x)-g(x)\big]}{\Delta x} \\[5pt] &= \lim_{\Delta x\to 0} \frac{f(x+\Delta x)-f(x)}{\Delta x} \cdot \underbrace{\lim_{\Delta x\to 0} g(x+\Delta x)}_\text{See the note below.} + \lim_{\Delta x\to 0} f(x) \cdot \lim_{\Delta x\to 0} \frac{g(x+\Delta x)-g(x)}{\Delta x} \\[5pt] &= f'(x)g(x)+f(x)g'(x). \end{align}$$ The fact that $$ \lim_{\Delta x\to0} g(x+\Delta x) = g(x) $$ follows from the fact that differentiable functions are continuous.

Linear approximations
By definition, if $$ f, g: \mathbb{R} \to \mathbb{R} $$ are differentiable at $$ x $$, then we can write linear approximations: $$ f(x+h) = f(x) + f'(x)h + \varepsilon_1(h) $$ and $$ g(x+h) = g(x) + g'(x)h + \varepsilon_2(h), $$ where the error terms are small with respect to h: that is, $ \lim_{h \to 0} \frac{\varepsilon_1(h)}{h} = \lim_{h \to 0} \frac{\varepsilon_2(h)}{h} = 0, $ also written $$\varepsilon_1, \varepsilon_2 \sim o(h)$$. Then: $$ \begin{align} f(x+h)g(x+h) - f(x)g(x) &= (f(x) + f'(x)h +\varepsilon_1(h))(g(x) + g'(x)h + \varepsilon_2(h)) - f(x)g(x) \\[.5em] &= f(x)g(x) + f'(x)g(x)h + f(x)g'(x)h -f(x)g(x) + \text{error terms} \\[.5em] &= f'(x)g(x)h + f(x)g'(x)h + o(h). \end{align} $$ The "error terms" consist of items such as $$f(x)\varepsilon_2(h), f'(x)g'(x)h^2$$ and $$hf'(x)\varepsilon_1(h)$$ which are easily seen to have magnitude $$o(h).$$ Dividing by $$ h $$ and taking the limit $$ h\to 0 $$ gives the result.

Quarter squares
This proof uses the chain rule and the quarter square function $$q(x)=\tfrac14x^2$$ with derivative $$q'(x) = \tfrac12 x$$. We have: $$uv=q(u+v)-q(u-v), $$ and differentiating both sides gives: $$\begin{align} f' &= q'(u+v)(u'+v') - q'(u-v)(u'-v') \\[4pt] &= \left(\tfrac12(u+v)(u'+v')\right) - \left(\tfrac12(u-v)(u'-v')\right) \\[4pt] &= \tfrac12(uu' + vu' + uv' + vv') - \tfrac12(uu' - vu' - uv' + vv') \\[4pt] &= vu'+uv'. \end{align}$$

Multivariable chain rule
The product rule can be considered a special case of the chain rule for several variables, applied to the multiplication function $$m(u,v) = uv$$: $$ {d (uv) \over dx} = \frac{\partial(uv)}{\partial u}\frac{du}{dx}+\frac{\partial (uv)}{\partial v}\frac{dv}{dx} = v \frac{du}{dx} + u \frac{dv}{dx}. $$

Non-standard analysis
Let u and v be continuous functions in x, and let dx, du and dv be infinitesimals within the framework of non-standard analysis, specifically the hyperreal numbers. Using st to denote the standard part function that associates to a finite hyperreal number the real infinitely close to it, this gives $$\begin{align} \frac{d(uv)}{dx} &= \operatorname{st}\left(\frac{(u + du)(v + dv) - uv}{dx}\right) \\ &= \operatorname{st}\left(\frac{uv + u \cdot dv + v \cdot du + du \cdot dv -uv}{dx}\right) \\ &= \operatorname{st}\left(\frac{u \cdot dv + v \cdot du + du \cdot dv}{dx}\right) \\ &= \operatorname{st}\left(u \frac{dv}{dx} + (v + dv) \frac{du}{dx}\right) \\ &= u \frac{dv}{dx} + v \frac{du}{dx}. \end{align}$$ This was essentially Leibniz's proof exploiting the transcendental law of homogeneity (in place of the standard part above).

Smooth infinitesimal analysis
In the context of Lawvere's approach to infinitesimals, let $$dx$$ be a nilsquare infinitesimal. Then $$du = u'\ dx$$ and $$dv = v'\ dx$$, so that $$\begin{align} d(uv) & = (u + du)(v + dv) -uv \\ & = uv + u \cdot dv + v \cdot du + du \cdot dv - uv \\ & = u \cdot dv + v \cdot du + du \cdot dv \\ & = u \cdot dv + v \cdot du \end{align}$$ since $$du \, dv = u' v' (dx)^2 = 0.$$ Dividing by $$dx$$ then gives $$\frac{d(uv)}{dx} = u \frac{dv}{dx} + v \frac{du}{dx}$$ or $$(uv)' = u \cdot v' + v \cdot u'$$.

Logarithmic differentiation
Let $$h(x) = f(x) g(x)$$. Taking the absolute value of each function and the natural log of both sides of the equation, $$\ln|h(x)| = \ln|f(x) g(x)| $$ Applying properties of the absolute value and logarithms, $$\ln|h(x)| = \ln|f(x)| + \ln|g(x)| $$ Taking the logarithmic derivative of both sides and then solving for $$h'(x) $$: $$\frac{h'(x)}{h(x)} = \frac{f'(x)}{f(x)} + \frac{g'(x)}{g(x)}$$ Solving for $$h'(x)$$ and substituting back $$f(x) g(x)$$ for $$h(x)$$ gives: $$\begin{align} h'(x) &= h(x)\left(\frac{f'(x)}{f(x)} + \frac{g'(x)}{g(x)}\right) \\ &= f(x) g(x)\left(\frac{f'(x)}{f(x)} + \frac{g'(x)}{g(x)}\right) \\ &= f'(x) g(x) + f(x) g'(x). \end{align}$$ Note: Taking the absolute value of the functions is necessary for the logarithmic differentiation of functions that may have negative values, as logarithms are only real-valued for positive arguments. This works because $$\tfrac{d}{dx}(\ln |u|) = \tfrac{u'}{u}$$, which justifies taking the absolute value of the functions for logarithmic differentiation.

Product of more than two factors
The product rule can be generalized to products of more than two factors. For example, for three factors we have $$\frac{d(uvw)}{dx} = \frac{du}{dx}vw + u\frac{dv}{dx}w + uv\frac{dw}{dx}.$$ For a collection of functions $$f_1, \dots, f_k$$, we have $$\frac{d}{dx} \left [ \prod_{i=1}^k f_i(x) \right ] = \sum_{i=1}^k \left(\left(\frac{d}{dx} f_i(x) \right) \prod_{j=1,j\ne i}^k f_j(x) \right) = \left( \prod_{i=1}^k f_i(x) \right) \left( \sum_{i=1}^k \frac{f'_i(x)}{f_i(x)} \right).$$

The logarithmic derivative provides a simpler expression of the last form, as well as a direct proof that does not involve any recursion. The logarithmic derivative  of a function $f$, denoted here $Logder(f)$, is the derivative of the logarithm of the function. It follows that $$\operatorname{Logder}(f)=\frac {f'}f.$$ Using that the logarithm of a product is the sum of the logarithms of the factors, the sum rule for derivatives gives immediately $$\operatorname{Logder}(f_1\cdots f_k)= \sum_{i=1}^k\operatorname{Logder}(f_i).$$ The last above expression of the derivative of a product is obtained by multiplying both members of this equation by the product of the $$f_i.$$

Higher derivatives
It can also be generalized to the general Leibniz rule for the nth derivative of a product of two factors, by symbolically expanding according to the binomial theorem: $$d^n(uv) = \sum_{k=0}^n {n \choose k} \cdot d^{(n-k)}(u)\cdot d^{(k)}(v).$$

Applied at a specific point x, the above formula gives: $$(uv)^{(n)}(x) = \sum_{k=0}^n {n \choose k} \cdot u^{(n-k)}(x)\cdot v^{(k)}(x).$$

Furthermore, for the nth derivative of an arbitrary number of factors, one has a similar formula with multinomial coefficients: $$\left(\prod_{i=1}^kf_i\right)^{\!\!(n)}=\sum_{j_1+j_2+\cdots+j_k=n}{n\choose j_1,j_2,\ldots,j_k}\prod_{i=1}^kf_i^{(j_i)}.$$

