text:
User:Duesouthband3/sandbox

Due South Band

“Family first”. “It has to be fun or we can’t do it”. “Equal partners”. All requirements for playing music agreed upon by members of the Due South Band.

Due South is a bluegrass trio that began playing their brand of bluegrass after coming together in the spring of 2009. Composed of members Bennie Rose on guitar and lead vocals, David Thompson on mandolin and Dan Kilgore on upright bass. Due South enjoyed playing for audiences from Atlanta to Savannah.

Bennie Rose is a native of Pike County Kentucky who learned to play guitar at a young age. Influenced by the traditional music of the area his style shows a hint of rock and roll with a touch boogie woogie. His style makes for a hard driving edge to bluegrass standards as well as some newer song selections. Rose plays a 1959 Martin D-18 purchased for him by his father at a barber shop in Kentucky when Bennie was just age thirteen. It is still the only guitar he owns.

David Thompson is a native of Spalding County Ga whose early musical influences included his maternal grandfather who played old time fiddle. David plays an “A” style mandolin he constructed himself. Thompson plays with a more aggressive style and adds tremendously to the drive behind Due South’s unique sound.

Dan Kilgore is from Upson County Ga and plays upright bass in the trio. Dan’s talent of walking up and down the strings on the Bass is unparalleled and contributes to the unique sound of the Band. Dan plays a 1950’s King Bass and tells a few jokes between songs. He is also the Sherriff of Upson County Ga.

Audiences are often surprised that so much sound can come from only three musicians. We like to think it is because we enjoy the music so much and like to share it with those willing to listen.

title: User:Duesouthband3/sandbox


text:
Talk:beater

RFV discussion: May 2007–September 2007
The Harry Potter sense: another Harry Potter coinage without independent use. Dmcdevit·t 21:50, 22 May 2007 (UTC)

RFV failed, sense removed. —Ruakh TALK 07:45, 15 September 2007 (UTC)

RFV discussion: September 2011–March 2012
Rfv-sense copycat. I can't see it in any other online dictionary, and I have never come across it. SemperBlotto 21:27, 9 September 2011 (UTC)

I hear it very often used to express frustration over someone who plagiarized their idea or imitated their style. It is slang and I don't know if it has been used in printed works.Gtroy 21:45, 9 September 2011 (UTC)
 * I can't think of a way to search for this that would avoid the other meanings of beater. Maybe something like "you're such a beater", would that work? Mglovesfun (talk) 23:21, 10 September 2011 (UTC)
 * I'm not sure, maybe "He is such a beater"?Gtroy 20:36, 13 September 2011 (UTC)
 * I wouldn't understand this sense if I saw or heard it. Could you cite any actual usage that makes clear the extra sense as a synonym of plagiarist?    D b f  i  r  s   10:16, 16 September 2011 (UTC)

 * RFV-failed. - -sche (discuss) 07:14, 1 March 2012 (UTC)

source: wikimedia

license: Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/4.0/

title: Talk:beater


text:
Talk:Invincible (Skillet album)

Fair use rationale for Image:Skillet Invincible.jpg
Image:Skillet Invincible.jpg is being used on this article. I notice the image page specifies that the image is being used under fair use but there is no explanation or rationale as to why its use in this Wikipedia article constitutes fair use. In addition to the boilerplate fair use template, you must also write out on the image description page a specific explanation or rationale for why using this image in each article is consistent with fair use.

BetacommandBot (talk) 05:42, 24 January 2008 (UTC)

title: Talk:Invincible (Skillet album)


text:
Samuel Carte

Samuel Carte (21 October 1652 – 16 April 1740) was an English antiquarian and clergyman of the Church of England. After attending Magdalen College, Oxford, he held many ecclesiastical positions in his adult life, publishing two sermons. He was an active local antiquarian researcher, assisting several fellow antiquarians in their histories of Leicestershire, and publishing several articles and one book on such subjects.

Biography
Samuel Carte was born on 21 October 1652 in Coventry, Warwickshire, to the local clothier, Thomas Carte. Here, he was educated at the local free school, after which he matriculated from Magdalen College, Oxford University on 10 June 1669, at the age of 17. He graduated with a BA in 1673 and MA in 1675. He received deacon's orders from the Bishop of Lichfield and Coventry, Thomas Wood, at Eccleshall on 21 September 1673. Following by being ordained a priest by the Bishop of London, Henry Compton, on 10 June 1677.

Carte's first ecclesiastical position was as a curate of Crick, Northamptonshire, appointed on 26 March 1678. On 29 September 1682, he was made a prebendary of Lichfield Cathedral, a position he held until shortly before his death. From 26 March 1684 to 2 February 1693, Carte was the vicar of Clifton-upon-Dunsmore, Warwickshire. In 1691, Carte returned to Coventry, where he served as the master of the local free school. He returned to his career as a clergyman, working as vicar of Dunchurch from 2 July 1697 to 8 June 1699. On 7 January 1699, he was appointed both as the Preacher throughout the diocese of Lincoln, and the rector of Eastwell, Leicestershire, the latter position he held up until his death. In 1700, he was made lecturer and sequestrator at Leicester Cathedral, where, on 21 October 1712, he was made vicar, holding the office until the year of his death.

Independent of these offices, Carte engaged in much theological and antiquarian work. As a clergyman, two of his sermons were published: A Dissuasive from Murmuring: Being a Sermon on 1 Cor. X. 10. (1694) and ''The Cure of Self-Conceit: A Sermon Preach'd at St. Martin's in Leicester on Sunday, May 13. 1705'' (1705). In the 1720s, he also engaged in a lengthy argument with John Jackson, a controversial unitarian cleric of Leicester Cathedral, concerning the validity of baptism by immersion and the Trinity. Contemporary antiquaries, including Browne Willis (for his 1718-19 Mitred Abbots) and John Throsby (for his 1791 History and Antiquities of Leicester), consulted Carte over the history of Leicestershire. After Carte's death, John Nichols used his work extensively for his History of Leicestershire (1795-1811). Carte's most important publication was antiquarian in nature: Tabula Chronologica Archiepiscopatuum et Episcopatuum in Anglia et Wallia [Chronological table of Archbishoprics and Bishoprics in England and Wales] (1714), a compilation of lists on the descent of the offices of archbishops and bishops. Carte also had a letter to Humfrey Wanley published in the Philosophical Transactions of the Royal Society.

Carte married Anne, together having thirteen children between 1686 and 1704, with Anne predeceasing Carte at a young age. Carte died in Leicester on 16 April 1740, and was buried at Leicester Cathedral next to his wife and two of his children. His will, written on 19 January 1736, put most of his possessions in the hands of his eldest son, Thomas Carte. Thomas left a large number of historical papers to the Bodleian Library upon his death, including several papers by his father, Samuel, such as his commonplace book and a "Chronological Account of Extraordinary Events in all Parts of the World". His grave reads:

"The remains of Samuel Carte, M.A. many years Vicar of this parish. He was a person of great learning, exemplary life and conversation, strict piety, sound judgment, orthodox principles, and a zealous able defender of the doctrine of the Holy Trinity He died April 16th, 1740, in the 87th year of his age, in full assurance of a joyful resurrection. Near this place lie interred Ann wife and Elizabeth daughter of the said Samuel Carte."

title: Samuel Carte


text:
Talk:Sina Khadempour

COI tag (May 2021)
Edits in May 2021 made by subject of draft, not yet reviewed for neutrality. Robert McClenon (talk) 17:13, 24 May 2021 (UTC)

title: Talk:Sina Khadempour


text:
Hans Christian

Hans Christian may refer to:

People

 * Hans Christian (musician) (born 1960), German-born musician and producer
 * Hans Christian Andersen (1805–1875), Danish author and poet
 * Hans Christian Gram (1853–1938), Danish bacteriologist
 * Hans Christian Ørsted (1777–1851), Danish physicist and chemist
 * Hans Christian Hansen (1906-1960), Danish Prime Minister
 * Hans-Christian Ströbele (born 1939), German politician and lawyer
 * Hans Christian Blech (1915–1993), German film, stage and television character actor
 * Hans Christian Doseth (1958–1984), Norwegian climber
 * Hans Christian Thoning (born 1952), Danish politician
 * Hans Christian Rosen (1794–1880), Norwegian military officer
 * Hans Christian Ægidius (1933–2008), Danish actor
 * Hans Christian Knuth (born 1940), German theologian
 * Hans Christian Friedrich (1925–1992), German botanist
 * Hans Christian Brandy (born 1958), German theologian

Other uses

 * Hans Christian Yachts

title: Hans Christian


text:
Wikipedia:Articles for deletion/Skunkweed

The result was Delete all. Xoloz 17:50, 20 August 2006 (UTC)

Skunkweed, History of the Beer Bong, and Chunky Butt Funky
Also Leon Waddy and Jay Brown (redirects to Skunkweed) and Category:Skunkweed albums.

Band of dubious notability + their two albums. Fireplace 23:44, 10 August 2006 (UTC)
 * Delete all Can't find anything that makes this band more than a local phenomena.  --Joelmills 01:03, 11 August 2006 (UTC)
 * Neutral Admittedly, they don't quite meet the letter of the notability guidelines in the albums project.  Not that I agree big label = notable, but you have to draw the line somewhere.  Don't forget to delete the two album cover images. --UnhandledException 04:21, 11 August 2006 (UTC)
 * Delete all - per nom and Joelmills. Zaxem 08:19, 20 August 2006 (UTC)


text:
User:Primus Peak/sandbox

<!-- EDIT BELOW THIS LINE --Your topic and then define it in the first sentence if you can. Say something important about the topic.

title: User:Primus Peak/sandbox


text:
Talk:2017 Puebla earthquake

SASMEX Advance Warning: the alert didn't really activated until 24 sec after initial detection
The initial section claims that SASMEX was able to give 20 seconds advance warning before the earthquake hit Mexico City. This didn't happen, the "Public Alert" wasn't activated until seconds after the quake actually started. The following video shows that the actual alert wasn't activated until 24 seconds after initial detection. https://www.youtube.com/watch?v=5tAVty-PvBw Danixdefcon5 (talk) 01:26, 28 September 2017 (UTC)

pdf
--Itu (talk) 19:31, 19 September 2017 (UTC)
 * https://earthquake.usgs.gov/realtime/product/losspager/us2000ar20/us/1505848038633/alertfatal.pdf
 * https://earthquake.usgs.gov/realtime/product/losspager/us2000ar20/us/1505848038633/alertecon.pdf

map switching
That implementation is very ugly. It should not use buttons, and at least they should align left. --Itu (talk) 21:32, 19 September 2017 (UTC)

Background section
I added an image, but its caption might need work. - Mardus /talk 22:17, 20 September 2017 (UTC)

Merge with the first earthquake from 11 days ago
This one would not have been so lethal if that one had not weakened the buildings before — Preceding unsigned comment added by Granito diaz (talk • contribs) 02:48, 21 September 2017 (UTC)
 * The other earthquake was way to the south, so that's very dubious. Regardless, a merge is definitely not warranted. These are two distinct earthquakes that affected different areas. Master of Time   ( talk ) 04:13, 21 September 2017 (UTC)
 * - appears to be a distinct event that isn't the same as this one. -- Fuzheado | Talk 16:14, 22 September 2017 (UTC)

?!
"In Mexico City, 22 of the fatalities occurred at the Enrique C. Rébsamen school, where the bodies of 26 students and four instructors were pulled from the rubble..." — This does not make much sense. By my reckoning, this means that 30 bodies were recovered at Enrique C. Rébsamen; so how could it be that they represent "22 of the fatalities"? Perhaps this has been updated clumsily? Kelisi (talk) 15:25, 21 September 2017 (UTC)

The "Aftershocks" section
This section only contains three graphics which, if I read the attributions correctly, were created by the submitter.

Doesn't that fall under "original research" which Wikipedia (or most of its editors) does not like to see in any entry/article?

