The surface area of a solid object is a measure of the total area that the surface of the object occupies. The mathematical definition of surface area in the presence of curved surfaces is more involved than the definition of arc length of one-dimensional curves, or of the surface area for polyhedra, for which the surface area is the sum of the areas of its faces. Smooth surfaces, such as a sphere, are assigned surface area using their representation as parametric surfaces; this definition of surface area is based on methods of infinitesimal calculus and involves partial derivatives and double integration. A general definition of surface area was sought by Henri Lebesgue and Hermann Minkowski at the turn of the twentieth century, their work led to the development of geometric measure theory, which studies various notions of surface area for irregular objects of any dimension. An important example is the Minkowski content of a surface. While the areas of many simple surfaces have been known since antiquity, a rigorous mathematical definition of area requires a great deal of care.

This should provide a function S ↦ A which assigns a positive real number to a certain class of surfaces that satisfies several natural requirements. The most fundamental property of the surface area is its additivity: the area of the whole is the sum of the areas of the parts. More rigorously, if a surface S is a union of finitely many pieces S1, …, Sr which do not overlap except at their boundaries A = A + ⋯ + A. Surface areas of flat polygonal shapes must agree with their geometrically defined area. Since surface area is a geometric notion, areas of congruent surfaces must be the same and the area must depend only on the shape of the surface, but not on its position and orientation in space; this means. These properties uniquely characterize surface area for a wide class of geometric surfaces called piecewise smooth; such surfaces consist of finitely many pieces that can be represented in the parametric form S D: r → = r →, ∈ D with a continuously differentiable function r →. The area of an individual piece is defined by the formula.

Thus the area of SD is obtained by integrating the length of the normal vector r → u × r → v to the surface over the appropriate region D in the parametric uv plane. The area of the whole surface is obtained by adding together the areas of the pieces, using additivity of surface area; the main formula can be specialized to different classes of surfaces, giving, in particular, formulas for areas of graphs z = f and surfaces of revolution. One of the subtleties of surface area, as compared to arc length of curves, is that surface area cannot be defined as the limit of areas of polyhedral shapes approximating a given smooth surface, it was demonstrated by Hermann Schwarz that for the cylinder, different choices of approximating flat surfaces can lead to different limiting values of the area. Various approaches to a general definition of surface area were developed in the late nineteenth and the early twentieth century by Henri Lebesgue and Hermann Minkowski. While for piecewise smooth surfaces there is a unique natural notion of surface area, if a surface is irregular, or rough it may not be possible to assign an area to it at all.

A typical example is given by a surface with spikes spread throughout in a dense fashion. Many surfaces of this type occur in the study of fractals. Extensions of the notion of area which fulfill its function and may be defined for badly irregular surfaces are studied in geometric measure theory. A specific example of such an extension is the Minkowski content of the surface; the below given formulas can be used to show that the surface area of a sphere and cylinder of the same radius and height are in the ratio 2: 3, as follows. Let the radius be r and the height be h. Sphere surface area = 4 π r 2 = × 2 Cylinder surface area = 2 π r = 2 π r ( 2

Daniel Brett Weiss is an American television producer and director. Along with his collaborator David Benioff, he is best known as co-creator and showrunner of Game of Thrones, the HBO adaptation of George R. R. Martin's series of books A Song of Ice and Fire. Weiss was raised in Chicago, Illinois, his family is Jewish. He graduated from Wesleyan University and earned a Master of Philosophy in Irish literature from Trinity College, where he wrote his thesis on James Joyce's Finnegans Wake, a Master of Fine Arts in creative writing from the Iowa Writers' Workshop. Weiss worked as personal assistant on films such as The Viking Sagas for New Line Cinema. For a brief period, Weiss worked as a personal assistant for musician Glenn Frey. Weiss went to Dublin in 1995 to study Anglo-Irish literature and met David Benioff, the screenwriter of Troy. Three years around 1998, they met again in Santa Monica, California. Weiss and Benioff co-wrote a screenplay for a film titled The Headmaster. In 2003, they were hired to collaborate on a new script of Orson Scott Card's book Ender's Game, in consultation with the then-designated director Wolfgang Petersen.