Higher partial derivatives
For partial derivatives, we have $${\partial^n \over \partial x_1\,\cdots\,\partial x_n} (uv) = \sum_S {\partial^{|S|} u \over \prod_{i\in S} \partial x_i} \cdot {\partial^{n-|S|} v \over \prod_{i\not\in S} \partial x_i}$$ where the index $S$ runs through all $2^{n}$ subsets of $\{1, ..., n\}$, and $|S|$ is the cardinality of $S$. For example, when $n = 3$, $$\begin{align} & {\partial^3 \over \partial x_1\,\partial x_2\,\partial x_3} (uv) \\[1ex] = {} & u \cdot{\partial^3 v \over \partial x_1\,\partial x_2\,\partial x_3} + {\partial u \over \partial x_1}\cdot{\partial^2 v \over \partial x_2\,\partial x_3} + {\partial u \over \partial x_2}\cdot{\partial^2 v \over \partial x_1\,\partial x_3} + {\partial u \over \partial x_3}\cdot{\partial^2 v \over \partial x_1\,\partial x_2} \\[1ex] & + {\partial^2 u \over \partial x_1\,\partial x_2}\cdot{\partial v \over \partial x_3} + {\partial^2 u \over \partial x_1\,\partial x_3}\cdot{\partial v \over \partial x_2} + {\partial^2 u \over \partial x_2\,\partial x_3}\cdot{\partial v \over \partial x_1} + {\partial^3 u \over \partial x_1\,\partial x_2\,\partial x_3}\cdot v. \\[-3ex]&\end{align}$$

Banach space
Suppose X, Y, and Z are Banach spaces (which includes Euclidean space) and B : X × Y → Z is a continuous bilinear operator. Then B is differentiable, and its derivative at the point (x,y) in X × Y is the linear map D(x,y)B : X × Y → Z given by $$ (D_\left( x,y \right)\,B)\left( u,v \right) = B\left( u,y \right) + B\left( x,v \right)\qquad\forall (u,v)\in X \times Y. $$

This result can be extended to more general topological vector spaces.

In vector calculus
The product rule extends to various product operations of vector functions on $$\mathbb{R}^n$$:

 * For scalar multiplication: $$(f \cdot \mathbf g)' = f'\cdot \mathbf g + f \cdot \mathbf g' $$
 * For dot product: $$(\mathbf f \cdot \mathbf g)' = \mathbf f' \cdot \mathbf g + \mathbf f \cdot \mathbf g' $$
 * For cross product of vector functions on $$\mathbb{R}^3$$: $$(\mathbf f \times \mathbf g)' = \mathbf f' \times \mathbf g + \mathbf f \times \mathbf g' $$

There are also analogues for other analogs of the derivative: if f and g are scalar fields then there is a product rule with the gradient: $$\nabla (f \cdot g) = \nabla f \cdot g + f \cdot \nabla g $$

Such a rule will hold for any continuous bilinear product operation. Let B : X × Y → Z be a continuous bilinear map between vector spaces, and let f and g be differentiable functions into X and Y, respectively. The only properties of multiplication used in the proof using the limit definition of derivative is that multiplication is continuous and bilinear. So for any continuous bilinear operation, $$H(f, g)' = H(f', g) + H(f, g').$$ This is also a special case of the product rule for bilinear maps in Banach space.

Derivations in abstract algebra and differential geometry
In abstract algebra, the product rule is the defining property of a derivation. In this terminology, the product rule states that the derivative operator is a derivation on functions.

In differential geometry, a tangent vector to a manifold M at a point p may be defined abstractly as an operator on real-valued functions which behaves like a directional derivative at p: that is, a linear functional v which is a derivation, $$v(fg) = v(f)\,g(p) + f(p) \, v(g).$$ Generalizing (and dualizing) the formulas of vector calculus to an n-dimensional manifold M, one may take differential forms of degrees k and l, denoted $$\alpha\in \Omega^k(M), \beta\in \Omega^\ell(M)$$, with the wedge or exterior product operation $$\alpha\wedge\beta\in \Omega^{k+\ell}(M)$$, as well as the exterior derivative $$d:\Omega^m(M)\to\Omega^{m+1}(M)$$. Then one has the graded Leibniz rule: $$d(\alpha\wedge\beta)= d\alpha \wedge \beta + (-1)^{k} \alpha\wedge d\beta.$$

Applications
Among the applications of the product rule is a proof that $$ {d \over dx} x^n = nx^{n-1}$$ when n is a positive integer (this rule is true even if n is not positive or is not an integer, but the proof of that must rely on other methods). The proof is by mathematical induction on the exponent n. If n = 0 then xn is constant and nxn − 1 = 0. The rule holds in that case because the derivative of a constant function is 0. If the rule holds for any particular exponent n, then for the next value, n + 1, we have $$\begin{align} \frac{d x^{n+1}}{dx} &{}= \frac{d}{dx} \left( x^n\cdot x\right) \\[1ex] &{}= x \frac{d}{dx} x^n + x^n \frac{d}{dx} x & \text{(the product rule is used here)} \\[1ex] &{}= x\left(n x^{n-1}\right) + x^n\cdot 1 & \text{(the induction hypothesis is used here)} \\[1ex] &{}= \left(n + 1\right) x^n. \end{align} $$ Therefore, if the proposition is true for n, it is true also for n + 1, and therefore for all natural n.

license: Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/4.0/

title: Product rule


text:
The Hee Bee Gee Bees

The Hee Bee Gee Bees was a fictitious pop group which parodied pop groups and performers in the early 1980s, consisting of Angus Deayton, Michael Fenton Stevens, and Philip Pope of the UK radio series Radio Active.

Their first single was "Meaningless Songs (in Very High Voices)" by the Hee Bee Gee Bees, a parody of the Bee Gees. The 'band' consisted of the three Cribb (Gibb) brothers; Garry (Barry), Norris (Maurice) and Dobbin (Robin), performed respectively by Deayton, Fenton Stevens, and Pope. It was written by Pope and Richard Curtis, released by Original Records in 1980, reached number two in the Australian singles chart and made an appearance on the UK Indie Charts.

Two albums were subsequently released under the Hee Bee Gee Bees name, featuring parodies of various pop groups that had originally been featured on Radio Active. Tracks included parodies of Supertramp ("Scatological Song" by Supertrash), Michael Jackson ("Up the Wall" by Jack Michaelson), Status Quo ("Boring Song" by Status Quid), The Police ("Too Depressed to Commit Suicide" by The PeeCees), David Bowie ("Quite Ahead of My Time" by David Bowwow), Gary Numan ("Are Trains Electric?" by Gary Inhuman) and others. The first album was recorded in the Strawberry Studios in Stockport, and featured 10cc and Sad Café studio musicians. The second album was recorded in and mixed at Silo Studios in London, with Les Davidson (guitar) Steve Shone (bass) and Dave Early (drums). The music was written by Pope, with lyrics by Deayton and other Radio Active scriptwriters.

Fenton Stevens and Pope later collaborated on the parody song "The Chicken Song", which reached number 1 in the UK in 1986.

title: The Hee Bee Gee Bees


text:
List of Formula One constructors

Formula One, abbreviated to F1, is the highest class of open-wheel racing defined by the Fédération Internationale de l'Automobile (FIA), motorsport's world governing body. The formula in the name alludes to a series of rules established by the FIA to which all participants and vehicles are required to conform. Each year, the F1 World Championship season is held, consisting of a series of races, known as Grands Prix, held usually on purpose-built circuits, and in a few cases on closed city streets. Constructors are awarded points based on the finishing position of each of their two drivers at each Grand Prix, and the constructor who accumulates the most points over each championship is crowned that year's World Constructors' Champion. As of the 2024 British Grand Prix, there have been 172 Formula One constructors who have raced at least one of the 1,113 FIA World Championship races since the first such event, the 1950 British Grand Prix.

Constructors are people or corporate entities which design key parts of Formula One cars that have competed or are intended to compete in the FIA World Championship. Since 1981, it has been a requirement that each competitor must have the exclusive rights to the use of certain key parts of their car; in 2018, these parts were the survival cell, the front impact structure, the roll structures and bodywork.

Ferrari holds the record for the most Constructors' and Drivers' Championships won with sixteen and fifteen, respectively. Ferrari also holds the record for the most wins by a constructor with, the most pole positions with , the most points with , and the most podiums with. Ferrari has also entered more Grands Prix than any other constructor with entries and also maintains the record for the most Grand Prix starts with. The most recent constructor to make their debut was RB, which debuted at the.