Just curious. 2600:8800:786:A300:C23F:D5FF:FEC4:D51D (talk) 06:41, 26 September 2017 (UTC)
 * The relevant policy says "Original images created by a Wikipedian are not considered original research, so long as they do not illustrate or introduce unpublished ideas or arguments, the core reason behind the NOR policy." --Joshualouie711talk 23:31, 29 September 2017 (UTC)

mexico aerthquake 2017
61 deaths — Preceding unsigned comment added by 2A02:120B:2C4A:69D0:34B3:4D2A:7E98:2B0A (talk) 13:14, 4 October 2017 (UTC)

Death toll
The article refers to 370 deaths in the lead section and the infobox and 371 in the table in the "Casualties" section. Most sources that I can find give 369 deaths as the total, including the AP source for citation #5. To complicate matters, the AP article breaks the total down into individual figures that add up to 370. The table is cited (#26) to a tweet from Luis Felipe Puente of the Proteccion Civil which gives a total of 361 and is clearly not the source for the figures shown. The difference between the 369 and 370 figure appears to be the number of fatalities in Morelos, either 74 or 75 depending on source. I am still trying to find a definitive breakdown, but no luck so far. I can say that there is virtually no support in reliable sources for 371 deaths, so we may have to live with a range. There is also even more variation in the number of injured, between 6,000 and over 7,000. I'll keep digging. Mikenorton (talk) 20:13, 23 May 2021 (UTC)

title: Talk:2017 Puebla earthquake


text:
Wikiversity:Motto contest/Round 3

These are the results of round 3 of the Motto contest.

Round 3
Please list your name by one motto and one slogan below.

In round 3, partly following the lead of the Wikiversity logo contest, similar mottos and slogans have been grouped together. This was done for one version in the group having support from at least two people. Some similar versions receiving support from at least one person are included (for a sense of breadth of interest).

To possibly create very good alternatives, new options similar in theme to existing ones are welcome, when signed with ones support. (Mottos and slogans with new themes can be added to the round 2 lists.)

Please share comments in the comment section that help explain the meaning and intent of mottos and slogans.

List of Mottos
The motto needs to be very short - for listing with logo. See wikimedia slogan/logo examples in the goals section above (and, if you wish, see Wikiversity logo contest proposals. Please list your support for one of the following:

"free learning" mottos

 * Because knowledge should be free. -- Support: Ross, Bounton, George D. Bozovic, RichMac, CSM, Kookykman, executivezen, Zerhynn,
 * Set Learning Free -- Support: stinkey91, Jimbobalina2005, Sam, Phaedrus86, Awolf002, CharlieCLC, Venkiteswaran,
 * Open Education -- Support: Aldenis, Roytales, samurainate, Necrocapo
 * Free thinking -- Support: Smurrayinchester,
 * Freedom of Learning -- Support: Mohan Gunti,
 * Learn Free, to be Free -- Support: nehamjoshi, Helios89,
 * Because thinking is free. -- Support: Jrothwell,
 * Education is freedom! Free education! -- Support: Paul Purcell,
 * Fast, Free, and Fun... The New Way to Learn -- Support: bendroz,
 * Free learning tools -- (used on Wikibooks main page with Wikiversity logo) -- Support: ?
 * Free learning materials and activities -- (used on Wikipedia main page with Wikiversity logo) -- Support: ?
 * The world's first free, open university -- (because we need to distinguish Wikiversity from other Wiki projects, all of which offer free knowledge, learning, and an open community). -- Support: anon.

"open/free learning community" mottos

 * An open learning community -- Support: Ron, Atfyfe, Dr-Now, Reswik, DouglasHolton, Dilaudid, Tohru,
 * Free learning community -- Support: Cormaggio, Smithgrrl, Hillgentleman, guillom, Taimz, sebmol

"three principles" mottos

 * Learn. Teach. Know. -- Support: Dev920, Rayc, Liondancer, Nazdude, Draicone (talk), Damien Black,
 * Read. Learn. Share. -- Support: Yazeed, ShAdIa, boba_girl, firass, amal, Scrappyjoe
 * We Learn, We Teach, We Build -- Support: Kiesa, Strathallen,
 * Learn. Teach. Share. -- Support: Talonhawk,
 * Read. Learn. Teach. -- Support: Mahdi7,
 * Citius, Altius, Fortius --Support: Ar.hhr,

"world wide wisdom"

 * World Wide Wisdom -- Support: TigreNoir, Halyks, E A Tarina, Mark Lewis,
 * Universe of Knowledge -- Support: Duarte.filho, Suto, mohanganeesh,
 * University anywhere, anytime -- Support: Sajid,

"free learning" mottos

 * Because knowledge should be free.
 * This one would be better if 'because' was dropped, 'because ' seems like we have to justify Wikiversity, "Knowledge should be free" is more powerful. CharlieCLC
 * Why even make it conditional? "Knowledge is free" is even more powerful. Pedro.Gonnet 09:33, 10 October 2006 (UTC)
 * Why not borrow from Shakespeare? "Thought is free" (The Tempest) --Lord Pheasant 03:05, 14 October 2006 (UTC)

 * Set Learning Free
 * Pithy and to the point. Although I like the "Three Nouns" approach, too, cleary we must have the word Free in the motto! Awolf002 00:00, 25 September 2006 (UTC)
 * I like the context free is used in, meaning available and accessible. CharlieCLC

 * Alternates (facilitation suggestions, based on various options above):
 * Free learning -- Support:
 * Open learning -- Support:
 * Free education -- Support:

 * Free learning tools
 * used on Wikibooks main page with Wikiversity logo -- selected by ?

 * Free learning materials and activities
 * used on Wikipedia with Wikiversity logo -- selected by ?

"three principles" mottos

 * Learn. Teach. Know.
 * I know the whole 'let's pretend we're an old European university by having a Latin motto' is pretty daft, but just in case anyone's interested, the Latin of the above could be  'cognosco. perdoceo. scio.' . --Sam 23:36, 12 September 2006 (UTC)

 * Read. Learn. Share.
 * Facilitation consideration: Any support for these syntheses of options (based on the list of mottos above)?
 * Freedom. Learning. Community. -- Support:
 * Openness. Learning. Community. -- Support:
 * Freedom. Education. Community. -- Support:
 * Openness. Education. Community. -- Support:
 * Freedom. Learning. Sharing. -- Support:
 * Openness. Learning. Sharing. -- Support:
 * Would I need to remove my vote from the origional motto in order to properly show my support for the synthesized motto? Talonhawk 03:43, 20 September 2006 (UTC)
 * Removing the original or strikethrough would work here (plus with your support, this one can go above). In round 3 here, I think it would be good to reorganize the layout a bit and create sets of similar mottos. Then, people could support a set and express preference for specific option(s) -- as in the logo contest for Wikiversity. --Reswik 14:35, 20 September 2006 (UTC)
 * Removing the original or strikethrough would work here (plus with your support, this one can go above). In round 3 here, I think it would be good to reorganize the layout a bit and create sets of similar mottos. Then, people could support a set and express preference for specific option(s) -- as in the logo contest for Wikiversity. --Reswik 14:35, 20 September 2006 (UTC)

 * We Learn, We Teach, We Build

 * Citius, Altius, Fortius
 * I'm not sure, but should this be 'Citius, altus, fortis' ('Quickly, noble, strong') -- could someone with more latin than me verify this please? --Sam 23:36, 12 September 2006 (UTC)
 * no, it's correct: "citius, altius, fortius" means "faster, higher, stronger", to represent the athlete's desire for continuous self-amelioration. --Nadira 10:23, 17 September 2006 (UTC)

 * Study A Little. Teach Alot. Change The World. --Adam 9:36, 22 October 2006

"open/free learning community" mottos

 * An open learning community
 * I switched my vote from "free learning community" to this for various reasons, including: 1) the appeal of the rich connotations of openness in the context of learning, in spite of my agreeing with the "free" side of debate in the politics of open source vs. free software; 2) the resonance with the slogan with most support below (so far) which uses "open."; 3) this offers our to-be-extensive project an opportunity to contribute (in discussions and on possible page(s) of explanation somewhere) to re-interpreting the implications of open, openness, and opening (in various senses such as development, awakening and liberation) in relation to the meaning of freedom in the free culture movement. Reswik 17:28, 12 September 2006 (UTC)

 * Free learning community

"world wide wisdom"

 * World Wide Wisdom
 * In Latin, it may be like this: "sapientis ubi gentium" or "sapiens ubi gentium" which means mostly "wisdom where in the world". Sblisesivdin 16:25, 7 October 2006 (UTC)

List of Slogans
This can be a short sentence or phrase - for listing at top of main page with "Welcome to Wikiversity". Please list your support by one option:

"opening minds..." slogan

 * Opening minds through open learning -- Support: Mos87, The Winged Self, Ross, Rayc, Dr-now, Liondancer@de, Reswik, Damien Black, Strathallen, Awolf002, CharlieCLC, AdamG

"learn and teach" slogans

 * Where teachers learn, and learners teach. -- Support: Deltinu, Jimbobalina2005, Dilaudid, Suto, Soahc_0
 * You can learn anything, teach anything, and know anything -- Support: Dev920, Nazdude,
 * Teach to Learn, Learn to Teach -- Support: OneWomanArmy, Bounton,
 * Because Learning Is Never Ending... -- Support: Juhi, Talonhawk,
 * Learning and teaching as one -- Support: D Platt,
 * Learn baby, learn! -- Lamarvelous 15:57, 20 October 2006 (UTC)

"free learning community " slogans

 * Free learning community -- Support: Cormaggio, Aldenis, George D. Bozovic, Hillgentleman, sebmol
 * the free learning network that you help create -- Support: Ron,

"freedom" and "free learning" slogans
Justice For All.
 * Free knowledge for everyone -- Support: Liju, Smithgrrl,
 * Reasonably free thinking, freely pursued. -- Support: Mirwin,
 * Join us for making Knowledge Valuable without Price -- Support: Neha, EulerGamma,
 * Learn free, think free. -- Support: stinkey91,
 * Universe of the free knowledge -- Support: Duarte.filho,
 * No need to earn to learn -- Support: Ajay,India,
 * Wikiversity - The Freedom of Development & Growth through Learning Support: Mohan Gunti,
 * Free, creative minds deserve free, creative learning experience -- Support: Saad Yousuf,
 * Education is freedom! Free education! -- Support: Paul Purcell,
 * People are the source of knowledge. This open forum, it's challenge. Support: --AaronG,

"wikiworms" slogan

 * All Wikiworms are welcomed! -- Support: Yazeed, ShAdIa, boba_girl,

"strength through wisdom..." slogan

 * Strength through wisdom, wisdom through study. -- Support: Nikola Enchev, Boris Hristov, Sam,

"opening minds..." slogan

 * Opening minds through open learning
 * See my comment about "openness" under "an open learning community" in the motto comment section. Reswik 15:57, 12 September 2006 (UTC)

The Revolution of Learning Has Begun.

"learn and teach" slogans

 * Where teachers learn, and learners teach.

 * You can learn anything, teach anything, and know anything

 * Teach to Learn, Learn to Teach

 * Because Learning Is Never Ending...