It was not used. Weiss's 2003 debut novel, Lucky Wander Boy, is themed around video games. In 2006, Weiss said he had a second novel finished that "needs a second draft"; that same year, Weiss completed a screenplay for a film adaptation of the video game series Halo, based on a script written by Alex Garland. However, director Neill Blomkamp declared the project dead in late 2007. Weiss worked on a script for a prequel to I Am Legend. However, in May 2011, director Francis Lawrence stated that he did not think the prequel was going to happen. Weiss collaborated with David Benioff on the HBO television series Game of Thrones, based on George R. R. Martin's book series A Song of Ice and Fire. Benioff and Weiss directed three episodes together. For the first two, they used a coin-flip to decide. Weiss received directing credit for "Two Swords", Season 4 episode 1, while Benioff was credited for "Walk of Punishment", Season 3 episode 3. Benioff and Weiss were both credited for co-directing the series finale, "The Iron Throne".

On July 19, 2017, Weiss announced that he and Benioff are going to begin production on another HBO series, titled Confederate, after the final season of Game of Thrones. Weiss and Benioff said, "We have discussed Confederate for years as a concept for a feature film - But our experience on Thrones has convinced us that no one provides a bigger, better storytelling canvas than HBO."On February 6, 2018, Disney announced that both Weiss and Benioff would write and produce a new series of Star Wars films after the finale season of Game of Thrones is completed, in 2019. In early 2019, Weiss and Benioff entered into an exclusive US$200 million deal with Netflix to produce several films and television shows for the streaming service. In late October 2019, it was reported that Weiss and Benioff had exited their deal with Disney due to their commitments to Netflix. Weiss and Benioff's first project on Netflix will be as directors of the stand-up comedy Leslie Jones: Time Machine. Weiss and his wife, Andrea Troyer, have two children.

List of awards and nominations received by Game of Thrones D. B. Weiss on IMDb

Gloria Bonilla-Santiago, born January 17, 1954 in Puerto Rico, is a Board of Governors Distinguished Service Professor at the Department of Public Policy and Administration and Director of the Center for Strategic Urban Community Leadership at Rutgers University-Camden. She is the founder of LEAP Academy University Charter School located in Camden, New Jersey. Created in 1997, LEAP is one of the 17 inaugural public charter schools in New Jersey; as the founder of LEAP, Bonilla-Santiago created a merit pay/pay-for-performance program designed to reward excellence in teaching and improve student academic outcomes. Bonilla-Santiago’s parents migrated to the United States from Puerto Rico, she received a B. A. from Glassboro State College, now known as Rowan University. S. W. from Rutgers University. A. in Philosophy and Ph. D. in Sociology from City University of New York. She has completed post-doctoral studies at the John F. Kennedy School of Government at Harvard University. Bonilla-Santiago is an expert in the fields of community development, migration of women, diversity management, organizational leadership and public policy.

She is considered a pioneer in the application of social capital to improve challenged schools and communities. Bonilla-Santiago is an advocate of merit-based pay for teachers, with success at LEAP Academy. Both teacher tenure and merit pay have been combined in a professional development and performance based compensation program that rewards teachers based on their ability to help students make significant academic gains as well as their willingness and contributions to school-wide success. On February 23, 2011 Santiago appeared on MSNBC Andrea Mitchell Reports to discuss merit pay for teachers. Bonilla-Santiago is a specialist columnist with the Huffington Post where she discusses educational issues such as charter schools and the need for increased science and technology education in elementary and secondary schools. In 1993 Bonilla-Santiago and the Rutgers-Camden Center for Strategic Urban Community Leadership developed the concept of an independently governed, public charter school that could improve education and opportunities for the children and families of Camden, NJ.

This concept led to Bonilla-Santiago’s creation of LEAP Academy University Charter School in 1997, one of 13 inaugural public charter schools in New Jersey. It was the first new school to open in Camden in 30 years; the academy opened as a K-5 charter school serving 324 students. Each subsequent year, it added a grade until reaching full enrollment of 780 students in grades K-12, it houses a 90 student preschool program. The school is founded on a model where the entire community, students and local organizations and businesses have a vested interest in student success; the academy was first housed in temporary modular units in Camden, is now located in two facilities. An elementary school building was opened in 1999 after a $7.5 million renovation grant from the Delaware River Port Authority. A high school building, was a combined effort. In 2003 LEAP was honored with the Pioneer Award and Exemplary School Award during the first New Jersey Charter Schools Recognition Ceremony sponsored by the New Jersey Department of Education Office of Innovative Programs.

