The cross product on the right-hand side of this expression is a surface normal determined by the parametrization.
This formula "defines" the integral on the left (note the dot and the vector notation for the surface element).
Various useful results for surface integrals can be derived using differential geometry and vector calculus, such as the divergence theorem, and its generalization, Stokes' theorem.
The discussion above defined the surface integral by using a parametrization of the surface "S". A given surface might have several parametrizations. For example, when the locations of the North Pole and South Pole are moved on a sphere, the latitude and longitude change for all the points on the sphere. A natural question is then whether the definition of the surface integral depends on the chosen parametrization. For integrals of scalar fields, the answer to this question is simple: the value of the surface integral will be the same no matter what parametrization one uses.
Integrals of vector fields are more complicated, because the surface normal is involved. Mathematicians have proved that given two parametrizations of the same surface, whose surface normals point in the same direction, both parametrizations give the same value for the surface integral. If, however, the normals for these parametrizations point in opposite directions, the value of the surface integral obtained using one parametrization is the negative of the one obtained via the other parametrization. It follows that given a surface, we do not need to stick to any unique parametrization; but, when integrating vector fields, we do need to decide in advance which direction the normal will point to, and then choose any parametrization consistent with that direction.
Another issue is that sometimes surfaces do not have parametrizations which cover the whole surface; this is true for example for the surface of a cylinder (of finite height). The obvious solution is then to split that surface in several pieces, calculate the surface integral on each piece, and then add them all up. This is indeed how things work, but when integrating vector fields, one needs to again be careful in choosing the normal-pointing vector for each piece of the surface, so that when the pieces are put back together, the results are consistent. For the cylinder, this means that if we decide that for the side region the normal will point out of the body, then for the top and bottom circular parts, the normal must point out of the body too.
Last, there are surfaces which do not have a surface normal at each point with consistent results (for example, the Möbius strip). If such a surface is split into pieces, on each piece a parametrization and corresponding surface normal is chosen, and the pieces are put back together, the normal vectors coming from different pieces cannot be reconciled. This means that at some junction between two pieces will have normal vectors pointing in opposite directions. Such a surface is called non-orientable. Vector fields cannot be integrated on non-orientable surfaces.
Philae (spacecraft)
Philae (before known as RoLand) is an ESA lander spacecraft. It went with the Rosetta spacecraft. It landed on comet 67P/Churyumov–Gerasimenkoon on 12 November 2014 to analyze it.
The name Philae was chosen through a contest organized by ESA, won by fifteen year old Serena Olga Vismara from Milan. The name comes from the orbiter name, already defined as Rosetta, which is the name of the stele that allowed Jean-François Champollion to understand ancient Egypt hieroglyphics. In order to complete the translation, he needed to use also the monuments on the island of Philae, the same way Philae lander will help to complete the work of the Rosetta lander.
Americana
Americana is about the United States, for example, its culture, geography, history etc.
FC Pinzgau Saalfelden
FC Pinzgau Saalfelden is an Austrian football club from Saalfelden in the federal state of Salzburg. The club plays in the Regionalliga West which is the third level in Austrian football. The second team is playing in the 2.Klasse Südwest of the Salzburger Fußballverband.
The club was founded on 23 May 2007. The new club was formed out of "1. Saalfeldner SK" (founded 1947) and "ESV Saalfelden" (founded 1952). The club was the first of the later players for the Austrian national football team Franz Bacher and Wolfgang Feiersinger. Feiersinger was also Championsleague winner with Borussia Dortmund.
Narrandera, New South Wales
Narrandera ( ) is a town in southern New South Wales, Australia. It is near the junction of the Newell and Sturt Highways, on the banks of the Murrumbidgee River. In 2006 there were 3,961 people living in Narrandera.
Narrandera is a river town with a long history. Captain Charles Sturt, the famous explorer, was the first European to see the area that later became known as Narrandera. However Sturt, who passed through on 12 December 1829, was not the first explorer to see the Murrumbidgee River. The upper Murrumbidgee, the "Big Water", was first discovered in April 1821, by Charles Throsby. The name "Narrandera" probably comes from Wiradjuri word "nharrang", meaning "frill-necked lizard".
The Narrungderra were the local indigenous people. They were all but destroyed by settlement, disease brought by European settlers, and clashes with the settlers. The last major battle between the indigenous people and European settlers took place near Massacre Island, and is said to have left only one survivor.
Narrandera is first recorded as a pastoral station or "run" (Narrandera Run) in 1848. The property was owned by Mr Edward Flood covered about .
The township developed in the early 1860s. Gillinbah Post Office opened nearby on 1 March 1859 and was replaced by the Narrandera office in 1861. A Gillenbah office was open from 1881 to 1892 and from 1906 to 1941.
The Borough of Narrandera began on 17 March 1885. The centenary of Local Government in Narrandera was celebrated in 1985.
The town of Narrandera is at the centre of a productive farming region. Its attractive tree-lined streets contrast with the open plains that surround it.
The Narrandera Memorial Gardens include the unusual Hankinson Fountain. Manufactured by the Royal Doulton Company of England, the ceramic fountain is one of only two known in the world, the other in India. It was given to the people of Narrandera by Alderman and Mrs Hankinson in 1922 in honour of locals who served in World War I. The Hankinson Fountain was smashed by an act of vandalism in 1971 and was restored by Charles Pearce.
To the east of Narrandera farmers grow cereal crops, sheep and wool. To the west, is the Murrumbidgee Irrigation Area (MIA) fed by water from the Burrinjuck Dam. The MIA is a region where farmer grow rice and other cereals with irrigation, as well as citrus, wine grapes and potatoes.
The town also has number of historic attractions, such as a fig tree on the corner of King and Cadell Streets, which is thought to be 150 years old; and the Mon Repos, a residence built in a Queen Anne-style, which was built in the 1890s.
Narrandera's Irrigation Canal which carries water to the Murrumbidgee Irrigation Area flows through the town. It begins west at Berembed Weir where water is taken from the Murrumbidgee River. The canal follows the natural bed of Bundidgerry Creek and in places is very wide and has no levee banks. Lake Talbot was formed in 1924 when the bank of the Irrigation Canal gave way, flooding the river flat between the canal and Bundidgerry Hill. The shallow lake was allowed to remain and became an important recreational feature of the town. The eastern end of the lake is used by water skiiers. The western end is a wetland habitat for native fauna. A wetland has been made near Lizard Drive, only from the Murrumbidgee River. The Narrandera Wetland is a collection point for storm water run-off from the town.