Terminology
In Formula One racing the terms "constructor" and "entrant" have specific and differing meanings. An entrant is the person or corporate entity that registers a car and driver for a race, and is then responsible for preparing and maintaining that car during the race weekend. As a result of this preparation role and active involvement in the running of the race, the term "team" has become commonly applied to an entrant organisation. Statisticians do not always agree on how to count statistics related to these entities.

Constructors
Under Article 6.3 of the FIA Sporting Regulations, "A constructor is the person (including any corporate or unincorporated body) which designs the Listed Parts set out in Appendix 6. The make of an engine or chassis is the name attributed to it by its constructor." These "listed parts" include the survival cell, the front impact structure, the roll structures and bodywork. However, if the chassis and engine are made by different entities, the constructor comprises both (e.g. McLaren-Mercedes, Lotus-Climax etc.), with the name of the chassis constructor being placed before that of the engine constructor. As both chassis and engine are included in the constructor name, chassis run with different engines are counted as two separate constructors and score points separately. This occurred for the last time in the season when the Tyrrell team ran their chassis powered by both Ford and Renault engines, scored points with both engines and thus finishing 9th as Tyrrell-Ford and 10th as Tyrrell-Renault in the World Constructors' Championship.

Under article 6.2 of the FIA sporting regulations, "The title of Formula One World Champion Constructor will be awarded to the competitor which has scored the highest number of points". From the inaugural season of the World Constructors' Championship in up until the  season only the highest-scoring driver in each race for each constructor contributed points towards the World Constructors' Championship (then officially as the International Cup for Formula One Constructors ); since the  season points from all cars entered by each constructor have counted towards their championship total.

Teams
Since the season the FIA have required that Formula One entrants own the intellectual rights to the chassis that they enter, and so the distinction between the terms "entrant" and "constructor", and hence also "team", have become less pronounced, though the intellectual rights of engines may still be owned by a different entity. That season also saw the International Cup for Formula One Constructors be officially renamed to the World Constructors' Championship.

Before this time, constructors were free to sell their chassis to as many other teams as they liked. Brabham and Lotus chassis were used extensively by other teams during the 1960s and 1970s and several quite competitive privateer teams never built their own chassis. Rob Walker Racing Team was the most successful example, being responsible for the first victories in Formula One for both Cooper and Lotus. The concept of a "works" or "factory" team (i.e. the official team of the company producing the cars, as opposed to a customer team which buys them off the shelf) therefore applied to chassis in the same way as it does in rallying and sports car racing.

There have been some recent exceptions where a specialist company, not itself entered in the championship, has been commissioned to design and build a chassis for a team, e.g. Lola built cars for the Larrousse team (-) and the Scuderia Italia team and Dallara built cars for the Scuderia Italia team (-). Larousse had their points from the season erased after the FIA decided that they had falsely nominated themselves and not Lola as the chassis constructor. In, the new Arrows team which had been established by former Shadow personnel was sued by Shadow on the grounds that the Arrows FA/1 car was a copy of Shadow's DN9 – a view upheld by the UK High Court, which placed a ban on Arrows racing the FA/1. There have been more recent cases with Ligier (1995), Sauber (2004), Scuderia Toro Rosso (2006–2007) and Super Aguri (2007–2008) where teams have been accused of using a chassis produced by another constructor (respectively Benetton, Ferrari, Red Bull Racing and Honda). No action was taken against any of these teams, the sporting authorities being satisfied in each case that the team owned the intellectual property to the chassis they raced.

From the middle of the season (the 1973 Belgian Grand Prix) until the end of the  season, each team had permanent racing numbers from race to race throughout the season. Between the and  seasons the numbers were based on the teams' finishing positions in the 1973 Constructors' Championship (with slight modifications, e.g. Ferrari's traditional numbers were 11–12 until  and 27–28 from  onwards) and each team only changed numbers if they had the driver who had won the World Drivers' Championship in the previous season – the winning driver taking the number 1 and his teammate the number 2, and the team that had previously had those numbers switching to the newly vacated ones. Between and  the numbers were based on the teams' finishing positions in the Constructors' Championship from the previous season, with numbers 1 and 2 assigned to the defending champion and his teammate. During the period of 1974–1995 Tyrrell was the only team to keep the same numbers (3 and 4) every season. Since, racing numbers have been assigned to drivers instead of teams.

The number of cars entered by one team into a single race was not strictly limited in the 1950s and early 1960s. Since the season teams were generally allowed to enter only two regular cars, with the third car reserved for an occasional driver. This rule was further promoted in the season when the permanent racing numbers were assigned to each team in pairs, with the third car having the racing number out of the pair. Entering more than three cars was exceptionally tolerated, most notably regarding the BRM team in the and  seasons. However, many teams during this period entered only two cars, e.g. Ferrari have entered no more than two cars (with one exception at the 1976 Italian Grand Prix in connection with Lauda's comeback) every season since. Since the season the FIA have required that teams enter no more than two cars for a race; during this season Renault became the last team to have entered three cars for a race at the 1985 German Grand Prix, but only two of their cars were eligible for championship points.

Team's nationality
Unlike drivers who are required to compete in the FIA Formula One World Championship under the nationality of their passport, the FIA's International Sporting Code states that teams competing in the FIA Formula One World Championship shall compete under the nationality of their parent National Automobile Club that issued their FIA racing licence. On the basis of this regulation, despite the fact that most current teams are based in the UK, this country is officially represented in Formula One only by teams holding a racing licence issued by the British National Sporting Authority.

Teams take the nationality of their parent National Automobile Club that issued their licence for the period of validity of that licence and the change of the nationality is allowed. Several teams changed their nationality during their competition in Formula One, some of them even twice (e.g. Shadow in from American to British, Benetton in  from British to Italian, Red Bull in  from British to Austrian, Renault in  from French to British and in  back to French ). At the 1997 German Grand Prix Benetton became the only team to have achieved victories while racing under two different nationalities. The team's nationality, determined by a racing licence that a team holds, subsequently determines a national anthem played after a race on the podium in honour of a winning team following a national anthem played in honour of a winning driver.

Before the arrival of sponsorship liveries in the season the team's nationality also determined the colour of a car entered by the team; thus, Italian teams' cars were rosso corsa red, French were bleu de France blue, and British (with several exceptions, such as cars entered by teams Rob Walker, Brabham and McLaren ) were British racing green. Since the licence is given to a team and not to a constructor, privateer teams entering cars built by constructors from another country before the 1968 season painted cars in the national colour of their home country, e.g. the French Guy Ligier's privateer team entered cars painted in bleu de France blue in and  seasons despite the fact that they were built by the British constructor Cooper.

Relating to the team's nationality because of teams' bases in Britain several mistakes occurred on official entry lists issued by or podium ceremonies organized by the FIA or race organisers, e.g. Wolf holding the Canadian nationality and Shadow (in ) and Penske  both holding the American nationality all identified as the British teams by official entry lists, or the British national anthem played on the podium in honour of the Irish-licensed Jordan team and the Austrian-licensed Red Bull team when they achieved their maiden victories at the 1998 Belgian Grand Prix and 2009 Chinese Grand Prix respectively.

Constructors for the 2024 season

 * Correct as of the

Note: Until a works team of every constructor was licensed in the country where it was really based. In 1965 Japanese-licensed Honda moved their works team from Tokyo, Japan to Amsterdam, Netherlands, followed in by the American-licensed Anglo American Racers team which was based in Rye, East Sussex, United Kingdom. Since the early 2000s most teams have been based in the United Kingdom, and either licensed there or in another country, with the rest based in Italy (Maranello and Faenza) and Switzerland (Hinwil).

Key: Licensed in = Country in which the works team of respective constructor is licensed; Races Entered = Number of individual races entered; Races Started = Number of individual races started; Drivers = Number of drivers; Total Entries = Total number of race entries; Wins = Number of races won; Points = Number of World Constructors' Championship points scored; Poles = Number of pole positions; FL = Number of fastest laps; Podiums = Number of podium finishes; WCC = World Constructors' Championships won; WDC = World Drivers' Championships won.

Former constructors
Key: Licensed in = Country in which the works team of respective constructor was licensed; Races Entered = Number of individual races entered; Races Started = Number of individual races started; Drivers = Number of drivers; Total Entries = Total number of race entries; Wins = Number of races won; Points = Number of Constructors' Championship points scored; Poles = Number of pole positions; FL = Number of fastest laps; Podiums. = Number of podium finishes; WCC = Constructors' Championships won; WDC = Drivers' Championships won.

