In vector calculus, the gradient of a scalar-valued differentiable function f of several variables, f: R n → R, is the vector field, or more a vector-valued function ∇ f: R n → R n, whose value at a point p is the vector whose components are the partial derivatives of f at p: ∇ f =. The gradient is related to the derivative, but it is not itself a derivative: the value of the gradient at a point is a tangent vector – a vector at each point, they are related in that the dot product of the gradient of f at a point p with another tangent vector v equals the directional derivative of f at p of the function along v. See § Definition and relationship with the derivative; the nabla symbol, a character that looks like an upside down triangle, shown above is called Del, the vector differential operator. The gradient can be interpreted as the "direction and rate of fastest increase". If at a point p, the gradient of a function of several variables is not the zero vector, the direction of the gradient is the direction of fastest increase of the function at p, its magnitude is the rate of increase in that direction.

Conversely, the gradient at a point is the zero vector if and only if the derivative vanishes at that point. The gradient thus plays a fundamental role in optimization theory, where it is used to maximize a function by gradient ascent; the gradient admits multiple generalizations to more general functions on manifolds. Consider a room where the temperature is given by a scalar field, T, so at each point the temperature is T. At each point in the room, the gradient of T at that point will show the direction in which the temperature rises most quickly; the magnitude of the gradient will determine. Consider a surface whose height above sea level at point is H; the gradient of H at a point is a vector pointing in the direction of the steepest slope or grade at that point. The steepness of the slope at that point is given by the magnitude of the gradient vector; the gradient can be used to measure how a scalar field changes in other directions, rather than just the direction of greatest change, by taking a dot product.

Suppose that the steepest slope on a hill is 40%. If a road goes directly up the hill the steepest slope on the road will be 40%. If, the road goes around the hill at an angle it will have a shallower slope. For example, if the angle between the road and the uphill direction, projected onto the horizontal plane, is 60° the steepest slope along the road will be 20%, 40% times the cosine of 60°; this observation can be mathematically stated. If the hill height function H is differentiable the gradient of H dotted with a unit vector gives the slope of the hill in the direction of the vector. More when H is differentiable, the dot product of the gradient of H with a given unit vector is equal to the directional derivative of H in the direction of that unit vector; the gradient of a scalar function f is denoted ∇f or ∇→f where ∇ denotes the vector differential operator, del. The notation grad f is commonly used to represent the gradient; the gradient of f is defined as the unique vector field whose dot product with any vector v at each point x is the directional derivative of f along v. That is, ⋅ v = D v f.

Formally, the gradient is dual to the derivative. When a function depends on a parameter such as time, the gradient refers to the vector of its spatial derivatives only; the magnitude and direction of the gradient vector are independent of the particular coordinate representation. In the three-dimensional Cartesian coordinate system with a Euclidean metric, the gradient, if it exists, is given by: ∇ f = ∂ f ∂ x i + ∂ f ∂ y j + ∂ f ∂ z k, where i, j, k are the standard unit vectors i

Disco is the first remix album by English synth-pop duo Pet Shop Boys, released on 17 November 1986 by Parlophone in the United Kingdom and by EMI America Records in the United States. Disco consists of remixes of tracks from its respective B-sides; the album includes remixes by Shep Pettibone and the Pet Shop Boys themselves. According to interviews, the album was released to showcase music the duo deemed non-radio friendly. With the exception of "Suburbia", all the mixes on Disco were either rare import remixes or unreleased. In The Night - originally intended for the b-side of the original 12" 1985 release of Opportunities (the sleeve of which lists'In The Night' but in fact played the 7" version; the sleeve cover is a still of Chris Lowe from the promo video of "Paninaro", directed by Pet Shop Boys themselves. The Disco mix of "Suburbia" can be found on the "Suburbia" 12" and the 2001 two-disc re-release of Please; the original version of "In the Night" was the B-side to the original release of "Opportunities".

Arthur Baker's Extended Mix of "In the Night" was used as the theme for the BBC's The Clothes Show. Stuart Price, producer of the Pet Shop Boys albums Electric and Super, cites Disco as a major influence on his musical taste as a teenager. "In the Night" – 6:28 "Suburbia" – 8:56 "Opportunities" – 5:34 "Paninaro" – 8:35 "Love Comes Quickly" – 7:38 "West End Girls" – 9:03 Neil Tennant Chris Lowe Andy Richards – Fairlight on track 2 Gary Barnacle – Saxophone on track 2 Blue Weaver and Khris Kallis – Additional keyboards on track 3 Adrien Cook – Fairlight on track 4 Andy Mackay – Saxophone on track 5