"free learning community " slogans

 * Free learning community
 * It certainly is simple, straight forward, and accurate. If nothing else better comes up, why not stick with simplisty? --  Zanimum 16:41, 29 September 2006 (UTC)

"freedom" and "free learning" slogans

 * Free knowledge for everyone

 * Join us for making Knowledge Valuable without Price

 * People are the source of knowledge. This open forum, it's challenge.
 * ...There is one BIG threat to this web site that is common to all open forum Wiki's; achieving credibility. This motto is my suggestion because one many levels there is a need for truth and knowledge. This open forum, in my opinion, will be it's ultimate test as laymen and academics confront information together in the wonderful project.--AaronG

"wikiworms" slogan

 * All Wikiworms are welcomed!
 * Um, no. --  Zanimum 16:39, 29 September 2006 (UTC)

"strength through wisdom..." slogan

 * Strength through wisdom, wisdom through study
 * The idea behind this is not solely from "V for Vendetta" :), but rather my own country's (Bulgaria) slogan: "Strength through Unity". Since I'm more a cosmopolitain than a nationalist, I think that "Wisdom through study" is entirely based on worldwide progress through enriching knowledge, which basically my own idea on development of humanity as a whole. Nikola Enchev 13.09.'06.

title: Wikiversity:Motto contest/Round 3


text:
Talk:Acts Retirement-Life Communities

External links modified
Hello fellow Wikipedians,

I have just modified one external link on ACTS Retirement-Life Communities. Please take a moment to review my edit. If you have any questions, or need the bot to ignore the links, or the page altogether, please visit this simple FaQ for additional information. I made the following changes:
 * Added archive https://web.archive.org/web/20080123230529/http://www.helpguide.org/elder/nursing_homes_skilled_nursing_facilities.htm to http://www.helpguide.org/elder/nursing_homes_skilled_nursing_facilities.htm

Cheers.— InternetArchiveBot  (Report bug) 22:27, 23 June 2017 (UTC)

title: Talk:Acts Retirement-Life Communities


text:
Calculus on Euclidean space

In mathematics, calculus on Euclidean space is a generalization of calculus of functions in one or several variables to calculus of functions on Euclidean space $$\mathbb{R}^n$$ as well as a finite-dimensional real vector space. This calculus is also known as advanced calculus, especially in the United States. It is similar to multivariable calculus but is somewhat more sophisticated in that it uses linear algebra (or some functional analysis) more extensively and covers some concepts from differential geometry such as differential forms and Stokes' formula in terms of differential forms. This extensive use of linear algebra also allows a natural generalization of multivariable calculus to calculus on Banach spaces or topological vector spaces.

Calculus on Euclidean space is also a local model of calculus on manifolds, a theory of functions on manifolds.

Functions in one real variable
This section is a brief review of function theory in one-variable calculus.

A real-valued function $$f : \mathbb{R} \to \mathbb{R}$$ is continuous at $$a$$ if it is approximately constant near $$a$$; i.e.,
 * $$\lim_{h \to 0} (f(a + h) - f(a)) = 0.$$

In contrast, the function $$f$$ is differentiable at $$a$$ if it is approximately linear near $$a$$; i.e., there is some real number $$\lambda$$ such that
 * $$\lim_{h \to 0} \frac{f(a + h) - f(a) - \lambda h}{h} = 0.$$

(For simplicity, suppose $$f(a) = 0$$. Then the above means that $$f(a + h) = \lambda h + g(a, h)$$ where $$g(a, h)$$ goes to 0 faster than h going to 0 and, in that sense, $$f(a + h)$$ behaves like $$\lambda h$$.)

The number $$\lambda$$ depends on $$a$$ and thus is denoted as $$f'(a)$$. If $$f$$ is differentiable on an open interval $$U$$ and if $$f'$$ is a continuous function on $$U$$, then $$f$$ is called a C1 function. More generally, $$f$$ is called a Ck function if its derivative $$f'$$ is Ck-1 function. Taylor's theorem states that a Ck function is precisely a function that can be approximated by a polynomial of degree k.

If $$f : \mathbb{R} \to \mathbb{R}$$ is a C1 function and $$f'(a) \ne 0$$ for some $$a$$, then either $$f'(a) > 0$$ or $$f'(a) < 0$$; i.e., either $$f$$ is strictly increasing or strictly decreasing in some open interval containing a. In particular, $$f : f^{-1}(U) \to U$$ is bijective for some open interval $$U$$ containing $$f(a)$$. The inverse function theorem then says that the inverse function $$f^{-1}$$ is differentiable on U with the derivatives: for $$y \in U$$
 * $$(f^{-1})'(y) = {1 \over f'(f^{-1}(y))}.$$

Derivative of a map and chain rule
For functions $$f$$ defined in the plane or more generally on an Euclidean space $$\mathbb{R}^n$$, it is necessary to consider functions that are vector-valued or matrix-valued. It is also conceptually helpful to do this in an invariant manner (i.e., a coordinate-free way). Derivatives of such maps at a point are then vectors or linear maps, not real numbers.

Let $$f : X \to Y$$ be a map from an open subset $$X$$ of $$\mathbb{R}^n$$ to an open subset $$Y$$ of $$\mathbb{R}^m$$. Then the map $$f$$ is said to be differentiable at a point $$x$$ in $$X$$ if there exists a (necessarily unique) linear transformation $$f'(x) : \mathbb{R}^n \to \mathbb{R}^m$$, called the derivative of $$f$$ at $$x$$, such that
 * $$\lim_{ h \to 0 } \frac{1}{|h|} |f(x + h) - f(x) - f'(x)h| = 0$$

where $$f'(x)h$$ is the application of the linear transformation $$f'(x)$$ to $$h$$. If $$f$$ is differentiable at $$x$$, then it is continuous at $$x$$ since
 * $$|f(x + h) - f(x)| \le (|h|^{-1}|f(x + h) - f(x) - f'(x)h|) |h| + |f'(x)h| \to 0$$ as $$h \to 0$$.

As in the one-variable case, there is

This is proved exactly as for functions in one variable. Indeed, with the notation $$\widetilde{h} = f(x + h) - f(x)$$, we have:
 * $$\begin{align}

& \frac{1}{|h|} |g(f(x + h)) - g(y) - g'(y) f'(x) h| \\ & \le \frac{1}{|h|}  |g(y + \widetilde{h}) - g(y) - g'(y)\widetilde{h}| +  \frac{1}{|h|} |g'(y)(f(x+h) - f(x) - f'(x) h)|. \end{align}$$ Here, since $$f$$ is differentiable at $$x$$, the second term on the right goes to zero as $$h \to 0$$. As for the first term, it can be written as:
 * $$\begin{cases}

\frac{|\widetilde{h}|}{|h|} |g(y+ \widetilde{h}) - g(y) - g'(y)\widetilde{h}|/|\widetilde{h}|, & \widetilde{h} \neq 0, \\ 0, & \widetilde{h} = 0. \end{cases}$$ Now, by the argument showing the continuity of $$f$$ at $$x$$, we see $$\frac{|\widetilde{h}|}{|h|}$$ is bounded. Also, $$\widetilde{h} \to 0$$ as $$h \to 0$$ since $$f$$ is continuous at $$x$$. Hence, the first term also goes to zero as $$h \to 0$$ by the differentiability of $$g$$ at $$y$$. $$\square$$

The map $$f$$ as above is called continuously differentiable or $$C^1$$ if it is differentiable on the domain and also the derivatives vary continuously; i.e., $$x \mapsto f'(x)$$ is continuous.

As a linear transformation, $$f'(x)$$ is represented by an $$m \times n$$-matrix, called the Jacobian matrix $$Jf(x)$$ of $$f$$ at $$x$$ and we write it as:
 * $$(Jf)(x) = \begin{bmatrix}

\frac{\partial f_1}{\partial x_1}(x) & \cdots & \frac{\partial f_1}{\partial x_n}(x) \\ \vdots & \ddots & \vdots \\ \frac{\partial f_m}{\partial x_1}(x) & \cdots & \frac{\partial f_m}{\partial x_n}(x) \end{bmatrix}.$$

Taking $$h$$ to be $$h e_j$$, $$h$$ a real number and $$e_j = (0, \cdots, 1, \cdots, 0)$$ the j-th standard basis element, we see that the differentiability of $$f$$ at $$x$$ implies:
 * $$\lim_{h \to 0} \frac{f_i(x + h e_j) - f_i(x)}{h} = \frac{\partial f_i}{\partial x_j}(x)$$

where $$f_i$$ denotes the i-th component of $$f$$. That is, each component of $$f$$ is differentiable at $$x$$ in each variable with the derivative $$\frac{\partial f_i}{\partial x_j}(x)$$. In terms of Jacobian matrices, the chain rule says $$J(g \circ f)(x) = Jg(y) Jf(x)$$; i.e., as $$(g \circ f)_i = g_i \circ f$$,
 * $$\frac{\partial (g_i \circ f)}{\partial x_j}(x) = \frac{\partial g_i}{\partial y_1} (y) \frac{\partial f_1}{\partial x_j}(x) + \cdots + \frac{\partial g_i}{\partial y_m} (y) \frac{\partial f_m}{\partial x_j}(x),$$

which is the form of the chain rule that is often stated.

A partial converse to the above holds. Namely, if the partial derivatives $${\partial f_i}/{\partial x_j}$$ are all defined and continuous, then $$f$$ is continuously differentiable. This is a consequence of the mean value inequality:

(This version of mean value inequality follows from mean value inequality in applied to the function $$[0, 1] \to \mathbb{R}^m, \, t \mapsto f(x + ty) - tv$$, where the proof on mean value inequality is given.)

Indeed, let $$g(x) = (Jf)(x)$$. We note that, if $$y = y_i e_i$$, then
 * $$\frac{d}{dt}f(x + ty) = \frac{\partial f}{\partial x_i}(x+ty)y = g(x + ty)(y_i e_i).$$

For simplicity, assume $$n = 2$$ (the argument for the general case is similar). Then, by mean value inequality, with the operator norm $$\| \cdot \|$$,
 * $$\begin{align}

&|\Delta_y f (x) - g(x)y| \\ &\le |\Delta_{y_1 e_1} f(x_1, x_2 + y_2) - g(x)(y_1 e_1)| + |\Delta_{y_2 e_2} f(x_1, x_2) - g(x)(y_2 e_2)| \\ &\le |y_1| \sup_{0 < t < 1}\|g(x_1 + t y_1, x_2 + y_2) - g(x)\| + |y_2| \sup_{0 < t < 1}\|g(x_1, x_2 + ty_2) - g(x)\|, \end{align} $$ which implies $$|\Delta_y f (x) - g(x)y|/|y| \to 0$$ as required. $$\square$$

Example: Let $$U$$ be the set of all invertible real square matrices of size n. Note $$U$$ can be identified as an open subset of $$\mathbb{R}^{n^2}$$ with coordinates $$x_{ij}, 0 \le i, j \ne n$$. Consider the function $$f(g) = g^{-1}$$ = the inverse matrix of $$g$$ defined on $$U$$. To guess its derivatives, assume $$f$$ is differentiable and consider the curve $$c(t) = ge^{tg^{-1}h}$$ where $$e^A$$ means the matrix exponential of $$A$$. By the chain rule applied to $$f(c(t)) = e^{-t g^{-1}h} g^{-1} $$, we have:
 * $$f'(c(t)) \circ c'(t) = -g^{-1}h e^{-t g^{-1}h} g^{-1}$$.

Taking $$t = 0$$, we get:
 * $$f'(g) h = -g^{-1}h g^{-1}$$.

Now, we then have:
 * $$\|(g+h)^{-1} - g^{-1} + g^{-1}h g^{-1}\| \le \| (g+h)^{-1} \| \|h\| \|g^{-1} h g^{-1}\|.$$

Since the operator norm is equivalent to the Euclidean norm on $$\mathbb{R}^{n^2}$$ (any norms are equivalent to each other), this implies $$f$$ is differentiable. Finally, from the formula for $$f'$$, we see the partial derivatives of $$f$$ are smooth (infinitely differentiable); whence, $$f$$ is smooth too.

Higher derivatives and Taylor formula
If $$f : X \to \mathbb{R}^m$$ is differentiable where $$X \subset \mathbb{R}^n$$ is an open subset, then the derivatives determine the map $$f' : X \to \operatorname{Hom}(\mathbb{R}^n, \mathbb{R}^m)$$, where $$\operatorname{Hom}$$ stands for homomorphisms between vector spaces; i.e., linear maps. If $$f'$$ is differentiable, then $$f'' : X \to \operatorname{Hom}(\mathbb{R}^n, \operatorname{Hom}(\mathbb{R}^n, \mathbb{R}^m))$$. Here, the codomain of $$f''$$ can be identified with the space of bilinear maps by:
 * $$\operatorname{Hom}(\mathbb{R}^n, \operatorname{Hom}(\mathbb{R}^n, \mathbb{R}^m)) \overset{\varphi}\underset{\sim}\to \{ (\mathbb{R}^n)^2 \to \mathbb{R}^m \text{ bilinear}\}$$

where $$\varphi(g)(x, y) = g(x)y$$ and $$\varphi$$ is bijective with the inverse $$\psi$$ given by $$(\psi(g)x)y = g(x, y)$$. In general, $$f^{(k)} = (f^{(k-1)})'$$ is a map from $$X$$ to the space of $$k$$-multilinear maps $$(\mathbb{R}^n)^k \to \mathbb{R}^m$$.