Indianapolis 500 only
The following are constructors whose only participation was in the Indianapolis 500 from 1950 to 1960 when the race was part of the Formula One World Drivers' Championship. All were based in the United States.

 * Adams
 * Bromme
 * Christensen
 * Emil Diedt
 * Frank DelRoy
 * Dunn
 * Elder
 * Quin Epperly
 * Wayne Ewing
 * Hall
 * Eddie Kuzma
 * Langley
 * Lesovsky
 * Carl Marchese
 * Meskowski
 * Lou Moore
 * Nichels
 * Olson
 * Pankratz
 * Pawl
 * Phillips
 * Rae
 * Schroeder
 * Sherman
 * Russ Snowberger
 * Stevens
 * Sutton
 * Trevis
 * Turner
 * A. J. Watson
 * Wetteroth

Privateer teams
From the inaugural 1950 British Grand Prix until the 1981 Spanish Grand Prix numerous privateer teams entered cars, built by another companies as their constructors, in World Championship events. Some of them, such as Tyrrell and Williams, later began to build their own chassis and thus became constructors as well as works teams. At the 1981 Spanish Grand Prix the Equipe Banco Occidental team became the last privateer team to have entered a car for a race alongside a works team when they entered a Williams car alongside the Williams works team. During the period of the – seasons, privateer teams won 20 World Championship races in total. Only once (the Matra International team in ) a privateer team helped a constructor (Matra) to win the World Constructors' Championship and a driver (Jackie Stewart) to win the World Drivers' Championship. The following are privateer teams which never built their own chassis, and thus were not constructors:
 * 🇬🇧 AE Moss
 * 🇬🇧 Bernard White Racing (–)
 * 🇮🇹 BMS Scuderia Italia (–)
 * 🇬🇧 British Formula One Racing Team
 * 🇬🇧 BS Fabrications (–)
 * 🇺🇸 Camoradi International (–)
 * 🇬🇧 DW Racing Enterprises (–)
 * 🇧🇪 Ecurie Belge (–)
 * 🇫🇷 Ecurie Bleue (, –)
 * 🇸🇪 Ecurie Bonnier (–, –)
 * 🇬🇧 Ecurie Ecosse (–)
 * 🇨🇭 Ecurie Espadon (–)
 * 🇫🇷 Ecurie Lutetia
 * 🇳🇱 Ecurie Maarsbergen (–)
 * 🇫🇷 Ecurie Rosier (–)
 * 🇨🇭 Enrico Platé (–)
 * 🇪🇸 Equipe Banco Occidental
 * 🇬🇧 Equipe Moss
 * 🇧🇷 Escuderia Bandeirantes (–)
 * 🇮🇹 FISA
 * 🇬🇧 FR Gerard Cars (–, –, –)
 * 🇬🇧 Goldie Hexagon Racing
 * 🇦🇺 John Willment Automobiles (–)
 * 🇦🇷 Juan Manuel Fangio
 * 🇬🇧 Matra International (–)
 * 🇺🇸 Mecom Racing Team
 * 🇺🇸 North American Racing Team (–, )
 * 🇿🇦 Otelle Nucci (–, )
 * 🇬🇧 Reg Parnell Racing (, –)
 * 🇬🇧 Rob Walker Racing Team (–, –)
 * 🇮🇹 Scuderia Achille Varzi
 * 🇮🇹 Scuderia Ambrosiana (–, )
 * 🇮🇹 Scuderia Centro Sud (–, –)
 * 🇨🇭 Scuderia Filipinetti (–)
 * 🇬🇧 Scuderia Franera
 * 🇮🇹 Scuderia Sant'Ambroeus
 * 🇦🇷 Scuderia Sud Americana
 * 🇨🇭 Silvio Moser Racing Team (, –)
 * 🇬🇧 T.A.S.O. Mathieson
 * Team Gunston (–,, –, –)

Privateer teams by number of wins
* All constructor's wins

** First win for the constructor

*** Team's only championship race

title: List of Formula One constructors


text:
List of Formula One circuits

Formula One, abbreviated to F1, is currently the highest class of open-wheeled auto racing defined by the Fédération Internationale de l'Automobile (FIA), motorsport's world governing body. The "formula" in the name refers to a set of rules to which all participants and vehicles must conform. The Formula One World Championship season consists of a series of races, known as Grands Prix, usually held on purpose-built circuits, and in a few cases on closed city streets. The results of each race are combined to determine two annual championships, one for drivers (World Drivers' Championship), and one for constructors (World Constructors' Championship).

This list is for the circuits that hosted World Championship races from 1950 until now. The terms "Formula One race" and "World Championship race" were not always synonymous throughout history – see for a detailed explanation.

History
The first World Championship Grand Prix was held in at Silverstone; since then  77 circuits in total have hosted a Grand Prix. A lot of classic (older) circuits have hosted Grands Prix using different configurations throughout their history: Nürburgring, Spa-Francorchamps, Monza, etc. Taking Nürburgring as an example, the first World Championship race there used the 22.835 km configuration, but concerns over safety meant that more recent Grands Prix have used a shorter, safer circuit. Formula One circuits were predominantly in Europe during the early years of the championship; as the sport has expanded, so has the location of its circuits. The expansion into Asia and America has been a recent occurrence. Of the 20 circuits that hosted a Grand Prix in, nearly half were not on the calendar before. The Autodromo Nazionale Monza has hosted the most World Championship races; the only season it did not host a race was in, when the Italian Grand Prix was held at the Autodromo Enzo e Dino Ferrari. The Las Vegas Strip Circuit became the 77th circuit to host a Grand Prix, when it held the Las Vegas Grand Prix in 2023; this is the latest addition to this list. The longest circuit to have hosted a Grand Prix is the Pescara Circuit, which hosted the 1957 Pescara Grand Prix: the 25.800 km long circuit in Pescara, Italy, held the annual Coppa Acerbo race, and in it was the only time that this race was included as part of the World Championship, a race which Stirling Moss won.

As some circuits have hosted Grands Prix using different configurations, the most recent configuration used is listed in the table below.

Circuits

 * The "Circuit" column uses the name contemporary to the last time the circuit was used in Formula One.
 * The "Map" column shows a diagram of the latest configuration on current tracks and the last configuration used on past tracks.
 * The "Type" column refers to the type of circuit: "street" is a circuit held on closed city streets, "road" refers to a mixture of public roads and a permanent track, and "race" is a permanent facility.
 * The "Last length used" shows the track length for the configuration that was used last time the Formula One race was held on a given track.
 * The "Direction" column shows the direction for the configuration that was used last time the Formula One race was held on a given track.
 * The turns column shows the number of turns for the configuration that was used the last time a Formula One race was held on a given track.

title: List of Formula One circuits


text:
Port Hope, Ontario

Port Hope is a municipality in Southern Ontario, Canada, approximately 109 km east of Toronto and about 159 km west of Kingston. It is located at the mouth of the Ganaraska River on the north shore of Lake Ontario, in the west end of Northumberland County. The private Trinity College School opened here in 1868.

History
The Cayuga people, one of the Six Nations of the Iroquois Confederacy, migrated as United Empire Loyalists to the Port Hope area from New York state in 1779. They had been forced from their ancestral homeland in the Finger Lakes region, south of the Great Lakes, after having fought for King George III as Loyalists during the American Revolution. Great Britain had ceded their lands, along with all other territory in the Thirteen Colonies east of the Mississippi River, after the United States won independence.

In 1793, other Loyalists from the northern colonies became the first permanent settlers of European heritage in Port Hope, as the Crown granted them land as compensation for being forced to leave the colonies (much of their property was confiscated by rebel governments) and as payment for military service. The new colonists called the settlement Smith's Creek after a former fur trader. They developed mills and a town plot by the turn of the century.

After the War of 1812, the Crown tried to recruit more British settlers, and townspeople wanted a new name. After a brief fling with the name Toronto, the village was renamed in 1817 as Port Hope, after the Township of Hope of which it was a part. That was the namesake of Colonel Henry Hope, lieutenant governor of the Province of Quebec. The post office dates from 1820. In 1834 Port Hope was incorporated as a town.

Relatively slow growth from 1881 to 1951 resulted in much of the town's 19th century architecture surviving. In the early 21st century, Port Hope's downtown is celebrated as the best-preserved 19th-century streetscape in the province of Ontario. The town's local chapter of the Architectural Conservancy of Ontario and the Heritage Port Hope Advisory Committee are very active and advise on the restoration and preservation of architecturally or historically significant buildings.