The Battle of Kobryn took place on September 11–23, 1920, during the Polish–Soviet War. Polish Fourth Army, commanded by General Leonard Skierski defeated Soviet forces in the area of Kobryn. After the Battle of Warsaw, the Fourth Army was in early September 1920 transported eastwards, to guard the front along the rail line Bialystok – Brzesc nad Bugiem – Wlodawa; the Army consisted of the following units: 15th Infantry Division, 14th Infantry Division, 11th Infantry Division, 16th Infantry Division. Facing them was newly created Soviet Fourth Army, consisting of two rifle divisions, 17th Cavalry Division; these units were supported by 19th and 55th Rifle Divisions, which had completed their concentration in the first half of September. After additional reinforcements brought from Russia, the Soviets had a numerical advantage along the frontline. General Skierski, aware of the Soviet plans, decided to forestall their advance, ordered the 14th I. D. to capture Zhabinka on September 8. In the night of September –11, the Polish 11th I.

D. destroyed the Soviet forces near Malaryta, after this victory, Skierski ordered an assault on Kobryn, defended by the Soviet 57th Rifle Division. The Poles entered the town in the morning of September 12, but the Soviets concentrated three divisions, in the night of September 15–16, attacked a gap between the Polish 14th and 16th I. D; the enemy captured a bridge over the Mukhavets River, attacked the Polish 57th Infantry Regiment. Heavy fighting ensued. On September 16, regiments of Polish 16th I. D. were forced to retreat, but on the next day, after reinforcements had been brought, Polish soldiers recaptured the lost territory. In two days of fighting, the Poles lost 500 soldiers dead and wounded, to draw the Soviets away from Kobryn, General Skierski created the Operational Group of General Michal Milewski to carry out a raid on Pruzhany; the town was captured in the night of September 18–19, but fighting there continued until September 22. Fighting over Kobryn and Pruzana tied down the whole Soviet Fourth Army, which removed the threat of the capture of Brest Fortress, allowed the Polish Army headquarters to prepare an operational plan for the Battle of the Niemen River.

Furthermore, the Soviet 12th Army, fighting in Volhynia, had to retreat, as its wing came under Polish pressure. The battle is commemorated on the Tomb of the Unknown Soldier, with the inscription "KOBRYN 14 – 15 IX 1920". Odziemkowski, J.. Leksykon wojny polsko–rosyjskiej 1919–1920. Warsaw: RYTM

For the U. S. Army general, see Wallace H. Nutting Wallace Nutting was a U. S. minister, photographer and antiquarian, most famous for his landscape photos of New England. He was an accomplished author, furniture maker, antiques expert and collector, his atmospheric photographs helped spur the Colonial Revival style. He was born in Rockbottom, Massachusetts, on Sunday, November 17, 1861, the second child of Albion and Elizabeth Nutting, his father was killed in the Civil War. The family was descended from John Nutting, who came from England in 1639 and was killed by Indians during a raid against Groton, Massachusetts; the Indians severed John Nutting's head and put it on a pole to discourage others from settling in the area. Wallace Nutting graduated from high school in Maine, he studied at Phillips Exeter Academy, Harvard University, Hartford Theological Seminary and Union Theological Seminary. He graduated from Harvard with the class of 1887. Nutting earned a Doctor of Divinity degree from Whitman College in 1893.

He received an honorary doctor of Humanities from Washington and Jefferson College in 1938. On June 5, 1888 he married Mariet Griswold in Massachusetts, they had no children. Nutting began his career as a Congregational minister in several towns including Minneapolis and Providence and Fryeburg, but he was forced to retire at age 43 because of poor health, he suffered from neurasthenia, turned to bicycling as a means of relaxation and improving his health. It was on these bicycle rides in the countryside. In 1904 he opened the Wallace Nutting Art Prints Studio on East 23rd Street in New York. After a year he moved his business to a farm in Connecticut, he called this place "Nuttinghame". In 1912 he moved the photography studio to Framingham, Massachusetts, in a home he called "Nuttingholme"; that year he published a catalog of prints, 97 pages and included about 900 images. By 1915, Nutting claimed to be earning $1,000 per day. Nutting's photographs ranged in price to suit a variety of tastes, his catalog included pastoral scenes such as views of abbeys, bridges, mountains and winding roads.

One of his most common themes was "Colonials", which were photos of women in traditional 18th-Century roles. These were traditional pictures of femininity pictured indoors in front of a chest, chair, or looking glass, his prints sold from $1.25 to $20. Nutting authored several books about the scenic beauties of New England, the United Kingdom, Ireland. In the peak of his business he employed about two-hundred colorists. By his own account, Wallace Nutting sold ten million pictures. Wallace Nutting's colorists painted the photographs; these colorists would sometimes sign Wallace Nutting's name on the photos, why the signatures vary. Nutting had a collection of furniture; this experience led him to start a business selling reproduction furniture. His expertise in this field led him to author a guidebook to American Windsor furniture in 1917. By 1918 his mail-order catalog offered dozens of Windsor chairs in several historic styles. Wallace Nutting died at his home at 24 Vernon St. Framingham, Massachusetts on Saturday, July 19, 1941, at age 79.

