The Viking Age is a period in the history of the Scandinavians, during which they expanded and built settlements throughout Europe and beyond after the main European Migration Period. As such the Viking Age applies not only to their homeland of Scandinavia, but to any place settled by Scandinavians during the period; the Scandinavians of the Viking Age are referred to as Vikings or Norsemen, although few of them were Vikings in the technical sense. It was preceded by the Germanic Iron Age, it is the period of history when Scandinavian Norsemen explored Europe by its seas and rivers for trade, raids and conquest. In this period, voyaging from their homelands in Denmark and Sweden the Norsemen settled in the present-day Faroe Islands, Norse Greenland, the Netherlands, Normandy, Scotland, Wales, the Isle of Man, Ukraine and Turkey, as well as initiating the process of consolidation that resulted in the formation of the present day Scandinavian countries. Viking travellers and colonists were seen at many points in history as brutal raiders.

Many historical documents suggest that their invasion of other countries was retaliation for the encroachment upon tribal lands by Christian missionaries, by the Saxon Wars prosecuted by Charlemagne and his kin to the south, or were motivated by overpopulation, trade inequities, the lack of viable farmland in their homeland. Information about the Viking Age is drawn from what was written about the Vikings by their enemies, primary sources of archaeology, supplemented with secondary sources such as the Icelandic Sagas. In England, the beginning of the Viking Age is dated to 8 June 793, when Vikings destroyed the abbey on Lindisfarne, a centre of learning on an island off the northeast coast of England in Northumberland. Monks were killed in the abbey, thrown into the sea to drown, or carried away as slaves along with the church treasures, giving rise to the traditional prayer—A furore Normannorum libera nos, Domine, "Free us from the fury of the Northmen, Lord."Three Viking ships had beached in Weymouth Bay four years earlier, but that incursion may have been a trading expedition that went wrong rather than a piratical raid.

Lindisfarne was different. The Viking devastation of Northumbria's Holy Island was reported by the Northumbrian scholar Alcuin of York, who wrote: "Never before in Britain has such a terror appeared". Vikings were portrayed as wholly bloodthirsty by their enemies. In medieval English chronicles, they are described as "wolves among sheep"; the first challenges to the many anti-Viking images in Britain emerged in the 17th century. Pioneering scholarly works on the Viking Age reached a small readership in Britain. Linguistics traced the Viking Age origins of rural proverbs. New dictionaries of the Old Norse language enabled more Victorians to read the Icelandic Sagas. In Scandinavia, the 17th-century Danish scholars Thomas Bartholin and Ole Worm and Swedish scholar Olaus Rudbeck were the first to use runic inscriptions and Icelandic Sagas as primary historical sources. During the Enlightenment and Nordic Renaissance, historians such as the Icelandic-Norwegian Thormodus Torfæus, Danish-Norwegian Ludvig Holberg, Swedish Olof von Dalin developed a more "rational" and "pragmatic" approach to historical scholarship.

By the latter half of the 18th century, while the Icelandic sagas were still used as important historical sources, the Viking Age had again come to be regarded as a barbaric and uncivilised period in the history of the Nordic countries. Scholars outside Scandinavia did not begin to extensively reassess the achievements of the Vikings until the 1890s, recognising their artistry, technological skills, seamanship; until the history of the Viking Age had been based on Icelandic Sagas, the history of the Danes written by Saxo Grammaticus, the Kievan Rus's Primary Chronicle, Cogad Gáedel re Gallaib. Today, most scholars take these texts as sources not to be understood and are relying more on concrete archaeological findings and other direct scientific disciplines and methods; the Vikings who invaded western and eastern Europe were pagans from the same area as present-day Denmark and Sweden. They settled in the Faroe Islands, Iceland, peripheral Scotland and Canada, their North Germanic language, Old Norse, became the mother-tongue of present-day Scandinavian languages.