Just as $$f'(x)$$ is represented by a matrix (Jacobian matrix), when $$m = 1$$ (a bilinear map is a bilinear form), the bilinear form $$f(x)$$ is represented by a matrix called the Hessian matrix of $$f$$ at $$x$$; namely, the square matrix $$H$$ of size $$n$$ such that $$f(x)(y, z) = (Hy, z)$$, where the paring refers to an inner product of $$\mathbb{R}^n$$, and $$H$$ is none other than the Jacobian matrix of $$f' : X \to (\mathbb{R}^n)^* \simeq \mathbb{R}^n$$. The $$(i, j)$$-th entry of $$H$$ is thus given explicitly as $$H_{ij} = \frac{\partial^2 f}{\partial x_i \partial x_j}(x)$$.

Moreover, if $$f''$$ exists and is continuous, then the matrix $$H$$ is symmetric, the fact known as the symmetry of second derivatives. This is seen using the mean value inequality. For vectors $$u, v$$ in $$\mathbb{R}^n$$, using mean value inequality twice, we have:
 * $$|\Delta_v \Delta_u f(x) - f(x)(u, v)| \le \sup_{0 < t_1, t_2 < 1} | f(x + t_1 u + t_2 v)(u, v) - f''(x)(u, v) |,$$

which says
 * $$f''(x)(u, v) = \lim_{s, t \to 0} (\Delta_{tv} \Delta_{su} f(x) - f(x))/(st).$$

Since the right-hand side is symmetric in $$u, v$$, so is the left-hand side: $$f(x)(u, v) = f(x)(v, u)$$. By induction, if $$f$$ is $$C^k$$, then the k-multilinear map $$f^{(k)}(x)$$ is symmetric; i.e., the order of taking partial derivatives does not matter.

As in the case of one variable, the Taylor series expansion can then be proved by integration by parts:

 * $$f(z+(h,k))=\sum_{a+b<n} \partial_x^a\partial_y^b f(z){h^a k^b\over a! b!} + n\int_0^1 (1-t)^{n-1} \sum_{a+b=n} \partial_x^a\partial_y^b f(z+t(h,k)){h^a k^b\over a! b!} \, dt.$$

Taylor's formula has an effect of dividing a function by variables, which can be illustrated by the next typical theoretical use of the formula.

Example: Let $$T : \mathcal{S} \to \mathcal{S}$$ be a linear map between the vector space $$\mathcal{S}$$ of smooth functions on $$\mathbb{R}^n$$ with rapidly decreasing derivatives; i.e., $$\sup |x^{\beta} \partial^{\alpha} \varphi| < \infty$$ for any multi-index $$\alpha, \beta$$. (The space $$\mathcal{S}$$ is called a Schwartz space.) For each $$\varphi$$ in $$\mathcal{S}$$, Taylor's formula implies we can write:
 * $$\varphi - \psi \varphi(y) = \sum_{j=1}^n (x_j - y_j) \varphi_j$$

with $$\varphi_j \in \mathcal{S}$$, where $$\psi$$ is a smooth function with compact support and $$\psi(y) = 1$$. Now, assume $$T$$ commutes with coordinates; i.e., $$T (x_j \varphi) = x_j T\varphi$$. Then
 * $$T\varphi - \varphi(y) T\psi = \sum_{j=1}^n (x_j - y_j) T\varphi_j$$.

Evaluating the above at $$y$$, we get $$T\varphi(y) = \varphi(y) T\psi(y).$$ In other words, $$T$$ is a multiplication by some function $$m$$; i.e., $$T\varphi = m \varphi$$. Now, assume further that $$T$$ commutes with partial differentiations. We then easily see that $$m$$ is a constant; $$T$$ is a multiplication by a constant.

(Aside: the above discussion almost proves the Fourier inversion formula. Indeed, let $$F, R : \mathcal{S} \to \mathcal{S}$$ be the Fourier transform and the reflection; i.e., $$(R \varphi)(x) = \varphi(-x)$$. Then, dealing directly with the integral that is involved, one can see $$T = RF^2$$ commutes with coordinates and partial differentiations; hence, $$T$$ is a multiplication by a constant. This is almost a proof since one still has to compute this constant.)

A partial converse to the Taylor formula also holds; see Borel's lemma and Whitney extension theorem.

Inverse function theorem and submersion theorem
A $$C^k$$-map with the $$C^k$$-inverse is called a $$C^k$$-diffeomorphism. Thus, the theorem says that, for a map $$f$$ satisfying the hypothesis at a point $$x$$, $$f$$ is a diffeomorphism near $$x, f(x).$$ For a proof, see.

The implicit function theorem says: given a map $$f : \mathbb{R}^n \times \mathbb{R}^m \to \mathbb{R}^m$$, if $$f(a, b) = 0$$, $$f$$ is $$C^k$$ in a neighborhood of $$(a, b)$$ and the derivative of $$y \mapsto f(a, y)$$ at $$b$$ is invertible, then there exists a differentiable map $$g : U \to V$$ for some neighborhoods $$U, V$$ of $$a, b$$ such that $$f(x, g(x)) = 0$$. The theorem follows from the inverse function theorem; see.

Another consequence is the submersion theorem.

Integrable functions on Euclidean spaces
A partition of an interval $$[a, b]$$ is a finite sequence $$a = t_0 \le t_1 \le \cdots \le t_k = b$$. A partition $$P$$ of a rectangle $$D$$ (product of intervals) in $$\mathbb{R}^n$$ then consists of partitions of the sides of $$D$$; i.e., if $$D = \prod_1^n [a_i, b_i]$$, then $$P$$ consists of $$P_1, \dots, P_n$$ such that $$P_i$$ is a partition of $$[a_i, b_i]$$.

Given a function $$f$$ on $$D$$, we then define the upper Riemann sum of it as:
 * $$U(f, P) = \sum_{Q \in P} (\sup_Q f) \operatorname{vol}(Q)$$

where The lower Riemann sum $$L(f, P)$$ of $$f$$ is then defined by replacing $$\sup$$ by $$\inf$$. Finally, the function $$f$$ is called integrable if it is bounded and $$\sup \{ L(f, P) \mid P \} = \inf \{ U(f, P) \mid P \}$$. In that case, the common value is denoted as $$\int_D f \, dx$$.
 * $$Q$$ is a partition element of $$P$$; i.e., $$Q = \prod_{i = 1}^n [t_{i, j_i}, t_{i, j_i+1}]$$ when $$P_i : a_i = t_{i, 0} \le \dots \cdots \le t_{i, k_i} = b_i$$ is a partition of $$[a_i, b_i]$$.
 * The volume $$\operatorname{vol}(Q)$$ of $$Q$$ is the usual Euclidean volume; i.e., $$\operatorname{vol}(Q) = \prod_1^n (t_{i, j_i+1} - t_{i, j_i})$$.

A subset of $$\mathbb{R}^n$$ is said to have measure zero if for each $$\epsilon > 0$$, there are some possibly infinitely many rectangles $$D_1, D_2, \dots, $$ whose union contains the set and $$\sum_i \operatorname{vol}(D_i) < \epsilon.$$

A key theorem is

The next theorem allows us to compute the integral of a function as the iteration of the integrals of the function in one-variables:

In particular, the order of integrations can be changed.

Finally, if $$M \subset \mathbb{R}^n$$ is a bounded open subset and $$f$$ a function on $$M$$, then we define $$\int_M f \, dx := \int_D \chi_M f \, dx$$ where $$D$$ is a closed rectangle containing $$M$$ and $$\chi_M$$ is the characteristic function on $$M$$; i.e., $$\chi_M(x) = 1$$ if $$x \in M$$ and $$=0$$ if $$x \not\in M,$$ provided $$\chi_M f$$ is integrable.

Surface integral
If a bounded surface $$M$$ in $$\mathbb{R}^3$$ is parametrized by $$\textbf{r} = \textbf{r}(u, v)$$ with domain $$D$$, then the surface integral of a measurable function $$F$$ on $$M$$ is defined and denoted as:
 * $$\int_M F \, dS := \int \int_D (F \circ \textbf{r}) | \textbf{r}_u \times \textbf{r}_v | \, du dv$$

If $$F : M \to \mathbb{R}^3$$ is vector-valued, then we define
 * $$\int_M F \cdot dS := \int_M (F \cdot \textbf{n}) \, dS$$

where $$\textbf{n}$$ is an outward unit normal vector to $$M$$. Since $$\textbf{n} = \frac{\textbf{r}_u \times \textbf{r}_v}{|\textbf{r}_u \times \textbf{r}_v|}$$, we have:
 * $$\int_M F \cdot dS = \int \int_D (F \circ \textbf{r}) \cdot (\textbf{r}_u \times \textbf{r}_v) \, du dv = \int \int_D \det(F \circ \textbf{r}, \textbf{r}_u, \textbf{r}_v) \, dudv.$$

Tangent vectors and vector fields
Let $$c : [0, 1] \to \mathbb{R}^n$$ be a differentiable curve. Then the tangent vector to the curve $$c$$ at $$t$$ is a vector $$v$$ at the point $$c(t)$$ whose components are given as:
 * $$v = (c_1'(t), \dots, c_n'(t))$$.

For example, if $$c(t) = (a \cos(t), a \sin(t), bt), a > 0, b > 0$$ is a helix, then the tangent vector at t is:
 * $$c'(t) = (-a \sin(t), a \cos(t), b).$$

It corresponds to the intuition that the a point on the helix moves up in a constant speed.

If $$M \subset \mathbb{R}^n$$ is a differentiable curve or surface, then the tangent space to $$M$$ at a point p is the set of all tangent vectors to the differentiable curves $$c: [0, 1] \to M$$ with $$c(0) = p$$.

A vector field X is an assignment to each point p in M a tangent vector $$X_p$$ to M at p such that the assignment varies smoothly.

Differential forms
The dual notion of a vector field is a differential form. Given an open subset $$M$$ in $$\mathbb{R}^n$$, by definition, a differential 1-form (often just 1-form) $$\omega$$ is an assignment to a point $$p$$ in $$M$$ a linear functional $$\omega_p$$ on the tangent space $$T_p M$$ to $$M$$ at $$p$$ such that the assignment varies smoothly. For a (real or complex-valued) smooth function $$f$$, define the 1-form $$df$$ by: for a tangent vector $$v$$ at $$p$$,
 * $$df_p(v) = v(f)$$

where $$v(f)$$ denotes the directional derivative of $$f$$ in the direction $$v$$ at $$p$$. For example, if $$x_i$$ is the $$i$$-th coordinate function, then $$dx_{i, p}(v) = v_i$$; i.e., $$dx_{i,p}$$ are the dual basis to the standard basis on $$T_p M$$. Then every differential 1-form $$\omega$$ can be written uniquely as
 * $$\omega = f_1 \, dx_1 + \cdots + f_n \, dx_n$$

for some smooth functions $$f_1, \dots, f_n$$ on $$M$$ (since, for every point $$p$$, the linear functional $$\omega_p$$ is a unique linear combination of $$dx_i$$ over real numbers). More generally, a differential k-form is an assignment to a point $$p$$ in $$M$$ a vector $$\omega_p$$ in the $$k$$-th exterior power $$\bigwedge^k T^*_p M$$ of the dual space $$T^*_p M$$ of $$T_p M$$ such that the assignment varies smoothly. In particular, a 0-form is the same as a smooth function. Also, any $$k$$-form $$\omega$$ can be written uniquely as:
 * $$\omega = \sum_{i_1 < \cdots < i_k} f_{i_1 \dots i_k} \, dx_{i_1} \wedge \cdots \wedge dx_{i_k}$$

for some smooth functions $$f_{i_1 \dots i_k}$$.

Like a smooth function, we can differentiate and integrate differential forms. If $$f$$ is a smooth function, then $$df$$ can be written as:
 * $$df = \sum_{i=1}^n \frac{\partial f}{\partial x_i} \, dx_i$$

since, for $$v = \partial / \partial x_j |_p$$, we have: $$df_p(v) = \frac{\partial f}{\partial x_j}(p) = \sum_{i=1}^n \frac{\partial f}{\partial x_i}(p) \, dx_i(v)$$. Note that, in the above expression, the left-hand side (whence the right-hand side) is independent of coordinates $$x_1, \dots, x_n$$; this property is called the invariance of differential.