The body was taken to Maine for burial. Wallace Nutting residence in Portsmouth, New Hampshire, is known as the Wentworth Gardner House. A Wallace Nutting Society exists for the study of his work. Today Nutting's furniture and photographs are collected. Houses owned and the subject of Nutting's preservation activities: Hazen-Spiller House Joseph Webb House Wentworth-Gardner House Iron Works House Works by or about Wallace Nutting at Internet Archive Wallace Nutting's Windsor Handbook photo of Wallace Nutting The Wallace Nutting Center The Wallace Nutting Library example of Wallace Nutting's photography The Winterthur Library Overview of an archival collection on Wallace Nutting

Kiyoshi Saitō was a sōsaku-hanga artist in 20th-century Japan. In 1938, he issued his first prints in his now famous "Winter in Aizu" series. Saitō was one of the first Japanese printmaking artists to have won at the São Paulo Biennale in 1951. Saitō's early works depict villages populated with local Japanese with a high degree of realism and three-dimensionality, his more mature works merge modern elements with Japanese tradition. His prints feature plant life flattened in two-dimensionality, he spent time in Paris, did a series there. Kiyoshi Saito’s woodblock prints titled “Autumn” are considered rare and valuable. Harada, Minoru; the Life and Works of Kiyoshi Saito. Tokyo: Abe Shuppan, 1990. Yanaizu Municipal Saito Kiyoshi Museum of Yanaizu in the prefecture Fukushima The Cats of Saito

Earl H. Norem, who signed his work Norem, was an American artist known for his painted covers for men's-adventure magazines published by Martin Goodman's Magazine Management Company and for Goodman's line of black-and-white comics magazines affiliated with his Marvel Comics division. Over his long career, Norem illustrated covers for novels and gaming books, as well as movie posters, baseball programs, trading cards. Norem was born on April 17, 1923, he saw military action in World War II with the 85th Regiment of the 10th Mountain Division. He trained in Colorado and Texas, fought the Germans in the Northern Apennine Mountains of Italy. By age 20, Norem was a squad leader and staff sergeant who in the Italian Campaign fought alongside famed skier Torger Tokle, whom he had seen ski jumping at Bear Mountain, New York when Norem was 12. After Tokle was killed in action on March 3, 1945, Norem was one of the men assigned to retrieve his body from the mountain. Norem himself was wounded going into the Po Valley, ending his military stint.

Upon returning to the US, Norem embarked on an illustration career. Norem throughout the 1950s and 1960s worked extensively for men's adventure magazines, producing covers and interior-art spreads. In addition, he produced illustrations for such magazines such as Reader's Digest and Stream, Real West, Discover, he worked on such Marvel Comics projects as Savage Sword of Conan, He-Man and the Masters of the Universe Magazine, Marvel Preview, Tales of the Zombie, Monsters Unleashed, Planet of the Apes, Rampaging Hulk, The Silver Surfer, storybooks featuring Spider-Man and the Fantastic Four. His Transformers work consisted of four Marvel Big Looker storybooks, some of which were adapted into "read along" storybooks: Battle for Cybertron, The Great Car Rally, Car Show Blow Up, The Story of Wheelie, The Wild Boy of Quintesson In addition to his work for Marvel, Norem painted illustrations and covers for the Worlds of Power, Wizards Warriors & You book series, Mars Attacks comics and trading cards, Charlton Comics' The Six Million Dollar Man.

The U. S. release of the Wizards Warriors & You series illustrated by Norem included covers in color. In 2013, he had contributed paintings to the company's "Mars Attacks: Invasion" card set. At the time of his death, he was working on a trading card assignment for Topps' Mars Attacks franchise the "Mars Attacks" Occupation" set being produced in 2015. Norem painted movie posters for Conforte Graphics. Norem favored painting in acrylics. Suffering from arthritis, he had retired as of 2005, only painting for his own amusement and for his grandchildren, he said in a 2005 interview, "All the contacts that I had in the commercial art field are either retired or dead, the younger art buyers don't want anything to do with an 81-year-old artist."According to posts on Facebook accounts attributed to Norem family-members, the artist died in Danbury, Connecticut on June 19, 2015, shortly after undergoing surgery. His family announced the news on Norem's Facebook fan page. O'Donoghue, Eamon, ed. "EarlNorem.com: The Unofficial Website of Illustrator Earl Norem".

Archived from the original on July 10, 2011. Retrieved July 30, 2012. CS1 maint: extra text: authors list "Earl Norem". American Art Archives. Archived from the original on May 25, 2011. Retrieved July 30, 2012