Stokes' theorem

In vector calculus, more differential geometry, Stokes' theorem is a statement about the integration of differential forms on manifolds, which both simplifies and generalizes several theorems from vector calculus. Stokes' theorem says that the integral of a differential form ω over the boundary of some orientable manifold Ω is equal to the integral of its exterior derivative dω over the whole of Ω, i.e. ∫ ∂ Ω ω = ∫ Ω d ω. Stokes' theorem was formulated in its modern form by Élie Cartan in 1945, following earlier work on the generalization of the theorems of vector calculus by Vito Volterra, Édouard Goursat, Henri Poincaré; this modern form of Stokes' theorem is a vast generalization of a classical result that Lord Kelvin communicated to George Stokes in a letter dated July 2, 1850. Stokes set the theorem as a question on the 1854 Smith's Prize exam, which led to the result bearing his name, it was first published by Hermann Hankel in 1861. This classical Kelvin–Stokes theorem relates the surface integral of the curl of a vector field F over a surface in Euclidean three-space to the line integral of the vector field over its boundary: Let γ: → R2 be a piecewise smooth Jordan plane curve.

The Jordan curve theorem implies that γ divides R2 into two components, a compact one and another, non-compact. Let D denote the compact part, bounded by γ and suppose ψ: D → R3 is smooth, with S:= ψ. If Γ is the space curve defined by Γ = ψ and F is a smooth vector field on R3, then: ∮ Γ ⁡ F ⋅ d Γ = ∬ S ∇ × F ⋅ d S This classical statement, along with the classical divergence theorem, the fundamental theorem of calculus, Green's theorem are special cases of the general formulation stated above; the fundamental theorem of calculus states that the integral of a function f over the interval can be calculated by finding an antiderivative F of f: ∫ a b f d x = F − F. Stokes' theorem is a vast generalization of this theorem in the following sense. By the choice of F, dF/dx = f. In the parlance of differential forms, this is saying that f dx is the exterior derivative of the 0-form, i.e. function, F: in other words, that dF = f dx. The general Stokes theorem applies to higher differential forms ω instead of just 0-forms such as F.

A closed interval is a simple example of a one-dimensional manifold with boundary. Its boundary is the set consisting of the two points a and b. Integrating f over the interval may be generalized to integrating forms on a higher-dimensional manifold. Two technical conditions are needed: the manifold has to be orientable, the form has to be compactly supported in order to give a well-defined integral; the two points a and b form the boundary of the closed interval. More Stokes' theorem applies to oriented manifolds M with boundary; the boundary ∂M of M is itself a manifold and inherits a natural orientation from that of M. For example, the natural orientation of the interval gives an orientation of the two boundary points. Intuitively, a inherits the opposite orientation as b. So, "integrating" F over two boundary points a, b is taking the difference F − F. In simpler terms, one can consider the points as boundaries of curves, as 0-dimensional boundaries of 1-dimensional manifolds. So, just as one can find the value of an integral over a 1-dimensional manifold by considering the anti-derivative at the 0-dimensional boundaries, one can generalize the fundamental theorem of calculus, with a few additional caveats, to deal with the value of integrals over n-dimensional manifolds by considering the antiderivative at the -dimensional boundaries of the manifold.

So the fundamental theorem reads: ∫ f d x = ∫ d F = ∫ − ∪ + F = F − F. Let Ω be an oriented smooth manifold with boundary of dimension n and let α be a smooth n-differential form, compactly supported on Ω. First, suppose that α is compactly supported in the domain of a single, oriented coordinate chart. In this case, we define the integral of α over Ω as ∫ Ω α = ∫ φ ∗ α, {\displaystyle \int _\alpha =\int _{\

ZOEgirl (album)

ZOEgirl is the debut studio album American Christian teen pop girl group ZOEgirl. It was released on August 2000 through Sparrow Records. "I Believe", "No You" and "Living For You" were released as radio singles. After the ZOEgirl group was formed, they began to work on their self-titled album at a fast pace, they temporarily separated during the Christmas holidays of 1999 to pray about the album and the message they should record on it. After this, they realized that they wanted to talk about their relationships with God and the romantic relationships that people may face in life. Producers remarked that paper, drafting the debut album's content, was scattered all around the studio; when the album's musical content was finished, ZOEgirl traveled to New York City to meet with a stylist for fashion suggestions regarding a photo shoot. After they agreed on what to wear, they traveled to Los Angeles, California for an entire photo shoot day, it is unknown whether the music video was recorded in this city, but the similarities between the album art and the music video seem to suggest this to be true.