The operation $$d$$ is called the exterior derivative and it extends to any differential forms inductively by the requirement (Leibniz rule)
 * $$d(\alpha \wedge \beta) = d \alpha \wedge \beta + (-1)^p \alpha \wedge d \beta.$$

where $$\alpha, \beta$$ are a p-form and a q-form.

The exterior derivative has the important property that $$d \circ d = 0$$; that is, the exterior derivative $$d$$ of a differential form $$d \omega$$ is zero. This property is a consequence of the symmetry of second derivatives (mixed partials are equal).

Boundary and orientation
A circle can be oriented clockwise or counterclockwise. Mathematically, we say that a subset $$M$$ of $$\mathbb{R}^n$$ is oriented if there is a consistent choice of normal vectors to $$M$$ that varies continuously. For example, a circle or, more generally, an n-sphere can be oriented; i.e., orientable. On the other hand, a Möbius strip (a surface obtained by identified by two opposite sides of the rectangle in a twisted way) cannot oriented: if we start with a normal vector and travel around the strip, the normal vector at end will point to the opposite direction.

The proposition is useful because it allows us to give an orientation by giving a volume form.

Integration of differential forms
If $$\omega = f \, dx_1 \wedge \cdots \wedge dx_n$$ is a differential n-form on an open subset M in $$\mathbb{R}^n$$ (any n-form is that form), then the integration of it over $$M$$ with the standard orientation is defined as:
 * $$\int_M \omega = \int_M f \, dx_1 \cdots dx_n.$$

If M is given the orientation opposite to the standard one, then $$\int_M \omega$$ is defined as the negative of the right-hand side.

Then we have the fundamental formula relating exterior derivative and integration:

Here is a sketch of proof of the formula. If $$f$$ is a smooth function on $$\mathbb{R}^n$$ with compact support, then we have:
 * $$\int d(f \omega) = 0$$

(since, by the fundamental theorem of calculus, the above can be evaluated on boundaries of the set containing the support.) On the other hand,
 * $$\int d(f \omega) = \int df \wedge \omega + \int f \, d\omega.$$

Let $$f$$ approach the characteristic function on $$M$$. Then the second term on the right goes to $$\int_M d \omega$$ while the first goes to $$-\int_{\partial M} \omega$$, by the argument similar to proving the fundamental theorem of calculus. $$\square$$

The formula generalizes the fundamental theorem of calculus as well as Stokes' theorem in multivariable calculus. Indeed, if $$M = [a, b]$$ is an interval and $$\omega = f$$, then $$d\omega = f' \, dx$$ and the formula says:
 * $$\int_M f' \, dx = f(b) - f(a)$$.

Similarly, if $$M$$ is an oriented bounded surface in $$\mathbb{R}^3$$ and $$\omega = f\,dx + g\,dy + h\,dz$$, then $$d(f\,dx) = df \wedge dx = \frac{\partial f}{\partial y} \, dy \wedge dx + \frac{\partial f}{\partial z} \,dz \wedge dx$$ and similarly for $$d(g\,dy)$$ and $$d(g\,dy)$$. Collecting the terms, we thus get:
 * $$d\omega = \left( \frac{\partial h}{\partial y} - \frac{\partial g}{\partial z} \right) dy \wedge dz + \left( \frac{\partial f}{\partial z} - \frac{\partial h}{\partial x} \right) dz \wedge dx + \left( \frac{\partial g}{\partial x} - \frac{\partial f}{\partial y} \right) dx \wedge dy.$$

Then, from the definition of the integration of $$\omega$$, we have $$\int_M d \omega = \int_M (\nabla \times F) \cdot dS$$ where $$F = (f, g, h)$$ is the vector-valued function and $$\nabla = \left( \frac{\partial}{\partial x}, \frac{\partial}{\partial y}, \frac{\partial}{\partial z} \right)$$. Hence, Stokes’ formula becomes
 * $$\int_M (\nabla \times F) \cdot dS = \int_{\partial M} (f\,dx + g\,dy + h\,dz),$$

which is the usual form of the Stokes' theorem on surfaces. Green’s theorem is also a special case of Stokes’ formula.

Stokes' formula also yields a general version of Cauchy's integral formula. To state and prove it, for the complex variable $$z = x + iy$$ and the conjugate $$\bar z$$, let us introduce the operators
 * $$\frac{\partial}{\partial z} = \frac{1}{2}\left( \frac{\partial}{\partial x} - i \frac{\partial}{\partial y} \right), \, \frac{\partial}{\partial \bar{z}} = \frac{1}{2}\left( \frac{\partial}{\partial x} + i \frac{\partial}{\partial y} \right).$$

In these notations, a function $$f$$ is holomorphic (complex-analytic) if and only if $$\frac{\partial f}{\partial \bar z} = 0$$ (the Cauchy–Riemann equations). Also, we have:
 * $$df = \frac{\partial f}{\partial z}dz + \frac{\partial f}{\partial \bar{z}}d \bar{z}.$$

Let $$D_{\epsilon} = \{ z \in \mathbb{C} \mid \epsilon < |z - z_0| < r \}$$ be a punctured disk with center $$z_0$$. Since $$1/(z - z_0)$$ is holomorphic on $$D_{\epsilon}$$, We have:
 * $$d \left( \frac{f}{z-z_0} dz \right) = \frac{\partial f}{\partial \bar z} \frac{d \bar{z} \wedge dz}{z - z_0} $$.

By Stokes’ formula,
 * $$\int_{D_{\epsilon}} \frac{\partial f}{\partial \bar z} \frac{d \bar{z} \wedge dz}{z - z_0} = \left( \int_{|z - z_0| = r} - \int_{|z - z_0| = \epsilon} \right) \frac{f}{z-z_0} dz.$$

Letting $$\epsilon \to 0$$ we then get:
 * $$2\pi i \, f(z_0) = \int_{|z - z_0| = r} \frac{f}{z-z_0} dz + \int_{|z - z_0| \le r} \frac{\partial f}{\partial \bar z} \frac{dz \wedge d \bar z}{z - z_0}.$$

Winding numbers and Poincaré lemma
A differential form $$\omega$$ is called closed if $$d\omega = 0$$ and is called exact if $$\omega = d\eta$$ for some differential form $$\eta$$ (often called a potential). Since $$d \circ d = 0$$, an exact form is closed. But the converse does not hold in general; there might be a non-exact closed form. A classic example of such a form is:
 * $$\omega = \frac{-y}{x^2 + y^2} + \frac{x}{x^2 + y^2}$$,

which is a differential form on $$\mathbb{R}^2 - 0$$. Suppose we switch to polar coordinates: $$x = r \cos \theta, y = r \sin \theta$$ where $$ r = \sqrt{x^2 + y^2}$$. Then
 * $$\omega = r^{-2}(-r \sin \theta \, dx + r \cos \theta \, dy) = d \theta.$$

This does not show that $$\omega$$ is exact: the trouble is that $$\theta$$ is not a well-defined continuous function on $$\mathbb{R}^2 - 0$$. Since any function $$f$$ on $$\mathbb{R}^2 - 0$$ with $$df = \omega$$ differ from $$\theta$$ by constant, this means that $$\omega$$ is not exact. The calculation, however, shows that $$\omega$$ is exact, for example, on $$\mathbb{R}^2 - \{ x = 0 \}$$ since we can take $$\theta = \arctan(y/x)$$ there.

There is a result (Poincaré lemma) that gives a condition that guarantees closed forms are exact. To state it, we need some notions from topology. Given two continuous maps $$f, g : X \to Y$$ between subsets of $$\mathbb{R}^m, \mathbb{R}^n$$ (or more generally topological spaces), a homotopy from $$f$$ to $$g$$ is a continuous function $$H : X \times [0, 1] \to Y$$ such that $$f(x) = H(x, 0)$$ and $$g(x) = H(x, 1)$$. Intuitively, a homotopy is a continuous variation of one function to another. A loop in a set $$X$$ is a curve whose starting point coincides with the end point; i.e., $$c : [0, 1] \to X$$ such that $$c(0) = c(1)$$. Then a subset of $$\mathbb{R}^n$$ is called simply connected if every loop is homotopic to a constant function. A typical example of a simply connected set is a disk $$D = \{ (x, y) \mid \sqrt{x^2 + y^2} \le r \} \subset \mathbb{R}^2$$. Indeed, given a loop $$c : [0, 1] \to D$$, we have the homotopy $$H : [0, 1]^2 \to D, \, H(x, t) = (1-t) c(x) + t c(0)$$ from $$c$$ to the constant function $$c(0)$$. A punctured disk, on the other hand, is not simply connected.

Moving frame
Vector fields $$E_1, \dots, E_3$$ on $$\mathbb{R}^3$$ are called a frame field if they are orthogonal to each other at each point; i.e., $$E_i \cdot E_j = \delta_{ij}$$ at each point. The basic example is the standard frame $$U_i$$; i.e., $$U_i(x)$$ is a standard basis for each point $$x$$ in $$\mathbb{R}^3$$. Another example is the cylindrical frame
 * $$E_1 = \cos \theta U_1 + \sin \theta U_2, \, E_2 = -\sin \theta U_1 + \cos \theta U_2, \, E_3 = U_3.$$

For the study of the geometry of a curve, the important frame to use is a Frenet frame $$T, N, B$$ on a unit-speed curve $$\beta : I \to \mathbb{R}^3$$ given as:

The Gauss–Bonnet theorem
The Gauss–Bonnet theorem relates the topology of a surface and its geometry.

Method of Lagrange multiplier
The set $$g^{-1}(0)$$ is usually called a constraint.

Example: Suppose we want to find the minimum distance between the circle $$x^2 + y^2 = 1$$ and the line $$x + y = 4$$. That means that we want to minimize the function $$f(x, y, u, v) = (x - u)^2 + (y - v)^2$$, the square distance between a point $$(x, y)$$ on the circle and a point $$(u, v)$$ on the line, under the constraint $$g = (x^2 + y^2 - 1, u + v - 4)$$. We have:
 * $$\nabla f = (2(x - u), 2(y - v), -2(x - u), -2(y - v)).$$
 * $$\nabla g_1 = (2x, 2y, 0, 0), \nabla g_2 = (0, 0, 1, 1).$$

Since the Jacobian matrix of $$g$$ has rank 2 everywhere on $$g^{-1}(0)$$, the Lagrange multiplier gives:
 * $$x - u = \lambda_1 x, \, y - v = \lambda_1 y, \, 2(x-u) = -\lambda_2, \, 2(y-v) = -\lambda_2.$$

If $$\lambda_1 = 0$$, then $$x = u, y = v$$, not possible. Thus, $$\lambda_1 \ne 0$$ and
 * $$x = \frac{x-u}{\lambda_1}, \, y = \frac{y-v}{\lambda_1}.$$

From this, it easily follows that $$x = y = 1/\sqrt{2}$$ and $$u = v = 2$$. Hence, the minimum distance is $$2\sqrt{2} - 1$$ (as a minimum distance clearly exists).

Here is an application to linear algebra. Let $$V$$ be a finite-dimensional real vector space and $$T : V \to V$$ a self-adjoint operator. We shall show $$V$$ has a basis consisting of eigenvectors of $$T$$ (i.e., $$T$$ is diagonalizable) by induction on the dimension of $$V$$. Choosing a basis on $$V$$ we can identify $$V = \mathbb{R}^n$$ and $$T$$ is represented by the matrix $$[a_{ij}]$$. Consider the function $$f(x) = (Tx, x)$$, where the bracket means the inner product. Then $$\nabla f = 2(\sum a_{1i} x_i, \dots, \sum a_{ni} x_i)$$. On the other hand, for $$g = \sum x_i^2 - 1$$, since $$g^{-1}(0)$$ is compact, $$f$$ attains a maximum or minimum at a point $$u$$ in $$g^{-1}(0)$$. Since $$\nabla g = 2(x_1, \dots, x_n)$$, by Lagrange multiplier, we find a real number $$\lambda$$ such that $$2 \sum_i a_{ji} u_i = 2 \lambda u_j, 1 \le j \le n.$$ But that means $$Tu = \lambda u$$. By inductive hypothesis, the self-adjoint operator $$T : W \to W$$, $$W$$ the orthogonal complement to $$u$$, has a basis consisting of eigenvectors. Hence, we are done. $$\square$$.