With over 270 heritage-designated buildings throughout the municipality, Port Hope has a higher per capita rate of preservation than any other town or city in Canada. Downtown businesses are regulated by the municipality to maintain the town's unique character. This special character makes Port Hope a destination for heritage tourism and people interested in architecture.

In 1978, eight members of the Satan's Choice Motorcycle Club were charged with a murder that occurred at Port Hope's Queen's Hotel. The trial and conviction of some members—the Port Hope 8 case—has been described as a miscarriage of justice.

On January 1, 2001, the original town amalgamated with Hope Township to form the Municipality of Port Hope and Hope, which was renamed to its current name in November of that same year. Prior to amalgamation, the town's census population was listed as 11,718 while the township's was 3,877.

The 2017 horror movie It, its 2019 sequel It Chapter Two , and the upcoming television series were all filmed in Port Hope, which portrayed the fictional town of Derry, Maine.

Radiation and cleanup
Port Hope is known for having the largest volume of historic low-level radioactive wastes in Canada. These wastes were initially created by Eldorado Mining and Refining Limited and its private sector predecessors, resulting from the refining of radium from pitchblende. Radium was used in radioluminescent paint (such as aircraft dials), and in early treatments for cancer.

During World War II, the Eldorado plant produced exponentially more uranium oxides, which the United States used in the Manhattan Project that created the first nuclear weapons. This plant, now under the ownership of Cameco, continues to produce uranium fuel for nuclear power plants.

In 2002, a large amount of contaminated soil was removed from beachfront areas. More recently, a testing program began of over 5,000 properties, with a plan to remove and store contaminated soil that had been used as landfill. Over a billion dollars is expected to be spent on the soil remediation project, the largest such cleanup in Canadian history.

Communities
Besides the town proper of Port Hope, the municipality of Port Hope comprises a number of villages and hamlets, including Campbellcroft, Canton, Dale, Davidson's Corners (partially), Decker Hollow (ghost town), Elizabethville, Garden Hill, Knoxville, Morrish, Osaca, Perrytown, Port Britain, Rossmount (partially), Tinkerville, Thomstown, Welcome, Wesleyville, and Zion.

Climate
Port Hope has a humid continental climate (Dfb) with warm summers and cold winters.

Demographics
In the 2021 Census of Population conducted by Statistics Canada, Port Hope had a population of $17,294$ living in $7,318$ of its $7,607$ total private dwellings, a change of NaN% from its 2016 population of $16,753$. With a land area of 278.8 km2, it had a population density of in 2021.

Mother tongue spoken:
 * English as first language: 94.7%
 * French as first language: 1.0%
 * English and French as first language: 0%
 * Other as first language: 4.3%

Economy
Downtown Port Hope offers shopping and a historic main street. Port Hope is served by a Via Rail station. It has a medical center, and a community health centre. It has had a daily newspaper since 1878, the Port Hope Evening Guide. Until 2007, this was part of the Osprey Media chain and subsequently a part of the Sun Media organization. In 2009 the newspaper was amalgamated with the Cobourg Daily Star and renamed as Northumberland Today.com. In November 2017 the newspaper was included in the large-scale closing of many local community newspapers throughout the province of Ontario.

Port Hope's Economic Development Strategic Plan aims to increase job growth at least as fast as population growth. The town has a variety of industries.

Arts and culture

The Ganaraska River (affectionately known as "The Ganny"), is well known to area anglers for annual salmon and trout runs. It has caused many historic floods, the most recent having occurred on March 21–22, 1980. Every April since until 2020, Port Hope has commemorated the flood with "Float Your Fanny Down the Ganny" ten kilometre boat race. "Participants range from serious paddlers navigating the cold, fast-moving water in kayaks and canoes, to the very entertaining 'crazy craft' paddlers, floating any combination of materials down the river in an attempt to reach the finish line." Due to the COVID-19 pandemic, the event was cancelled in 2020 and 2021, the first time in its history for such action.

Attractions
The Capitol Theatre is Canada's last functioning atmospheric theatre. The theatre's main auditorium is styled after an outdoor medieval courtyard and rolling clouds are projected onto the ceiling. The town spent in excess of three million dollars renovating and upgrading the theatre in 2004–2005. It is also used for live events by Port Hope Festival Theatre.

The Municipality of Port Hope is home to many heritage and cultural attractions, and events, including:

 * Float Your Fanny Down the Ganny—a water race commemorating the 1980 flood of the Ganaraska River
 * Ganaraska Forest Centre
 * Canadian Firefighters Museum (now demolished)
 * Port Hope Yacht Club
 * Port Hope Festival Theatre at the Capitol Theatre
 * La Jeunesse Youth Orchestra (3 concerts per year)
 * Port Hope and District Agricultural Fall Fair
 * The All Canadian Jazz Festival
 * Port Hope Farmers' Market (May to October)
 * Port Hope Christmas and Santa Claus Parade (includes Festival of Trees, Candlelight Walk to Memorial Park, and Carol Singing)
 * Port Hope Drive-In (Built in 1952, it is among the oldest Canadian drive-ins still operating)
 * Architectural Conservancy of Ontario Annual House Tour, Garden Tour, and Antiques and Artifacts Auction
 * Port Hope and District Historical Society Dorothy's House Museum
 * Port Hope Archives
 * Friends of Wesleyville Village
 * Beaches: West Beach (parking at the end of Marsh Street) East Beach (parking at the bottom of King Street at Madison Street)
 * Port Hope Waterfront Trail
 * Port Hope Golf and Country Club

Transportation
Highway 401 runs through the north end of Port Hope, with exits at County Road 2/Toronto Road (461) and Highway 28/Ontario Street (464).

Port Hope Transit provides local bus service, and VIA Rail provides passenger service from the Port Hope railway station along the Toronto-Montreal corridor. The station was built in 1856 for the Grand Trunk Railway and later CN Rail. It was restored in 1985.

Pleasure boats dock at the foot of John Street at Hayward Street and share the facilities with Cameco, which has berths for freighters servicing their manufacturing facilities at the mouth of the Ganaraska River.

Education
Public education in Port Hope is under the management of the Kawartha Pine Ridge District School Board, and Catholic education is by the Peterborough Victoria Northumberland and Clarington Catholic District School Board.

Elementary schools

 * St. Anthony's Elementary School, Catholic JK–8
 * Ganaraska Trail Public School, Public JK–6
 * North Hope Central School, Public JK–6
 * Beatrice Strong Public School, Public JK–6

High schools

 * Port Hope High School c. 1871, Public Gr 9-12 - opened in 1853 as Port Hope Grammar School
 * Dr M. S. Hawkins Senior Public School, Public Gr 7–8 (same building as Port Hope High School)
 * Port Hope High School Student to Work Transition Program (SWOT Campus), Public Grade 9–12
 * Trinity College School, Private Gr 5–12
 * Discovery Academy, International campus (not active)

Notable people

 * David Blackwood, artist.
 * Lew Cirne, pioneer of Application Performance Management, founder of Wily Technology and New Relic.
 * William Henry Draper, lawyer, judge, and politician.
 * Sue Gardner, executive director of the Wikimedia Foundation.
 * J.J. Hagerman, Colorado railroad and mining magnate who went on to become one of founders of New Mexico.
 * William Leonard Hunt ("The Great Farini"), entertainer.
 * Watson Kirkconnell (1895-1977), linguist, public intellectual, father of multiculturalism in Canada, and a highly important figure in both Canadian poetry and the culture of Canada.
 * Archibald Cameron Macdonell, commander of the 1st Canadian Division during the First World War.
 * Charles Vincent Massey, first Canadian-born Governor General of Canada.
 * Claire Mowat, writer.
 * Farley Mowat, conservationist and writer.
 * Alice Munro, author and Nobel Prize winner, lived in Port Hope.
 * Dennis O'Brien, NHL hockey player.
 * Shane O'Brien, NHL hockey player.
 * Cal Quantrill, Major League Baseball player.
 * Paul Quantrill, Major League Baseball player.
 * Jim Roberts, NHL hockey player.
 * Wade Rowland, writer and journalist.
 * Joseph M. Scriven, author of the hymn "What a Friend We Have in Jesus".
 * William Sims, U.S. Naval Admiral, awarded 1921 Pulitzer Prize for History.
 * Ron Smith, NHL hockey player.
 * Ambrose Thomas Stanton, Chief Medical Officer for the British colonies. Born in Kendal and attended Port Hope High School.
 * Paul Terbenche, NHL hockey player.
 * Arthur Trefusis Heneage Williams, politician.
 * Major-General Arthur Victor Seymour Williams.

title: Port Hope, Ontario


text:
Inverted question and exclamation marks

The inverted question mark, ¿, and inverted exclamation mark, ¡, are punctuation marks used to begin interrogative and exclamatory sentences or clauses in Spanish and some languages which have cultural ties with Spain, such as Asturian and Waray languages. The initial marks are mirrored at the end of the sentence or clause by the ordinary question mark, ?, or exclamation mark, !.