ZOEgirl was released to retail stores on August 15, 2000. It included a link to The music video for "I Believe" was released around that time and could be streamed online via this site. A higher quality version of the video was released on the VHS cassette of WOW Hits 2001; the album peaked at No. 173 on the Billboard 200, No. 11 on the Billboard Contemporary Christian Charts and No. 8 on the Billboard Heatseeker's Chart. As of October 2003, over 350,000 copies of the debut record were sold. ZOEgirl got mixed reviews. Jesus Freak Hideout noted that those who "like to dance but don't like the secular junk will want to check ZOEgirl out." Meanwhile, Christian Music Today pointed out that the album "still didn't seem to measure up to mainstream quality songwriting and production." This was echoed by American Culture Scope, stating that the "sweet and sour" album was "tasteless" and that "groups such as SHeDAISY Dixie Chicks much better."Nearly a decade after its release, ZOEgirl admits that their debut album was "bashed" by critics.

The album remained on the Nielsen SoundScan Top 40 Christian Album chart for nearly a year after its release. All the songs from ZOEgirl's debut album were written by at least one band member; some songs were written with the help of non-members. Several products related to this album were released. A book, titled Backstage Exclusive, was released in 2001, featuring full-colour photos and facts from ZOEgirl; the following remixes can be found on Mix of Life: "I Believe" "No You" "Anything Is Possible" "Living For You" Additionally, the following remixes can be found on Life's limited edition single. "I Believe" "Anything Is Possible" Karaoke tracks were released. Volume one of ZOEgirl's Open Mic Karaoke disc series features "I Believe" and "Anything Is Possible". Original Dance Praise, released in 2005, featured two songs from ZOEgirl's debut album: "I Believe" and "Suddenly"

2000–01 Isthmian League

The 2000–01 season was the 86th season of the Isthmian League, an English football competition featuring semi-professional and amateur clubs from London and South East England. The league consisted of four divisions; the Premier Division consisted of 22 clubs, including 18 clubs from the previous season and four new clubs: Croydon, promoted as champions of Division One Grays Athletic, promoted as runners-up in Division One Maidenhead United, promoted as third in Division One Sutton United, relegated from the Football ConferenceFarnborough Town won the division and returned to the Football Conference after two seasons spent in Isthmian League. Slough Town, Carshalton Athletic and Dulwich Hamlet finished bottom of the table and relegated to Division One. Three fixtures left unfulfilled due to bad weather conditions. Division One consisted of 22 clubs, including 17 clubs from the previous season and five new clubs: Three clubs relegated from the Premier Division: Aylesbury United Boreham Wood Walton & HershamTwo clubs promoted from Division Two: Ford United NorthwoodInitially, champions of Division Two Hemel Hempstead Town were to be promoted to Division One, but they were refused due to ground grading.

Leatherhead finished. Boreham Wood returned to the Premier Division at the first attempt. Bedford Town and Braintree Town get a promotion. Leatherhead finished in the relegation zone the second time in a row and were relegated along with Romford and Barton Rovers. Division Two consisted of 22 clubs, including 17 clubs from the previous season and five new teams: Two clubs relegated from Division One: Chertsey Town Leyton PennantThree clubs promoted from Division Three: East Thurrock United Great Wakering Rovers TilburyInitially, previous season champions of Division Two Hemel Hempstead Town were to be promoted to Division One, but they were refused due to ground grading and stayed in the division. Tooting & Mitcham United won the division and were promoted to Division One along with Windsor & Eton and Barking. Barking ceased to exist when they merged with East Ham United to form Barking & East Ham United, taken place of Barking in Division One. Division Three consisted of 22 clubs, including 17 clubs from the previous season and five new teams: Arlesey Town, promoted as champions of the Spartan South Midlands League Ashford Town, promoted as champions of the Combined Counties League Chalfont St Peter, relegated from Division Two Wingate & Finchley, relegated from Division Two Witham Town, relegated from Division TwoArlesey Town and Ashford Town both debuted in the league and achieved a promotion along with Lewes.

Two fixtures left unfulfilled due to bad weather conditions. Isthmian League 2000–01 Northern Premier League 2000–01 Southern Football League