Weak derivatives
Up to measure-zero sets, two functions can be determined to be equal or not by means of integration against other functions (called test functions). Namely, the following sometimes called the fundamental lemma of calculus of variations:

Given a continuous function $$f$$, by the lemma, a continuously differentiable function $$u$$ is such that $$\frac{\partial u}{\partial x_i} = f$$ if and only if
 * $$\int \frac{\partial u}{\partial x_i} \varphi \, dx = \int f \varphi \, dx$$

for every $$\varphi \in C_c^{\infty}(M)$$. But, by integration by parts, the partial derivative on the left-hand side of $$u$$ can be moved to that of $$\varphi$$; i.e.,
 * $$-\int u \frac{\partial \varphi}{\partial x_i} \, dx = \int f \varphi \, dx$$

where there is no boundary term since $$\varphi$$ has compact support. Now the key point is that this expression makes sense even if $$u$$ is not necessarily differentiable and thus can be used to give sense to a derivative of such a function.

Note each locally integrable function $$u$$ defines the linear functional $$\varphi \mapsto \int u \varphi \, dx$$ on $$C_c^{\infty}(M)$$ and, moreover, each locally integrable function can be identified with such linear functional, because of the early lemma. Hence, quite generally, if $$u$$ is a linear functional on $$C_c^{\infty}(M)$$, then we define $$\frac{\partial u}{\partial x_i}$$ to be the linear functional $$\varphi \mapsto -\left \langle u, \frac{\partial \varphi}{\partial x_i} \right\rangle$$ where the bracket means $$\langle \alpha, \varphi \rangle = \alpha(\varphi)$$. It is then called the weak derivative of $$u$$ with respect to $$x_i$$. If $$u$$ is continuously differentiable, then the weak derivate of it coincides with the usual one; i.e., the linear functional $$\frac{\partial u}{\partial x_i}$$ is the same as the linear functional determined by the usual partial derivative of $$u$$ with respect to $$x_i$$. A usual derivative is often then called a classical derivative. When a linear functional on $$C_c^{\infty}(M)$$ is continuous with respect to a certain topology on $$C_c^{\infty}(M)$$, such a linear functional is called a distribution, an example of a generalized function.

A classic example of a weak derivative is that of the Heaviside function $$H$$, the characteristic function on the interval $$(0, \infty)$$. For every test function $$\varphi$$, we have:
 * $$\langle H', \varphi \rangle = -\int_0^{\infty} \varphi' \, dx = \varphi(0).$$

Let $$\delta_a$$ denote the linear functional $$\varphi \mapsto \varphi(a)$$, called the Dirac delta function (although not exactly a function). Then the above can be written as:
 * $$H' = \delta_0.$$

Cauchy's integral formula has a similar interpretation in terms of weak derivatives. For the complex variable $$z = x + iy$$, let $$E_{z_0}(z) = \frac{1}{\pi (z - z_0)}$$. For a test function $$\varphi$$, if the disk $$| z - z_0 | \le r$$ contains the support of $$\varphi$$, by Cauchy's integral formula, we have:
 * $$\varphi(z_0) = {1 \over 2 \pi i} \int \frac{\partial \varphi}{\partial \bar z} \frac{dz \wedge d \bar z}{z - z_0}.$$

Since $$dz \wedge d \bar z = -2i dx \wedge dy$$, this means:
 * $$\varphi(z_0) = -\int E_{z_0} \frac{\partial \varphi}{\partial \bar z} dxdy = \left\langle \frac{\partial E_{z_0}}{\partial \bar z}, \varphi \right \rangle,$$

or
 * $$\frac{\partial E_{z_0}}{\partial \bar z} = \delta_{z_0}.$$

In general, a generalized function is called a fundamental solution for a linear partial differential operator if the application of the operator to it is the Dirac delta. Hence, the above says $$E_{z_0}$$ is the fundamental solution for the differential operator $$\partial/\partial \bar z$$.

Definition of a manifold

 * This section requires some background in general topology.

A manifold is a Hausdorff topological space that is locally modeled by an Euclidean space. By definition, an atlas of a topological space $$M$$ is a set of maps $$\varphi_i : U_i \to \mathbb{R}^n$$, called charts, such that By definition, a manifold is a second-countable Hausdorff topological space with a maximal atlas (called a differentiable structure); "maximal" means that it is not contained in strictly larger atlas. The dimension of the manifold $$M$$ is the dimension of the model Euclidean space $$\mathbb{R}^n$$; namely, $$n$$ and a manifold is called an n-manifold when it has dimension n. A function on a manifold $$M$$ is said to be smooth if $$f|_U \circ \varphi^{-1}$$ is smooth on $$\varphi(U)$$ for each chart $$\varphi : U \to \mathbb{R}^n$$ in the differentiable structure.
 * $$U_i$$ are an open cover of $$M$$; i.e., each $$U_i$$ is open and $$M = \cup_i U_i$$,
 * $$\varphi_i : U_i \to \varphi_i(U_i)$$ is a homeomorphism and
 * $$\varphi_j \circ \varphi_i^{-1} : \varphi_i(U_i \cap U_j) \to \varphi_j(U_i \cap U_j)$$ is smooth; thus a diffeomorphism.

A manifold is paracompact; this has an implication that it admits a partition of unity subordinate to a given open cover.

If $$\mathbb{R}^n$$ is replaced by an upper half-space $$\mathbb{H}^n$$, then we get the notion of a manifold-with-boundary. The set of points that map to the boundary of $$\mathbb{H}^n$$ under charts is denoted by $$\partial M$$ and is called the boundary of $$M$$. This boundary may not be the topological boundary of $$M$$. Since the interior of $$\mathbb{H}^n$$ is diffeomorphic to $$\mathbb{R}^n$$, a manifold is a manifold-with-boundary with empty boundary.

The next theorem furnishes many examples of manifolds.

For example, for $$g(x) = x_1^2 + \cdots + x_{n+1}^2 - 1$$, the derivative $$g'(x) = \begin{bmatrix}2 x_1 & 2 x_2 & \cdots & 2 x_{n+1}\end{bmatrix}$$ has rank one at every point $$p$$ in $$g^{-1}(0)$$. Hence, the n-sphere $$g^{-1}(0)$$ is an n-manifold.

The theorem is proved as a corollary of the inverse function theorem.

Many familiar manifolds are subsets of $$\mathbb{R}^n$$. The next theoretically important result says that there is no other kind of manifolds. An immersion is a smooth map whose differential is injective. An embedding is an immersion that is homeomorphic (thus diffeomorphic) to the image.

The proof that a manifold can be embedded into $$\mathbb{R}^N$$ for some N is considerably easier and can be readily given here. It is known that a manifold has a finite atlas $$\{ \varphi_i : U_i \to \mathbb{R}^n \mid 1 \le i \le r \}$$. Let $$\lambda_i$$ be smooth functions such that $$\operatorname{Supp}(\lambda_i) \subset U_i$$ and $$\{ \lambda_i = 1 \}$$ cover $$M$$ (e.g., a partition of unity). Consider the map
 * $$f = (\lambda_1 \varphi_1, \dots, \lambda_r \varphi_r, \lambda_1, \dots, \lambda_r) : M \to \mathbb{R}^{(k+1)r}$$

It is easy to see that $$f$$ is an injective immersion. It may not be an embedding. To fix that, we shall use:
 * $$(f, g) : M \to \mathbb{R}^{(k+1)r+1}$$

where $$g$$ is a smooth proper map. The existence of a smooth proper map is a consequence of a partition of unity. See for the rest of the proof in the case of an immersion. $$\square$$

Nash's embedding theorem says that, if $$M$$ is equipped with a Riemannian metric, then the embedding can be taken to be isometric with an expense of increasing $$2k$$; for this, see this T. Tao's blog.

Tubular neighborhood and transversality
A technically important result is:

This can be proved by putting a Riemannian metric on the manifold $$M$$. Indeed, the choice of metric makes the normal bundle $$\nu_i$$ a complementary bundle to $$TN$$; i.e., $$TM|_N$$ is the direct sum of $$TN$$ and $$\nu_N$$. Then, using the metric, we have the exponential map $$\exp : U \to V$$ for some neighborhood $$U$$ of $$N$$ in the normal bundle $$\nu_N$$ to some neighborhood $$V$$ of $$N$$ in $$M$$. The exponential map here may not be injective but it is possible to make it injective (thus diffeomorphic) by shrinking $$U$$ (for now, see see ).

Integration on manifolds and distribution densities
The starting point for the topic of integration on manifolds is that there is no invariant way to integrate functions on manifolds. This may be obvious if we asked: what is an integration of functions on a finite-dimensional real vector space? (In contrast, there is an invariant way to do differentiation since, by definition, a manifold comes with a differentiable structure). There are several ways to introduce integration theory to manifolds:
 * Integrate differential forms.
 * Do integration against some measure.
 * Equip a manifold with a Riemannian metric and do integration against such a metric.

For example, if a manifold is embedded into an Euclidean space $$\mathbb{R}^n$$, then it acquires the Lebesgue measure restricting from the ambient Euclidean space and then the second approach works. The first approach is fine in many situations but it requires the manifold to be oriented (and there is a non-orientable manifold that is not pathological). The third approach generalizes and that gives rise to the notion of a density.

Extensions to infinite-dimensional normed spaces
The notions like differentiability extend to normed spaces.

title: Calculus on Euclidean space


text:
Talk:Kokoda (film)

Fair use rationale for Image:Kokoda-0.jpg
Image:Kokoda-0.jpg is being used on this article. I notice the image page specifies that the image is being used under fair use but there is no explanation or rationale as to why its use in this Wikipedia article constitutes fair use. In addition to the boilerplate fair use template, you must also write out on the image description page a specific explanation or rationale for why using this image in each article is consistent with fair use.

BetacommandBot (talk) 21:57, 2 January 2008 (UTC) "It is not clear what happens to Sam" in the movie,as the unit moves out, you can see sam has climbed into the hollow of a decapitated tress and looks dead —Preceding unsigned comment added by <IP_ADDRESS> (talk) 01:40, 6 January 2009 (UTC)

title: Talk:Kokoda (film)


text:
Talk:David Zdrilic

External links modified
Hello fellow Wikipedians,

I have just modified 2 one external links on David Zdrilic. Please take a moment to review my edit. If you have any questions, or need the bot to ignore the links, or the page altogether, please visit this simple FaQ for additional information. I made the following changes:
 * Added archive https://web.archive.org/web/20080601063235/http://au.fourfourtwo.com:80/news/58870,zdrilic-set-for-move-to-the-rear.aspx to http://au.fourfourtwo.com/news/58870,zdrilic-set-for-move-to-the-rear.aspx
 * Added archive https://web.archive.org/web/20071025052643/http://au.fourfourtwo.com:80/news/59037,zdrilic-hits-back-at-critics.aspx to http://au.fourfourtwo.com/news/59037,zdrilic-hits-back-at-critics.aspx

Cheers.— InternetArchiveBot  (Report bug) 15:51, 8 December 2016 (UTC)

A Commons file used on this page or its Wikidata item has been nominated for speedy deletion
The following Wikimedia Commons file used on this page or its Wikidata item has been nominated for speedy deletion: You can see the reason for deletion at the file description page linked above. —Community Tech bot (talk) 02:52, 12 February 2020 (UTC)
 * SBS WC Russia.jpg

A Commons file used on this page or its Wikidata item has been nominated for deletion
The following Wikimedia Commons file used on this page or its Wikidata item has been nominated for deletion: Participate in the deletion discussion at the. —Community Tech bot (talk) 03:37, 12 February 2020 (UTC)
 * Qatar World Cup Qualifiers.jpg

A Commons file used on this page or its Wikidata item has been nominated for deletion
The following Wikimedia Commons file used on this page or its Wikidata item has been nominated for deletion: Participate in the deletion discussion at the. —Community Tech bot (talk) 18:22, 12 February 2020 (UTC)
 * Thursday FC.jpg

A Commons file used on this page or its Wikidata item has been nominated for deletion
The following Wikimedia Commons file used on this page or its Wikidata item has been nominated for deletion: Participate in the deletion discussion at the nomination page. —Community Tech bot (talk) 21:51, 12 February 2020 (UTC)
 * David Zdrilic at WC 2018.jpg

A Commons file used on this page or its Wikidata item has been nominated for speedy deletion
The following Wikimedia Commons file used on this page or its Wikidata item has been nominated for speedy deletion: You can see the reason for deletion at the file description page linked above. —Community Tech bot (talk) 17:36, 14 February 2020 (UTC)
 * David Zdrilic Beach Socceroos.jpg

title: Talk:David Zdrilic


text:
1949 Wimbledon Championships – Girls' singles

Christiane Mercelis defeated Susan Partridge in the final, 6–4, 6–2 to win the girls' singles tennis title at the 1949 Wimbledon Championships.


text:
Tess Feury

Tess Feury (born March 15, 1996) is an American rugby union player. She competed for the United States at the 2017 and 2021 Rugby World Cup's. She plays for the Leicester Tigers in the Premiership Women's Rugby competition.

