( where the first map is the gradient, the second is the curl, the third is the divergence ) serves as a nice quantification of the complicatedness of the underlying region U. These are the beginnings and main motivations of de Rham cohomology.

Paradoxically, it is possible to reduce the divergence of a laser beam by first expanding it with one convex lens, and then collimating it with a second convex lens whose focal point is coincident with that of the first lens.

Because tools are used extensively by both humans and wild chimpanzees, it is widely assumed that the first routine use of tools took place prior to the divergence between the two species.

Arising from the dominant discourse of the Jewish suffering during the years of Nazi domination, and building on the divergence of differential victimhoods brought to light by studies of the Roma and the mentally ill, who suffered massively under the eugenics programs of the Third Reich, the idea of a “ Gay Holocaust ” was first explored in the early 1970s.

This is the first instance where one may observe a direct divergence from the theological virtues and concerns that were previously exerted by Justice in the morality plays of the fifteenth century.

He declared that the first work worthy to be called opéra-comique was Philidor's 1759 Blaise le savetier, and he described the gradual divergence of Italian and French notions of comic opera, with verve, imagination and gaiety from Italian composers, and cleverness, common sense, good taste and wit from the French composers.

As the Celtic world established renewed contact with the Continent it became aware of the divergence ; the first clash over the matter came in Gaul in 602, when Columbanus resisted pressure from the local bishops to conform.

The divergence of opinion on this matter between Maurice and Van Oldenbarneveldt were the first signs that the two de facto leaders of the Dutch Republic were beginning to drift apart.

An early example, which also constituted the first demonstration of the exponential divergence of chaotic trajectories, was carried out by R. H. Miller in 1964.

The first may be more accurately called a " diverging universe " whereby two versions of the universe share a common history up to a point of divergence.

Kirche ), shows considerable divergence from the first edition ( 1850 ), and reveals an entire emancipation from Baur's method.

The divergence of " gruiforms " among " Metaves " and " Coronaves " is proposed to be the first divergence among Neoaves, far predating the Cretaceous – Paleogene extinction event c. 65 mya ( Houde 2009 ).

Since the first appearance of the term, it has also been used for other predictions of a similar nature, as in quantum electrodynamics and such cases as ultraviolet divergence.

They were the first vocal group to become country music stars ; a beginning of the divergence of country music from traditional folk music.

On integrating the first term to get the volume average and applying the divergence theorem to the second, this yields

Frank's divergence from contemporary photographic standards gave Frank difficulty at first in securing an American publisher.

Note that Green's first identity above is a special case of the more general identity derived from the divergence theorem by substituting:

Green ’ s second identity establishes a relationship between second and ( the divergence of ) first order derivatives of two scalar functions.

After liberals have gained power and realized their first reforms, one often sees a divergence within their ranks:

where is the positive divergence in the first component of W.

In the first four novels, set in the 22nd century, the point of divergence is the Franco – Prussian War of 1870, where Grimwood posits a reality where Napoléon III's France defeats Otto von Bismarck's Prussia, causing the German Empire never to form and the Second French Empire never to collapse.

First, Derbyshire introduces the idea of an infinite series and the ideas of convergence and divergence in these series in the first chapter, " Card Trick ".

But her prettiness was what he had noticed first, and all the other things had come afterward: cruelty, meanness, self-will.

There was an artificial lake just out of sight in the first stand of trees, fed by a half dozen springs that popped out of the ground above the hillside orchard.

The first part of the road was steep, but it leveled off after the second bend and curled gradually into the valley.

No one was behind it, but in the rear wall of the office I noticed, for the first time, a door which had been left partially open.

The herd was watered and then thrown onto a broad grass flat which was to be the first night's bedground.

Once again, Tom Horn was the first and most likely suspect, and he was brought in for questioning immediately.

For Matilda, it was the first she had known in many a night.

Even the knowledge that she was losing another boy, as a mother always does when a marriage is made, did not prevent her from having the first carefree, dreamless sleep that she had known since they dropped down the canyon and into Bear Valley, way, way back there when they were crossing those other mountains.

Stevens was grunting over the last empty pocket when Russ abruptly rose and lunged toward Carmer's hat, which had tumbled half-a-dozen feet away when he first fell.

The Indian's arm whipped sidewise -- there was a flash of amber and froth, the crash of the bottle shattering against the side of the first car.

It was her first smile.

At first, I thought he was out of his head, talking wildly like this.

Hell, I gave him the first decent job he ever had, six, seven -- how many years ago was it, Rob ''??

Miss Langford ( her first name was Evelyn ) was an attractive girl.

School began in August, the hottest part of the year, and for the first few days Miss Langford was very lenient with the children, letting them play a lot and the new ones sort of get acquainted with one another.

It was just as well that the ignorant Dandy enjoyed himself to the hilt that first evening, for the room was to become his prison cell.

`` Bastards '', he would say, `` all I did was put a beat to that Vivaldi stuff, and the first chair clobbered me ''!!

In 1961 the first important legislative victory of the Kennedy Administration came when the principle of national responsibility for local economic distress won out over a `` state's-responsibility '' proposal -- provision was made for payment for unemployment relief by nation-wide taxation rather than by a levy only on those states afflicted with manpower surplus.

The first systematic thinking about this Pandora's box within Pandora's boxes was done four years ago by Fred Ikle, a frail, meek-mannered Swiss-born sociologist.

The smell at first was more surprising than unpleasant.

His collaboration with Washington, begun when he was the general's aide during the Revolution, was resumed when he entered the first Cabinet as Secretary of the Treasury.