By 801, a strong central authority appears to have been established in Jutland, the Danes were beginning to look beyond their own territory for land and plunder. In Norway, mountainous terrain and fjords formed strong natural boundaries. Communities remained independent of each other, unlike the situation in lowland Denmark. By 800, some 30 small kingdoms existed in Norway; the sea was the easiest way of communication between the outside world. In the eighth century, Scandinavians began to build ships of war and send them on raiding expeditions which started the Viking Age; the North Sea rovers were traders, colonisers and plunderers. Many theories are posited for the cause of the Viking invasions. At the time, England and Ireland were vulnerable to attack, being divided into many different warring kingdoms in a state of internal disarray, while the Franks were well defended. Overpopulation near the Scandes, was influential. Technological advance like the use of ir

Great Brehat is a small fishing village on the Great Northern Peninsula of Newfoundland, located 10 km north of St. Anthony; as fishing has declined, the village has become a tourist attraction. Great Brehat is located on a small bay, Great Brehat Bay, on the eastern side of the Great Northern Peninsula. Flat Point Lookout is located at the northern end of the bay. Brehat Point forms the southern boundary of the bay; the black shales in the area may yield precious metals. The original settled indigenous peoples were of the Dorset culture, the area was visited by the Vikings, as their settlement at L'Anse aux Meadows was only 46 km away. By the sixteenth century the area was occupied by the Beothuk; when the French settled the northern peninsula of Newfoundland they established several fishing stations including Great Brehat. The people named their village after an island off the Brittany coast; the exact date of the French settlement is not known. In 1713, with the Treaty of Utrecht, the French ceded the peninsula to Great Britain.

Until the decline of the fishery in the 20th century, the primary catch was cod for export. At the beginning of the 20th century, a co-operative store was opened in the village with the assistance of the medical missionary Sir Wilfred Grenfell; the first road linking the coastal communities of the area was completed in 1971, with Great Brehat being the northern terminus. Great Brehat boasts a population of ninety-five. List of cities and towns in Newfoundland and Labrador

In mathematics, the vector flow refers to a set of related concepts of the flow determined by a vector field. These appear in a number of different contexts, including differential topology, Riemannian geometry and Lie group theory; these related concepts are explored in a spectrum of articles: exponential map matrix exponential exponential function infinitesimal generator integral curve one-parameter subgroup flow geodesic flow Hamiltonian flow Ricci flow Anosov flow injectivity radius Relevant concepts: Let V be a smooth vector field on a smooth manifold M. There is a unique maximal flow D → M whose infinitesimal generator is V. Here D ⊆ R × M is the flow domain. For each p ∈ M the map Dp → M is the unique maximal integral curve of V starting at p. A global flow is one whose flow domain is all of R × M. Global flows define smooth actions of R on M. A vector field is complete; every smooth vector field on a compact manifold without boundary is complete. Relevant concepts: The exponential map exp: TpM → Mis defined as exp = γ where γ: I → M is the unique geodesic passing through p at 0 and whose tangent vector at 0 is X.

Here I is the maximal open interval of R. Let M be a pseudo-Riemannian manifold and let p be a point in M. For every V in TpM there exists a unique geodesic γ: I → M for which γ = p and γ ˙ = V. Let Dp be the subset of TpM for which 1 lies in I. Relevant concepts: Every left-invariant vector field on a Lie group is complete; the integral curve starting at the identity is a one-parameter subgroup of G. There are one-to-one correspondences ⇔ ⇔ g = TeG. Let G be a Lie group and g its Lie algebra; the exponential map is a map exp: g → G given by exp = γ where γ is the integral curve starting at the identity in G generated by X. The exponential map is smooth. For a fixed X, the map t ↦ exp is the one-parameter subgroup of G generated by X; the exponential map restricts to a diffeomorphism from some neighborhood of 0 in g to a neighborhood of e in G. The image of the exponential map always lies in the connected component of the identity in G