Inverted marks are supported by various standards, including ISO-8859-1, Unicode, and HTML. They can be entered directly on keyboards designed for Spanish-speaking countries.

Usage

The inverted question mark ¿ is written before the first letter of an interrogative sentence or clause to indicate that a question follows. It is a rotated form of the standard symbol "?" recognized by speakers of other languages written with the Latin script. A normal question mark is written at the end of the sentence or clause.

Inverted punctuation is especially critical in Spanish (although Portuguese and Italian are also prone to the following syntax structure) since the syntax of the language means that both statements and questions or exclamations could have the same wording. "Do you like summer?" and "You like summer." are translated respectively as "¿Te gusta el verano?" and "Te gusta el verano." (There is not always a difference between the wording of a yes–no question and the corresponding statement in Spanish.)

In sentences that are both declarative and interrogative, the clause that asks a question is isolated with the starting-symbol inverted question mark, for example: "Si no puedes ir con ellos, ¿quieres ir con nosotros?" ("If you cannot go with them, would you like to go with us?"), not "¿Si no puedes ir con ellos, quieres ir con nosotros?" This helps to recognize questions and exclamations in long sentences.

Unlike the ending marks, which are printed along the baseline of the text, the inverted marks (¿ and ¡) descend below the line.

History
Inverted marks, simple in the era of hand typesetting, were originally recommended by the Real Academia Española (Royal Spanish Academy), in the second edition of the Ortografía de la lengua castellana (Orthography of the Castilian language) in 1754 recommending it as the symbol indicating the beginning of a question in written Spanish—e.g. "¿Cuántos años tienes?" ("How old are you?"; lit. 'How many years do you have?'). The Real Academia also ordered the same inverted-symbol system for statements of exclamation, using the symbols "¡" and "!".

These new rules were slowly adopted; there are 19th-century books in which the writer uses neither "¡" nor "¿".

Outside of the Spanish-speaking world, John Wilkins proposed using the inverted exclamation mark "¡" as a symbol at the end of a sentence to denote irony in 1668. He was one of many, including Desiderius Erasmus, who felt there was a need for such a punctuation mark, but Wilkins' proposal, as was true of the other attempts, failed to take hold.

Adoption
Some writers omit the inverted question mark in the case of a short unambiguous question such as: "Quién viene?" ("Who comes?"). This is the criterion in Galician and formerly in Catalan. Certain Catalan-language authorities, such as Joan Solà i Cortassa, insist that both the opening and closing question marks be used for clarity. The current Institute for Catalan Studies prescription is never to use the inverted marks for Catalan.

Some Spanish-language writers, among them Nobel laureate Pablo Neruda (1904–1973), refuse to use the inverted question mark.

It is common on social media to omit the inverted question mark since it saves typing time. Some also use the ending symbol for both beginning and ending, like this: "?Por qué dices eso?". Others may even use both the opening and closing question marks, but at the end of the sentence, giving "Por qué dices eso¿?" or "Por qué dices eso?¿".

Mixtures
It is acceptable in Spanish to begin a sentence with an opening inverted exclamation mark ("¡") and end it with a question mark ("?"), or vice versa, for statements that are questions but also have a clear sense of exclamation or surprise such as: ¡Y tú quién te crees? ("And who do you think you are?!"). Normally, four signs are used, always with one type in the outer side and the other in the inner side (nested) (¿¡Y tú quién te crees!?, ¡¿Y tú quién te crees?! )

Unicode 5.1 also includes, which is an inverted version of the interrobang, a nonstandard punctuation mark used to denote both excitement and a question in one glyph. It is also known as a "gnaborretni" (interrobang spelled backwards).

Encodings
¡ and ¿ are in the "Latin-1 Supplement" Unicode block,  which is inherited from ISO-8859-1:

Typing the character

¿ and ¡ are available in all keyboard layouts designed for Spanish-speaking countries. Smart phones typically offer these if you hold down or  in the on-screen keyboard. Auto-correct will often turn a normal mark typed at the start of a sentence to the inverted one.

On systems with an AltGr key (actual or emulated via right Alt key) and Extended ( or 'International') keyboard mapping set, the symbols can be accessed directly, though the sequence varies by OS and locality: for example on Windows and US-International, use and ; on ChromeOS with UK-Extended, use  and.

title: Inverted question and exclamation marks


text:
Heavenly Mountain Resort

Heavenly Mountain Resort is a ski resort located on the California–Nevada border in southeastern Lake Tahoe in the Sierra Nevada Mountain Range. It opened for business on December 15, 1955 and has 97 runs and 30 lifts that are spread between California and Nevada and four base facilities. The resort has 4800 acre within its permit area, with approximately 33% currently developed for skiing, boasting the highest elevation of the Lake Tahoe area resorts with a peak elevation of 10067 ft, and a peak lift-service elevation of 10040 ft.

Since 2002, Heavenly has been owned by Vail Resorts, which also operates Northstar California and Kirkwood Mountain Resort at Lake Tahoe and multiple other ski resorts in Colorado, British Columbia, Vermont, New Hampshire, Washington, Utah, Minnesota, Michigan, and Wisconsin (Vail, Breckenridge, Okemo, Mount Sunapee, Crested Butte, Stevens Pass, Keystone, Beaver Creek, Park City Resort (Canyons), Afton Alps, Mt. Brighton, and Wilmot Mountain, Whistler Blackcomb).

With an average of 360 in of snow annually, and one of America's largest snowmaking systems, their ski season usually runs from mid November to mid April.

Heavenly is notable as the resort where Congressman and singer Sonny Bono died after hitting a tree on January 5, 1998.

Ski Lifts
The lifts built by Doppelmayr are the Big Easy, Canyon Express, Comet Express, Dipper Express, Galaxy, Gondola, Gunbarrel Express, Mott Canyon, NorthBowl Express, Olympic Express, Powderbowl Express, Sky Express, Stagecoach Express, and Tamarack Express. The lifts built by Yan are Boulder, First Ride, Groove, and Patsy's. SLI built World Cup. The oldest lift is World Cup, having been built in 1969. Heavenly also has seven surface lifts. Mott Canyon also closes at 3:00 during the day thus its trails are closed too. Mostly high-speed detachable quads serve the resort. The longest lift is the Gondola by a significant amount. Two six-packs serve skiers; the Tamarack and Powderbowl Express chairs. Most of the lifts close at 4:00 PM with the exception of the Mott Canyon lift, which closes at 3:00, the Yan and SLI double lifts have a very similar appearance in chairs to lines, to terminals, only the bull wheels are different.

title: Heavenly Mountain Resort


text:
Phatthalung province

Phatthalung (พัทลุง, ) is one of the southern provinces (changwat) of Thailand. Neighboring provinces are (from north clockwise) Nakhon Si Thammarat, Songkhla, Satun, and Trang. Phatthalung is essentially a landlocked province, one of the only two in southern Thailand, the other being Yala.

Geography
The province is on the Malay Peninsula. It borders to the east the large and shallow Songkhla Lake, and to the west the Nakhon Si Thammarat mountain range. Khao Pu–Khao Ya National Park is at the border with Trang. Forests cover 628 km², or 16.3 percent of the province's area.

History
Phatthalung was formerly known as Mardelong (Jawi: مردلوڠ) in Malay, especially during the time when the region came under Malay-Muslim influence.

Phatthalung became one of twelve royal cities during the reign of King Ramathibodi I of the Ayutthaya Kingdom in the 14th century. At the end of the 18th century, King Rama I submitted the city to the Ministry of Defense, which was responsible for all the southern provinces. In 1896, during the administrative reforms of King Chulalongkorn, Phatthalung became part of the Monthon Nakhon Si Thammarat. In 1924, King Rama VI ordered to move the city of Phatthalung to the present-day Khuha Sawan Subdistrict.

Symbols
The provincial seal shows the 177 meter high Phu Khao Ok Thalu mountain, the symbol of the province. The provincial tree and flower is the sweet shorea (Shorea roxburghii).

Air
Phatthalung does not have an airport. The nearest airport is Trang Airport, which is 66 km from the center of Phatthalung.