Early life
Feury graduated from the Pennsylvania State University where she majored in Nursing and played for their women’s rugby team. She grew up playing rugby; her father was introduced to the sport when he attended Rutgers University. He started an American Flag Rugby Program which allowed his sons and daughter to play rugby.

Feury currently works as a Pediatric Intensive Care Nurse at Morristown Medical Center.

Rugby career
Feury captained the United States Girls team at the 2014 Summer Youth Olympics. She made her debut for the United States in 2016 against England at the Women's Super Series in Salt Lake City, Utah. She was selected for the Eagles 2017 Women's Rugby World Cup squad.

In 2022, she was named in the Eagles squad for the Pacific Four Series in New Zealand. She was selected in the Eagles squad for the 2021 Rugby World Cup in New Zealand.

Feury was named in the Eagles traveling squad for the 2023 Pacific Four Series. She signed with the Leicester Tigers for the 2023–24 Premiership Women's Rugby season.

title: Tess Feury


text:
Cookbook:Sunchoke

The sunchoke, often called the Jerusalem artichoke, is a tuber from a plant that is related to the sunflower. It is native to America, and has nothing to do with either Jerusalem or artichokes. The tubers are gnarly and uneven, vaguely resembling ginger root. The white flesh is nutty, sweet, and crunchy like chestnuts when raw. Baked in their skins, they become more like potatoes with a mild taste of artichoke hearts.

The sunchoke is widely grown in gardens in Texas and is harvested in the fall for highest quality. Widely available in supermarkets, its peak period is September through January, but often continues through the early spring.

Select sunchokes that are firm and free from mold and wrinkles. If left too long in the open, they become wrinkled and soft and can develop a bitter taste. Sunchokes vary in color from dark brown to light brown, similar to ginger.

These tubers need be refrigerated, unwashed, in a plastic bag for up to 1 week for successful storage.

Unlike most tubers, the tubers store the carbohydrate inulin (not to be confused with insulin) instead of starch. For that reason, they are an important source of fructose for industry.

History
Sunchokes were cultivated by the Native Americans (who called them "sun roots") long before the arrival of the Europeans. The French explorer Samuel de Champlain found them being grown at Cape Cod in 1605.

Cultivation
Sunchokes are easy to grow. In fact the problem for many people is to not grow them once they have been planted. For this reason, it is tempting to just leave them in place year after year and dig them as needed. But the quality of the tubers degrades with this treatment. Keeping the soil fertile is important. Every small piece of tuber will grow if left in the ground and they can become a persistent weed.

title: Cookbook:Sunchoke


text:
John Tufnell

John Joliffe Tufnell (1720–1794), was a British politician who sat in the House of Commons from 1754 to 1761. Tufnell was the son of Samuel Tufnell MP of Monken Hadley, Hertfordshire and Langleys, Essex and his wife Elizabeth Cressener, daughter of George Cressener of Earl's Colne, Essex and was born on 27 June 1720. He entered Middle Temple in 1737 and Corpus Christi College, Cambridge in 1738. He married Anna Meeke, daughter of William Meeke of Northallerton, Yorkshire on 2 April 1748.

Tufnell was elected as Member of Parliament for Beverley in a contest in the 1754 general election. Little is known of his contribution in parliament. He was listed under the heading “country gentlemen” and was counted as a Government supporter. He did not stand in 1761 when his brother George took over the seat.

Tufnell succeeded his father on 28 December 1758, inheriting Langleys in Great Waltham, Essex. He became one of the richest commoners in England, renting out property in Essex and the north. He died on 23 September 1794.

title: John Tufnell


text:
Buddhism/Schools/Vajrayāna

Vajrayana
Vajrayana (Diamond Vehicle) is a fast path to enlightenment, achieved through use of tantra techniques. It is a complex spiritual system. It is based on the teachings of Buddha, but were extended by the mahayana path of love and by a lot of techniques from Indian yoga. Monks developed in the Middle Ages a comprehensive system of teaching that is imparted in a long-term training and tailored to the needs of people in Tibet. Advanced monks and nuns is recommended a three-year meditation as a yogi in seclusion.

A major problem is the adaptation of Tibetan Buddhism in the modern world and in particular to the needs and possibilities of people in the west. The Dalai Lama tries to create a modern Buddhism based on the equalty of men and women, democracy and less strict sexual rules. He permits masturbation for monks and accepts homosexuality (Luise Rinser, Compassion as a way to peace, 1995, page 80).

In principle the Tibetan Buddhism is very simple. It consists of the Kundalini Yoga, the meditation in silence and the path of all-embracing love. With Kundalini Yoga the kundalini energy (enlightenment energy, happiness energy) is awakened through yoga postures, breathing exercises, mantras and visualizations. The center of tibetian meditation is the deity yoga. Through the visualization of various deities the energy channels are cleaned, the chakras are activated and the enlightenment consciousness is created. By special methods the ego will be dissolved and the Yogi enters into the bliss of nirvana.

A key element of Tibetan Buddhism is the work with an enlightened Master. The enlightened Master transmits enlightenment energy to his students after an initiation. The great advantage of Tibetan Buddhism is that there are many enlightened Masters who regularly travel to their western groups. Also possesses the Tibetan Buddhism with the Dalai Lama an inspiring leader who is often seen by Western television. The Dalai Lama represents the cooperation of all religions and initiates also people who do not belong to the Tibetan Buddhism.

In Tibetan Buddhism there are five major lineages: the Gelugpa (emphasis on compassion and spiritual training), the Kagyupa (emphasis on meditation), the Sakya (emphasis on the wisdom), the Bonpo (traditional shamanism) and the Nyingma (the non-dogmatic way). The Nyingma emphasize the spiritual development from the inside. They teach the individual path. Nyingma yogis may also live in a relationship.

The founder of Tibetan Buddhism is Padmasambhava. He lived some 1200 years ago as a yogi in India and was invited by the Tibetian King Trisong Detsen. He knew a lot of spiritual methods and taught a diverse mix of yoga techniques (Hatha Yoga, Karma Yoga, Mantra Yoga, Tantra Yoga and Guru Yoga). He said, "Practise so that you feel comfortable. Whenever there is a doubt on your way, then pray to the enlightened Masters, think for yourself and follow the voice of your inner wisdom. This is the voice of the enlightened Masters. If you do this, you will win on your spiritual path."

---> For advanced see also Wikipedia: Tantra techniques (Vajrayana) (Deity Yoga, Guru Yoga, Clear Light Yoga, Death Yoga, Tantric Sādhana)

title: Buddhism/Schools/Vajrayāna


text:
Holy Empire of Reunion

The Holy Empire of Reunion (Portuguese: Sacro Império de Reunião) is a micronation that was founded 28 August 1997 as a political and constitutional simulation.

History
It claims the territory of the French marine department of Réunion Island. Its creators were students of the Law School of the Pontifical Catholic University of Rio de Janeiro.

The micronation has been mentioned in media.

title: Holy Empire of Reunion


text:
Walrasian

Adjective

 * 1)  Of or pertaining to the theories of the 19th-century French economist.
 * 2)  Relating to a tatonnement process for achieving exchange equilibrium.
 * a Walrasian auction

title: Walrasian


text:
Waves/Single slit Diffraction

Diffraction Through a Single Slit
Figure 2.15: Schematic behavior when a plane wave impinges on a narrow slit and a broad slit.

How does all of this apply to the passage of waves through a slit? Imagine a plane wave of wavelength $$\lambda $$ impingent on a barrier with a slit or opening of width $$w$$. The barrier transforms the plane wave with infinite extent in the lateral direction into a beam with initial transverse dimensions equal to the width of the slit. The subsequent development of the beam is illustrated in figures 2.13 and 2.14, and schematically in figure 2.15. In particular, if the slit width is comparable to the wavelength, the beam spreads broadly as in figure 2.13. If the slit width is large compared to the wavelength, the beam doesn't spread as much, as figure 2.14 illustrates. Equation (2.22) gives us an approximate quantitative result for the spreading angle if $$w$$ is interpreted as the width of the slit.

One use of the above equation is in determining the maximum angular resolution of optical instruments such as telescopes. The primary lens or mirror can be thought of as a rather large “slit”. Light from a distant point source is essentially in the form of a plane wave when it arrives at the telescope. However, the light passed by the telescope is no longer a plane wave, but is a beam with a tendency to spread. The spreading angle $$\alpha_{max}$$ is given by equation (2.22), and the telescope cannot resolve objects with an angular separation less than $$\alpha_{max}$$. Replacing $$w$$ with the diameter of the lens or mirror in equation (2.22) thus yields the telescope's angular resolution. For instance, a moderate sized telescope with aperture $$1 \mbox{ m}$$ observing red light with $$\lambda \approx 6 \times 10^{-7} \mbox{ m}$$ has a maximum angular resolution of about $$3 \times 10^{-7} \mbox{ radians}$$.

title: Waves/Single slit Diffraction


text:
Wikipedia:Articles for deletion/Ian T. Sebàs

The result was   delete. Missvain (talk) 06:19, 15 February 2015 (UTC)

Ian T. Sebàs

 * – ( View AfD View log  Stats )

non-notable author, promotional content, probable COI, previously deleted Deunanknute (talk) 13:11, 8 February 2015 (UTC)

 * Delete A promotional article on a non-notable subject. Apart from his own website, the only reference is a page which merely includes his name in a list of people nominated for an award. (Created by a single purpose account with a clear conflict of interest.) The editor who uses the pseudonym "JamesBWatson" (talk) 17:27, 9 February 2015 (UTC)

 * Delete Same reasons as James Bentogoa (talk) 13:19, 11 February 2015 (UTC)
 * Note: This debate has been included in the list of United States of America-related deletion discussions. • Gene93k (talk) 14:39, 11 February 2015 (UTC)
 * Note: This debate has been included in the list of Caribbean-related deletion discussions. • Gene93k (talk) 14:39, 11 February 2015 (UTC)
 * Note: This debate has been included in the list of Authors-related deletion discussions. • Gene93k (talk) 14:39, 11 February 2015 (UTC)


text:
Sotades platypus

Sotades platypus is a species of beetle in the family Cerambycidae, and the only species in the genus Sotades. It was described by Pascoe in 1864.

title: Sotades platypus


text:
Doubrava

Doubrava may refer to:

Places in the Czech Republic

 * Doubrava (Karviná District), a municipality and village in the Moravian-Silesian Region
 * Doubrava (Aš), a village and part of Aš in the Karlovy Vary Region
 * Doubrava, a village and part of Chrášťany (České Budějovice District) in the South Bohemian Region
 * Doubrava, a village and part of Hořice in the Hradec Králové Region
 * Doubrava, a village and part of Kostomlátky in the Central Bohemian Region
 * Doubrava, a village and part of Lipová (Cheb District) in the Karlovy Vary Region
 * Doubrava, a village and part of Nýřany in the Plzeň Region
 * Doubrava, a village and part of Puclice in the Plzeň Region
 * Doubrava, a village and part of Vlachovo Březí in the South Bohemian Region
 * Doubrava, a village and part of Žďár (Mladá Boleslav District) in the Central Bohemian Region
 * Karlov t. Doubrava, a village and part of Vidice (Kutná Hora District) in the Central Bohemian Region