The Alfredo Santiago Endowed Scholarship was established in 1999 by Bonilla-Santiago in memory of her husband, Alfredo Santiago, to increase the number of students in Camden who attend college. It provides financial assistance to first-year undergraduate students who graduate from LEAP Academy University Charter School and enroll full-time at one of Rutgers University’s campuses. In 2002, a donation from former President/CEO of Commerce Bank, Vernon Hill, created the Commerce Alfredo Santiago Scholarship, which allowed the endowment to support students attending other colleges as well. Gloria Bonilla-Santiago is the author of three books, Breaking Ground and Barriers: Hispanic Women Developing Effective Leadership, Organizing Puerto Rican Migrant Farmworkers: The Experience of Puerto Ricans in New Jersey, The Miracle on Cooper Street: Lessons from an Inner City She has published numerous articles and monographs and contributed a chapter to Helping Battered Women, edited by A. Roberts. 2008 L’Oreal Paris Women Of Worth Award 2007 National Mujer Award from the National Hispana Leadership Institute 1993 Congressional Citation from Congressman Frank Pallone Jr. for contributions to the Hispanic Community 1992 Warren I.

Susman Award for Excellence in Teaching Website of Gloria Bonilla-Santiago, Ph. D. Rutgers University Centers for Excellence LEAP Academy

Charles Humphrey Keating, Jr. was an American athlete, real estate developer, banker and activist best known for his role in the savings and loan scandal of the late 1980s. Keating was a champion swimmer for the University of Cincinnati in the 1940s. From the late 1950s through the 1970s, he was a noted anti-pornography activist, founding the organization Citizens for Decent Literature and serving as a member on the 1969 President's Commission on Obscenity and Pornography. In the 1980s, Keating ran American Continental Corporation and the Lincoln Savings and Loan Association, took advantage of loosened restrictions on banking investments, his enterprises were investigated by federal regulators. His financial contributions to, requests for regulatory intervention from five sitting U. S. senators led to those legislators being dubbed the "Keating Five". When Lincoln failed in 1989 it cost the federal government over $3 billion and about 23,000 customers were left with worthless bonds. In the early 1990s, Keating was convicted in both federal and state courts of many counts of fraud and conspiracy.

He served four and a half years in prison before those convictions were overturned in 1996. In 1999, he pleaded guilty to a more limited set of wire fraud and bankruptcy fraud counts, was sentenced to the time he had served. Keating spent his final years in low-profile real estate activities until his death in 2014. Keating was born on December 4, 1923, in Cincinnati, into a devout Roman Catholic family, he was the son of Charles Humphrey Keating. He grew up in the Clifton neighborhoods of that city, his younger brother William was born in 1927. Their father managed a dairy. Charles Keating Sr. lost a leg in a hunting accident, fell into a long decline from Parkinson's disease around 1931, was nursed by his wife until his death in 1964. Keating became passionately involved in the sport, he attended St Xavier High School, where he was a good student, was on the swim team all four years, ran track and played football. In swimming he led the team to three Greater Catholic League championships, set several school records, was named all-state, was captain of the team in his senior year.

Keating graduated from St Xavier in 1941. After one semester at the University of Cincinnati in fall 1941, Keating left because of poor grades, although he advanced to the NCAA Men's Swimming and Diving Championships in 1942, finishing sixth in the 200 yard breaststroke, he enlisted in the United States Navy. He trained in the Navy Air Corps to become a carrier-based night fighter pilot flying F6F Hellcats. During World War II, Keating was stationed in the U. S. sometimes at Banana Creek in Florida, flew Hellcats to armed services swimming meets. He narrowly escaped serious injury one night at Naval Air Station Vero Beach when he neglected to lower the landing gear on his Hellcat and wrecked the plane in an unexpected belly landing. Due to additional training on new intercept methods and the vagaries of squadron transfers, the war ended before he would be deployed to any combat theater. Keating was ready to return to college after finishing his Navy service in 1945, his abilities as a swimmer made him an attractive recruit, despite his having dropped out earlier.

He cut a deal with the University of Cincinnati wherein it would accept for academic credit much of his Navy service he would take six months of liberal arts courses before entering its law school. Keating won the 200-yard breaststroke at the Ohio Intercollegiate Conference championship in 1945. On March 30, 1946, Keating competed in the 200-yard breaststroke at the NCAA Men's Swimming and Diving Championships, before a packed house of 2,500 spectators at Yale University's Payne Whitney Gymnasium. In an exciting, back-and-forth contest with Paul Murray of Cornell University and future coaching legend James Counsilman of Ohio State University, he prevailed by a foot to win the championship with a time of 2:26.2. This was the first national championship in any sport for the University of Cincinnati, he and teammate Roy Lagaly become the first-ever Bearcats to be named All-Americans. Keating was a natural leader and co-captain of the team with Lagaly. Of Keating, Lagaly said, "You could tell then he was going to be successful.