Rail
The main station in the province is the Phatthalung Railway Station.

Provincial government
Phatthalung is divided into 11 districts (amphoes). The districts are further divided into 65 subdistricts (tambons) and 626 villages (mubans).

Local government
As of 26 November 2019 there are: one Phatthalung Provincial Administration Organisation (ongkan borihan suan changwat) and 49 municipal (thesaban) areas in the province. Phatthalung has town (thesaban mueang) status. Further 48 subdistrict municipalities (thesaban tambon). The non-municipal areas are administered by 24 Subdistrict Administrative Organisations - SAO (ongkan borihan suan tambon).

Demographics
The majority of the province's populace are Thai Buddhists. Muslims account for 11.1 percent of the population. Many of Phatthalung's Muslims have some ethnic Malay ancestry, but over the centuries they had intermarried with the Thais and adopted Thai cultural norms.

Sights
Phraya Thukkharat (Chuai) Monument (อนุสาวรีย์พระยาทุกขราษฎร์ (ช่วย)) - Phraya Thukkharat was a former monk known as Phra Maha Chuai. During the Nine Armies War in the reign of King Rama I, then Phra Maha Chuai had assisted Phraya Phatthalung, who had led a force of villagers to defeat an invading Burmese army. Later, when he had left the monkhood, he was awarded the royal title Phraya Thukkharat and was an assistant to the city's ruler.

Culture
Manora or Nora (มโนราห์หรือโนรา) A local performing art in the south. It was an influence from the south of India, together with Lakhon Chatri (theatrical show by males). However, some dancing patterns were changed to match folk cultures in each province. There are 12 major dancing patterns. The patterns are done to lyrics sung by the dancers, either impromptu or composed in advance.

Talung (ตะลุง) *shadow plays) is a popular folk performance of the south. Talung puppets are made of dried cattle hide, cut beautifully into the characters of each shadow play. The puppets are usually painted black and each of them is held firmly between split bamboo slats called "mai tap". A puppet's mouth and hands will move in accordance with the narration. A Talung ensemble comprises the puppet masters (who are also the vocalists) and a band, totalling no more than eight persons. The musical instruments include pipes, drums, phon (a special kind of drum), and a gong.

Human achievement index 2022
Since 2003, United Nations Development Programme (UNDP) in Thailand has tracked progress on human development at sub-national level using the Human achievement index (HAI), a composite index covering all the eight key areas of human development. National Economic and Social Development Board (NESDB) has taken over this task since 2017.

title: Phatthalung province


text:
À

À, à (a-grave) is a letter of the Catalan, Emilian-Romagnol, French, Italian, Maltese,  Occitan, Portuguese, Sardinian, Scottish Gaelic, Vietnamese, and Welsh languages consisting of the letter A of the ISO basic Latin alphabet and a grave accent. À is also used in Pinyin transliteration. In most languages, it represents the vowel a. This letter is also a letter in Taos to indicate a mid tone.

In accounting or invoices, à abbreviates "at a rate of": "5 apples à $1" (one dollar each). That usage is based upon the French preposition à and has evolved into the at sign (@). Sometimes, it is part of a surname: Thomas à Kempis, Mary Anne à Beckett.

Emilian-Romagnol
À is used in Emilian to represent short stressed [a], e.g. Bolognese dialect sacàtt [saˈkatː] "sack".

French
The grave accent is used in the French language to differentiate homophones, e.g. the.F.SG and  there.

Portuguese
À is used in Portuguese to represent a contraction of the feminine singular definite article A with the preposition A:
 * Ele foi à praia.
 * He went to the beach.

Scottish Gaelic
In early orthographic descriptions of Scottish Gaelic from the 18th and 19th centuries, à is the only way to represent a long [a]; later forms of Scottish Gaelic also used the acute accent [á] to indicate a longer [a] sound.

Character mappings
Microsoft Windows users can type an "à" by pressing Alt33 or Alt224 on the numeric pad of the keyboard. "À" can be typed by pressing Alt192. On a Mac, you hold Option, and then let go and type a. Similarly on a GNU/Linux system, where the Compose key can be configured.

title: À


text:
Á

Á, á (a-acute) is a letter of the Chinese (Pinyin), Blackfoot, Czech, Dutch, Faroese, Filipino, Galician, Hungarian, Icelandic, Irish, Karakalpak, Lakota, Navajo, Occitan, Portuguese, Sámi, Slovak, Spanish, Vietnamese, Welsh and Western Apache languages as a variant of the letter a. It is sometimes confused with à; e.g. "5 pommes á €1", which is supposed to be written as "5 pommes à €1" (meaning "5 apples at 1 euro each" in French).

Chinese
In Chinese pinyin á is the yángpíng tone (陽平/阳平 "high-rising tone") of "a".

Czech
Á is the 2nd letter of the Czech language and represents the sound.

Dutch
In Dutch, the Á is used to put emphasis on an "a", either in a long "a" form like in háár ("hair"), or in a short form like in kán (the verb "can").

Faroese
Á is the 2nd letter of the Faroese alphabet and represents or.

Filipino
In Filipino, á is an accented letter and has no direct equivalent other than a. An example is the first "a" as in "baka" ("cow") and "tatawid" ("to cross").

Hungarian
In Hungarian, Á is the second letter of Hungarian representing.

Icelandic
Á is the second letter of the Icelandic alphabet and represents (as in "ow").

Irish
In Irish, á is called a fada ("long a"), pronounced and appears in words such as slán ("goodbye"). It is the only diacritic used in Modern Irish, since the decline of the dot above many letters in the Irish language. Fada is only used on vowel letters i.e. á, é, í, ó, ú. It symbolises a lengthening of the vowel.

Karakalpak
Á is defined as the second letter and represents.

Kazakh
In the 2018 amends of Kazakh alphabet list, Á is defined as the second letter and represents. It has been replaced by Ä|Ä ä in the 2019 amends, and matches Cyrillic alphabet Ә, 2017 version Aʼ and Arabic ٵ.

Portuguese
In Portuguese, á is used to mark a stressed in words whose stressed syllable is in an abnormal location within the word, as in lá (there) and rápido (rapid, fast). If the location of the stressed syllable is predictable, the acute accent is not used. Á contrasts with â, pronounced.

Scottish Gaelic
Á was once used in Scottish Gaelic, but has now been largely superseded by à. It can still be seen in certain writings, but it is no longer used in standard orthography.

Slovak
Á is the 2nd letter in the Slovak alphabet and it represents.

Spanish
In Spanish, á is an accented letter. There is no alphabetical or phonological difference between a and á; both sound like /a/, both are considered the same letter, and both have the same value in the Spanish alphabetical order. The accent indicates the stressed syllable in words with irregular stress patterns. It can also be used to "break up" a diphthong or to avoid what would otherwise be homonyms, although this does not happen with á, because a is a strong vowel and usually does not become a semivowel in a diphthong. See Diacritic and Acute accent for more details.

Vietnamese
In the Vietnamese alphabet, á is the sắc tone (high-rising tone) of a. "Á", as a shorthand for "châu Á", is also the Vietnamese word for "Asia".

Welsh
In Welsh, word stress usually falls on the penultimate syllable, but one way of indicating stress on a final (short) vowel is through the use of the acute accent. The acute accent on a is often found in verbal nouns and borrowed words, for example, casáu "to hate", caniatáu  "to allow", carafán  "caravan".

title: Á


text:
Â

Â, â (a-circumflex) is a letter of the Inari Sami, Skolt Sami, Romanian, Vietnamese and Mizo alphabets. This letter also appears in French, Friulian, Frisian, Portuguese, Turkish, Walloon, and Welsh languages as a variant of the letter "a". It is included in some romanization systems for Khmer, Persian, Balinese, Sasak, Russian, and Ukrainian.

Berber languages
"â" can be used in Berber Latin alphabet to represent.

Emilian-Romagnol
Â is used to represent [aː] in Emilian dialects, as in Bolognese câna [kaːna] "cane".

Faroese
Johan Henrik Schrøter, who translated the Gospel of Matthew into Faroese in 1823, used â to denote a non-syllabic a, as in the following example:

Â is not used in modern Faroese, however.

French
$⟨â⟩$, in the French language, is used as the letter $⟨a⟩$ with a circumflex accent. It is a remnant of Old French, where the vowel was followed, with some exceptions, by the consonant $⟨s⟩$. For example, the modern form bâton (stick) comes from the Old French baston. Phonetically, $⟨â⟩$ is traditionally pronounced as, but is nowadays rarely distinguished from in many dialects such as in Parisian French. However, the traditional $⟨â⟩$ is still pronounced this way in Québecois French or Canadian French, which is known to resemble the phonetics of the Old French accent, and is widely spoken by French Canadians, the majority of whom live in the province of Québec.