People

 * Jaroslav Doubrava, a Czech composer

title: Doubrava


text:
Talk:David McCreery

WikiProject class rating
This article was automatically assessed because at least one WikiProject had rated the article as stub, and the rating on other projects was brought up to Stub class. BetacommandBot 05:56, 8 September 2007 (UTC)

External links modified
Hello fellow Wikipedians,

I have just modified 1 one external link on David McCreery. Please take a moment to review my edit. If you have any questions, or need the bot to ignore the links, or the page altogether, please visit this simple FaQ for additional information. I made the following changes:
 * Added archive https://web.archive.org/web/20150110050747/http://www.newsabahtimes.com.my/nstweb/fullstory/60122 to http://www.newsabahtimes.com.my/nstweb/fullstory/60122

Cheers.— InternetArchiveBot  (Report bug) 06:14, 8 December 2016 (UTC)

External links modified
Hello fellow Wikipedians,

I have just modified one external link on David McCreery. Please take a moment to review my edit. If you have any questions, or need the bot to ignore the links, or the page altogether, please visit this simple FaQ for additional information. I made the following changes:
 * Added archive https://web.archive.org/web/20160314021758/http://irishfa.com/international/squad-profiles/most-capped-players/ to http://www.irishfa.com/international/squad-profiles/most-capped-players/

Cheers.— InternetArchiveBot  (Report bug) 06:50, 6 September 2017 (UTC)

External links modified
Hello fellow Wikipedians,

I have just modified one external link on David McCreery. Please take a moment to review my edit. If you have any questions, or need the bot to ignore the links, or the page altogether, please visit this simple FaQ for additional information. I made the following changes:
 * Added archive https://archive.is/20130722050213/http://borneoinsider.com/2013/06/19/illness-forces-mccreery-to-quit-sabahawks/ to http://borneoinsider.com/2013/06/19/illness-forces-mccreery-to-quit-sabahawks/

Cheers.— InternetArchiveBot  (Report bug) 11:24, 3 December 2017 (UTC)

External links modified
Hello fellow Wikipedians,

I have just modified one external link on David McCreery. Please take a moment to review my edit. If you have any questions, or need the bot to ignore the links, or the page altogether, please visit this simple FaQ for additional information. I made the following changes:
 * Added archive https://web.archive.org/web/20130627091036/http://www.nst.com.my/sports/soccer/mccreery-parts-company-with-sabah-1.304420 to http://www.nst.com.my/sports/soccer/mccreery-parts-company-with-sabah-1.304420

Cheers.— InternetArchiveBot  (Report bug) 23:43, 1 January 2018 (UTC)

title: Talk:David McCreery


text:
Onafhankelijkheidsplein

The Onafhankelijkheidsplein (, Independence Square) is a town square in Paramaribo, the capital of Suriname. The square is situated in the historical inner city, near the Suriname River and the Presidential Palace of Suriname.

Before the independence of Suriname from the Netherlands in 1975, it was also called the Oranjeplein (Orange Square) and the Gouvernementsplein (Government Square).

Buiten-Sociëteit Het Park is located on the square. On 1 August 1996, the building which housed the National Assembly of Suriname burnt down. The National Assembly moved to the former park house.

Other buildings on the square include the Ministry of Finance and the Congress Hall.

title: Onafhankelijkheidsplein


text:
First Aid/Appendix D: Notes for First Aid Instructors

Education
The teaching of first aid is in itself a valuable safety measure. Educating even a few people in first aid improves the safety of everyone they come into contact with.

Teaching first aid also provides an opportunity for teaching safety, prevention and risk reduction measures at the same time, as part of the first aid curriculum.

A person trained in first aid is more motivated to avoid personal injury, because their first aid training gives them a greater appreciation of the potential serious consequences.

Last but not least, someone trained in first aid can be taught to appreciate the importance of safety, prevention and risk reduction. This makes them an ideal advocate for spreading safety awareness to others in their organization or family. This "peer to peer" education can be very valuable.


text:
GCSE Science/Circuits

GCSE Science/Electricity

You've probably done quite a lot of work in basic electrical circuits in lower school, but we will revise them here so don't worry if you've forgotten them.

Before we start you need to know a whole host of circuit symbols.

Q1)See how many of the following you can name:

You should already know about half of them. For the others see here. You need to learn all these symbols by heart, but don't worry about that just yet. It's much easier to learn the symbols if you know what the components actually do, and we will be looking at that later on.

What is a circuit?
A circuit is a loop in which electricity can flow. Consider a simple circuit with a cell, a switch and a bulb. The current flows from the cell, via the closed switch to the bulb. The bulb lights because the electricity carries energy. But even after the bulb there current still needs to go around the rest of the loop back to the cell. If there is a break anywhere in the loop, the bulb will not light. This is a concept that is not all that difficult, yet most people do not understand it.

Ask a selection of adults you know and many will say: "Electricity flows from the cell to the bulb, where it gets used up"

Q2) What is wrong with the above statement?

Q3) Look at the selection of circuits below. What is wrong with each of them?

What we have been looking at so far has been largely revision. You should already be familiar with what a circuit is and how it works. Let's now go on to some more advanced work.

<< Electrolysis | Current, voltage resistance and Ohm's law

title: GCSE Science/Circuits


text:
Talk:Dave Waterbury

Cleanup
Article needs a lot of citations and cleanup, which I or others will do. Give it some time, please. SteakSpecial (talk) 00:31, 1 June 2010 (UTC)

title: Talk:Dave Waterbury


text:
margaritomancy

Etymology
.

Noun

 * 1) Divination by pearls.
 * 2) * 1931 Lock tr. de Givry Picture Museum of Sorcery
 * Very curious was margaritomancy, or divination by a pearl. "The pearl is to be enchanted," says de l'Ancre, "and shut in a pot; then if it chance that the name of the thief be pronounced the pearl will leap, striking against the pot."
 * 1) * 1931 Lock tr. de Givry Picture Museum of Sorcery
 * Very curious was margaritomancy, or divination by a pearl. "The pearl is to be enchanted," says de l'Ancre, "and shut in a pot; then if it chance that the name of the thief be pronounced the pearl will leap, striking against the pot."

title: margaritomancy


text:
Addi Abagiè

Addi Abagiè is a reservoir located in the Sa’isi Tsa’ida Imba woreda of the Tigray Region in Ethiopia. The earthen dam that holds the reservoir was built in 1993.

Dam characteristics
Dam crest length: 176 metres. Exceptionally, among the many dams in Tigray, this dam was breached around the year 2000, due to high positioning of the spillway and hence overtopping of the dam. Around 2010 the breach was closed and the dam height increased. The dam is now operational.

Environment
The catchment of the reservoir is 8.77 km2 large, with a perimeter of 13 km and a length of 4700 metres. The reservoir suffers from rapid siltation. The lithology of the catchment is Enticho Sandstone and Precambrian metamorphic rocks.

title: Addi Abagiè


text:
History of Quebec and Canada/Study Guide

History of Quebec and Canada Study Guide

This guide has been prepared for Quebec High School students enrolled in the Secondary IV History of Quebec and Canada Course. Topics emphasized in these courses include the French empire in America, Canadian society under the French regime, the Conquest and the early days of British rule, the beginnings of responsible government, Quebec and Confederation, industrial development, and contemporary Quebec.

How to use the Study Guide

1-Go through each of the Lessons in History starting with Module 1.

2-Complete the Topic Test Questions at the end of each topic and the Synthesis Test Questions at the end of each Module.

3-Go back over the Lesson until you're able to answer the questions.

4-Once you have completed the Module, test yourself with the Self Scoring Module Tests.

5-If you're still having trouble, try the Test+Tutorial, where you can get "hints" and references to the correct answers.

Lessons in History

 * Module 1
 * Module 2
 * Module 3
 * Module 4
 * Module 5
 * Module 6
 * Module 7


text:
User:KrakatoaKatie

I don't know how many Wiki-user pages I have, but I'm all for single sign-on.

Here's my user page on en.

title: User:KrakatoaKatie


text:
Mark Hubbard

Mark Hubbard may refer to:
 * Mark Hubbard (musician), American gospel musician
 * Mark "Monk" Hubbard (1970–2018), American skateboarder
 * Mark Hubbard (golfer) (born 1989), American golfer

title: Mark Hubbard


text:
Πυθώ

Etymology
Probably from, from , a reference to the monsters who inhabited the caverns, but it has also been linked to.

Proper noun

 * 1) Pytho; the ancient name of Delphi
 * 2) Python, the monstrous snake said to have been slain by Pythian Apollo at Delphi

title: Πυθώ


text:
Mary Day

Mary Day may refer to:

People

 * Mary Day (dance teacher) (1910–2006), founder of The Washington Ballet
 * Mary Anna Day (1852–1924), American botanist and librarian
 * Mary E. Day, in the 2005 Supreme Court opinion Varian v. Delfino
 * Mary L. Day (1836–?), American memoirist
 * Mary Gage Day (1857–1935), American physician and medical writer
 * Mary Louise Day (1968–2017), teenage girl who mysteriously disappeared from her home

Days

 * Marian feast days

title: Mary Day


text:
sarjan

Etymology
From, from , , , , , , from , , , , , from , accusative of , from , present participle of.

Noun

 * 1)  non-commissioned officer in several armed forces

Alternative forms

 * sèrsan

title: sarjan


text:
1963 Memphis State Tigers football team

The 1963 Memphis State Tigers football team represented Memphis State College (now known as the University of Memphis) as an independent during the 1963 NCAA University Division football season. In its sixth season under head coach Billy J. Murphy, the team compiled a 9–0–1 record and outscored opponents by a total of 199 to 52. Richard Saccoccia was the team captain. The team played its home games at Crump Stadium in Memphis, Tennessee.

The team's statistical leaders included Russell Vollmer with 466 passing yards, fullback Dave Casinelli with 1,016 rushing yards and 84 points scored, and Bob Sherlag with 183 receiving yards.

title: 1963 Memphis State Tigers football team


text:
User:Quosgtl/sandbox

Biography
Sali Hughes is a Welsh journalist, writer and broadcaster. Her early career was spent in print media, but as well as continuing to write for magazines and newspapers she is now also a regular contributor on TV and Radio.

Hughes has written features, opinion columns and celebrity interviews for Grazia, ELLE, Red, The Guardian, Glamour, Stylist, Shortlist, Top Santé, NOW and Cosmopolitan. Since January 2011 she has also been the Guardian’s resident beauty columnist, featuring weekly in their Saturday magazine and online video tutorials.

Hughes is a regular contributor on Sky News’ Press Preview and from June 2012 has been a regular press reviewer on ITV’s Daybreak. She has also regularly appeared on BBC Radio 5Live reviewing television, music and entertainment and has contributed to BBC Radio Four’s Woman’s Hour, The Chris Evans Show on BBC R2 and on news and entertainment programmes on ITV, BBC, Channel 4 and Sky.

title: User:Quosgtl/sandbox


text:
1987 Colgate Red Raiders football team

The 1987 Colgate Red Raiders football team was an American football team that represented Colgate University during the 1987 NCAA Division I-AA football season. Colgate finished third in the Colonial League.

In its 12th and final season under head coach Frederick Dunlap, the team compiled a 7–4 record. Kenny Gamble and Kyle Warwick were the team captains.

Despite posting a winning record, the Red Raiders finished exactly even on the scoresheet, scoring 236 points and allowing 236 points over the course of the season. Colgate's 2–2 conference record placed third in the six-team Colonial League standings.

The Red Raiders spent two weeks in the national top 20 rankings, reaching No. 13 in the poll released September 22 and No. 11 on September 29, but then fell out of the rankings and were not ranked at season's end.

The team played its home games at Andy Kerr Stadium in Hamilton, New York.

title: 1987 Colgate Red Raiders football team