He was ambitious. Whatever he did, he did all the way." Keating followed this by swimming for Cincinnati Gym, finishing second to future Olympic gold medalist Joseph Verdeur in the 220-yard breaststroke at the April 1946 national AAU championships. Keating received his law degree from the University of Cincinnati College of Law in 1948, would be named a member of the university's Athletic Hall of Fame. Charles Keating was a long-time supporter of U. S. swimming and beginning in 1969 he and his brother William donated $600,000 to St. Xavier High School in Cincinnati to build a state-of-the-art competition pool; the school's swimming team went on to win many state titles. St. Xavier named the Keating Natatorium after the brothers' father, inducted Charles Keating into its initial Athletic Hall of Fame class in 1985; the University of Cincinnati's 2006 athletic building is named the Keating Aquatic Center, in honor of William Keating, donations from the Keating family used to construct it. Charles Keating funded.

The 53rd Street Crosstown Line was a surface transit line in Manhattan, New York City, United States, running along 53rd Street in Midtown, between Sixth Avenue and Ninth Avenue. It served as a connection between north-south lines during the times that they were not affiliated with the 59th Street Crosstown Line; the street is no longer used by buses. The tracks in 53rd Street were added in 1894, when the Broadway and Columbus Avenue Line, a cable car line, opened. However, the Metropolitan had a lease on the Central Park and East River Railroad, the owner of the 59th Street Crosstown Line, used 59th Street as a connection in the Sixth and Amsterdam Avenues Line. After the Broadway and Columbus Avenue Line was converted to electricity in 1901, those cars were moved to 59th Street, a new service - the 53rd Street Line - was added; the route began at Sixth Avenue and 50th Street and traveled north on Sixth Avenue for three blocks, west on 53rd Street to Ninth Avenue, north on Ninth Avenue to 54th Street.

Effective August 6, 1908, the CPN&ER was separated from the bankrupt Metropolitan, three lines - the Broadway and Amsterdam Avenue Line and Columbus Avenue Line, Sixth and Amsterdam Avenues Line - were placed on 53rd. The Ninth Avenue Railroad, which owned the trackage on 53rd Street between Seventh and Ninth Avenues, was split from New York Railways, the Metropolitan's successor, in 1919; the Broadway and Amsterdam Avenue Line and Sixth and Amsterdam Avenues Line were discontinued, while the Broadway and Columbus Avenue Line remained, using trackage rights over the Ninth Avenue's trackage on 53rd Street and Ninth Avenue south of Broadway to reach Columbus Avenue. The Ninth Avenue Railroad brought back the old 53rd Street shuttle to Sixth Avenue. On February 12, 1936, New York City Omnibus Corporation buses replaced the trolleys on the Broadway and Columbus Avenue Line, ending use of the 53rd Street tracks; the replacement bus - now the M7 - used 54th Street eastbound and 55th Street westbound until after the 59th Street Crosstown Line was replaced by buses

The 2017 Tipperary Senior Football Championship was the 127th edition of the Tipperary GAA's premier club Gaelic football tournament for senior graded teams in County Tipperary, Ireland. The tournament consisted of 16 teams with the winner going on to represent Tipperary in the Munster Senior Club Football Championship; the championship began with a Group stage before proceeding to a knock-out format. Loughmore-Castleiney were the defending champions after they defeated Moyle Rovers in the previous years final; this was Fethard's first year in the senior grade after claiming the 2016 Tipperary Intermediate Football Championship title, however they will drop back to the I. F. C. for 2018 after losing the Relegation final. They will be replaced by I. F. C. Champions Kiladangan; the following teams have changed division since the 2016 championship season. The 16 teams are drawn into random groups of four; the top two teams automatically enter the knock-out stages. Should a team finish third or fourth in their group but win their regional championship, they will be re-instated into the county series in a Preliminary Quarter-Final in a random draw against any of the second placed teams.

West Tipperary S. F. C. Final: Arravale Rovers 4-10, 2-10 Galtee Rovers, Annacarty, 2/9/2017,Mid Tipperary S. F. C. Final: Loughmore/Castleiney 0-17, 0-6 Upperchurch/Drombane, Littleton, 9/9/2017,South Tipperary S. F. C. Final: Clonmel Commercials 0-8, 0-8 Moyle Rovers, Clonmel, 7/1/2018, All 16 teams enter the competition at this stage; the top 2 teams in each group go into the Quarter-Finals while the bottom team of each group will enter a Relegation Playoff. Arravale Rovers won the West Tipperary S. F. C. in 2017, thus qualified for the Knock-out stage despite finishing third in Group B. Tipperary GAA Official Site