In Maghreb French, $⟨â⟩$ is used to transcribe the Arabic consonant $⟨ع⟩$, whose pronunciation is close to a non-syllabic.

Friulian
Â is used to represent the sound.

Inari Sami
Â is used to represent the sound.

Italian
Â is occasionally used to represent the sound in words like amâr, a poetic contraction of amarono (they loved).

Khmer
Â is used in the UNGEGN romanization system to represent the sound in Khmer.

Persian
Â is used in the romanization of Persian to represent the sound in words such as Fârs.

Portuguese
In Portuguese, â is used to mark a stressed in words whose stressed syllable is nasal and in an unpredictable location within the word, as in "lâmina" (blade) and "âmbar" (amber). Where the location of the stressed syllable is predictable, such as in "ando" (I walk), the circumflex accent is not used. Â contrasts with á, pronounced.

Romanian
Â is the 3rd letter of the Romanian alphabet and represents, which is also represented in Romanian as letter î. The difference between the two is that â is used in the middle of the word, as in "România", while î is used at the beginning and at the ends: "înțelegere" (understanding), "a urî" (to hate). A compound word starting with the letter î will retain it, even if it goes in the middle of the word: compare "înțelegere" (understanding) with "neînțelegere" (misunderstanding). However, if a suffix is added, the î changes into â, as in the example: "a urî" (to hate), "urât" (hated). Another grapheme  in Romanian with diacritic is <ă>.

Russian
Â is used in the ISO 9:1995 system of Russian transliteration as the letter Я.

Serbo-Croatian
In all standard varieties of Serbo-Croatian, "â" is not a letter but simply an "a" with the circumflex that denotes vowel length. It is used only occasionally and then disambiguates homographs, which differ only by syllable length. That is most common in the plural genitive case and so it is also called "genitive sign": "Ja sam sâm" (I am alone).

Sicilian
Â is used to represent [aː] in Sicilian, as in the preposition pâ [paː] "for the".

Turkish
Â is used to indicate the consonant before "a" is palatalized, as in "kâr" (profit). It is also used to indicate in words for which the long vowel changes the meaning, as in "adet" (pieces) and "âdet" (tradition) / "hala" (aunt) and "hâlâ" (still).

Ukrainian
Â is used in the ISO 9:1995 system of Ukrainian transliteration to represent the letter Я.

Vietnamese
Â is the 3rd letter of the Vietnamese alphabet and represents. â is a higher vowel than plain a. In Vietnamese phonology, diacritics can be added to form five forms to represent five tones of â:
 * Ầ ầ
 * Ẩ ẩ
 * Ẫ ẫ
 * Ấ ấ
 * Ậ ậ

Welsh
In Welsh, â is used to represent long stressed a when, without the circumflex, the vowel would be pronounced as short, e.g., âr  "arable", as opposed to ar  "on"; or gwâr  "civilised, humane", rather than gwar  "nape of the neck". It is often found in final syllables where two adjacent a letters combine to produce a long stressed vowel. This commonly happens when a verb stem ending in stressed a combines with the nominalising suffix -ad, as in caniata- + -ad giving caniatâd "permission",  and also when a singular noun ending in a receives the plural suffix -au, as in drama + -au becoming dramâu  "dramas, plays". It is also useful in writing borrowed words with final stress, e.g. brigâd "brigade".

A circumflex is also used in the word â, which is both a preposition, meaning "with, by means of, as", and the third person non-past singular of the verbal noun mynd, "go". This distinguishes it in writing from the similarly pronounced a, meaning "and; whether; who, which, that".

Windows Alt Key codes
Source:

TeX and LaTeX
Â and â are obtained by the commands \^A and \^a.

In encoding mismatches
In a common example of mojibake, the capital Â is sometimes seen on webpages when the page has been encoded in UTF-8 and decoded using ISO 8859-1 or Windows-1252, two encodings which are commonly referred to as Western or Western European. In UTF-8, the copyright symbol (©) is encoded with the hexadecimal bytes. In the older Western encoding standards, however, the © symbol is simply. If a browser is given the bytes, intended to display © in UTF-8, but is led to parse the bytes according to one of the Western encodings, it will interpret the bytes   as two separate characters. corresponds to Â, as seen in the chart above, and  devolves to the © symbol, so the result seen by the person reading the page is  —that is, the correct © symbol but with an Â prepended. Characters with Unicode code points from  to   have UTF-8 encodings that are identical to their Western encodings but preceded by the byte , so that when any of these characters is encoded in UTF-8 and viewed in a Western encoding, an Â will appear before it.

title: Â


text:
Ã

A with tilde (majuscule: Ã, minuscule: ã) is a letter of the Latin alphabet formed by addition of the tilde diacritic over the letter A. It is used in Portuguese, Guaraní, Kashubian, Taa, Aromanian, and Vietnamese. In the past, it was also used in Greenlandic.

Usage
In Portuguese, it represents a nasal near-open central vowel ([ɐ̃]), though it varies from near-open to mid-central vowel according to dialect. It also appears as a part of the diphthongs ãe, pronounced as /ɐ̃j̃/, and ão, pronounced as /ɐ̃w̃/.

In Kashubian, the letter is generally pronounced as nasalized open front unrounded vowel ([ã]). In the dialect present in counties of Puck and Wejherowo, it is pronounced as nasalized open-mid front unrounded vowel ([ɛ̃]). Its the 3rd letter of the Kashubian alphabet.

In Guaraní and Taa, it is pronounced as nasalized open front unrounded vowel ([ã]).

In Aromanian, it is pronounced as mid-central vowel ([ə]) or close central unrounded vowel ([ɨ]).

In Vietnamese, it is pronounced as long open front unrounded vowel ([aː]) in a high breaking-rising tone.

It is also used as a phonetic symbol in the International Phonetic Alphabet, where its lower case (ã), represents the nasalized open front unrounded vowel.

Formerly, the letter was also used in the Greenlandic to represent long open front unrounded vowel ([aː]) next to a geminated consonant, but now it is replaced with Aa.

The letter is also used in Belter Creole, a constructed language made by Nick Farmer for The Expanse television sci-fi series. In the language, it represents the nasalized near-open front unrounded vowel ([æ̃]) sound, for example in the word shãsa, which means chance.

title: Ã


text: User talk:68.4.122.197

title: User talk:68.4.122.197


text:
È

È, è (e-grave) is a letter of the Latin alphabet. In English, è is formed with an addition of a grave accent onto the letter E and is sometimes used in the past tense or past participle forms of verbs in poetic texts to indicate that the final syllable should be pronounced separately. For example, blessèd would indicate the pronunciation  rather than. It also occurs in loanwords such as Italian caffè.

In other languages

 * In Emilian, è is used to represent [ɛː], e.g. lèt [lɛːt] "bed".
 * In French, it always represents a sound of letter e when this is at the end of a syllable.
 * È means "is" in modern Italian, e.g. il cane è piccolo meaning "the dog is small". It is derived from Latin ĕst and is accented to distinguish it from the conjunction e meaning "and". È is also used to mark a stressed at the end of a word only, as in caffè.
 * È (è) is used in Limburgish for the sound, like in the word 'Sjtèl'.
 * È in Norwegian (both Bokmål and Nynorsk) is used in some words to denote a longer vowel such as in karrière (career).
 * È (è) is also used in Macedonian Latin as an equivalent of the letter ye with grave (Ѐ, ѐ).
 * In Romagnol, it represents [ɛ], e.g. vècc' [vɛtʃː] "old men".
 * È (è) is used to mark the long vowel sounds and  in Scottish Gaelic.
 * È (è) is used in Vietnamese to represent the letter "E" with the dấu huyền tone. It can also combine with "Ê" to form "Ề".
 * È (è) is also used for an with a falling tone in pinyin, the Chinese language roman-alphabet transcription system.  The word 鄂, consisting only of this vowel, is an abbreviation for the Hubei province of the People's Republic of China.

Key strokes

 * Microsoft Windows users can type an "è" by pressing Alt232 or Alt38 on the numeric pad of the keyboard. "È" can be typed by pressing Alt200 or Alt12. Linux users may type an "è" with the Compose key (default is Right Alt) by pressing ComposeE` in sequence. "È" can be typed by pressing ComposeShift` in sequence.

title: È