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Consort Fu (Yuan)

Consort Fu (傅昭儀, personal name unknown) (died 21 February 2 BC ) was an imperial consort during the Han dynasty of China. She was a consort and a favourite of Emperor Yuan. She was known to be a domineering woman who wanted her son on the throne, and, failing that, wanted (and eventually was able to see) her grandson on the throne as Emperor Ai. During Emperor Ai's reign, she exerted heavy and frequent influence on his reign, that made her powerful and dominant and forcibly extracted empress dowager titles that she should not have properly possessed (since she was never an empress – and Emperor Cheng's wife, Grand Empress Dowager Wang Zhengjun was still alive) – which would bring her hatred from the Wang clan and eventually the desecration of her tomb after her death.

Family background
Consort Fu's father was from the Commandery of Henei (roughly modern Handan, Hebei) and died early. Her mother remarried a man named Zheng (鄭) after her father's death. When Consort Fu was young, she was a lady-in-waiting for Grand Empress Dowager Shangguan.

Years as imperial consort
It is not known when exactly Consort Fu became a consort to Emperor Yuan, but it is known that it happened during his years as crown prince – but after his eventual wife Empress Wang became a consort of his. She and Consort Feng Yuan became his favorite concubines, and each of them bore him a son. Consort Fu's son was Liu Kang (劉康), who later was created the Prince of Jiyang in April or May 41 BC, then Prince of Shanyang in 34 BC, then Prince of Dingtao in 27 BC. As Consort Fu's position was inferior to that of Empress Wang, and her son younger than Empress Wang's son Liu Ao (later Emperor Cheng), Prince Ao was created crown prince. However, Consort Fu was not content for her son merely to be an imperial prince, and a succession struggle would erupt.

As Crown Prince Ao grew older, Emperor Yuan became increasingly unhappy with his fitness as imperial heir and impressed with Consort Fu's son, Prince Kang. Several incidents led to this situation. One happened in 35 BC, when Emperor Yuan's youngest brother Prince Liu Jing of Zhongshan (中山王劉竟) died, Emperor Yuan became angry when he felt that the teenage Crown Prince Ao was insufficiently grieving – particularly because Princes Ao and Jing were of similar age and grew up together as playmates – and showing insufficient respect to Prince Jing. Prince Ao's head of household Shi Dan (史丹), a relative of Emperor Yuan's grandmother and a senior official respected by Emperor Yuan, managed to convince Emperor Yuan that Crown Prince Ao was trying to stop Emperor Yuan himself from over-grieving, but the seed of dissatisfaction was sown.

As the princes further grew, several things further led to an endearment between Emperor Yuan and Prince Kang. They shared affection and skills in music – particularly in the playing of drums. Prince Kang also showed high intelligence and diligence, while Crown Prince Ao was known for drinking and womanizing. When Emperor Yuan grew ill circa 35 BC – an illness that he would not recover from – Consort Fu and Prince Kang were often summoned to his sickbed to attend to him, while Empress Wang and Crown Prince Ao rarely were. In his illness, apparently encouraged by Consort Fu, Emperor Yuan reconsidered whether he should make Prince Kang his heir instead. Only the intercession of Shi Dan – who risked his life by stepping onto the carpet of the imperial bed chamber, an act that only the empress was allowed to do, at the pain of death – led Emperor Yuan to cease those thoughts. When Emperor Yuan died in July 33 BC, Crown Prince Ao ascended the throne (as Emperor Cheng).

While both Consorts Fu and Feng were imperial consorts, an incident occurred that would cause Consort Feng to be greatly praised, at Consort Fu's embarrassment. In 38 BC, Emperor Yuan was watching a fight between wild animals, accompanied by a number of his concubines (but apparently not by Empress Wang). A wild bear broke loose and tried to get out of the cage. All of the noble men and women who accompanied Emperor Yuan, including Consort Fu, fled, but Consort Feng stepped towards the wild bear, which, however, was killed at the last moment by imperial guards. Emperor Yuan asked her why she did so. Consort Feng replied that she had heard that bears, once they had caught one person, would attack just that person and no one else; because she was afraid that the bear would attack the emperor, she was willing to sacrifice herself. Consort Fu became greatly jealous of the praise that Consort Feng received for this incident.

As princess dowager
After Emperor Cheng's death, Prince Kang went to his Principality of Shanyang. Consort Fu received the title of princess dowager. She was not content with this status, however, and continued to wish for her son to become emperor. In 27 BC, Prince Kang's principality was moved to Dingtao, and Princess Dowager Fu went with him. When Prince Kang's consort Consort Ding gave birth to his son Liu Xin, it was Princess Dowager Fu – not Consort Ding – who raised him, nurturing him to be particularly close to her and dominated by her.

The fraternal relationship between Emperor Cheng and Prince Kang was quite affectionate, notwithstanding the succession struggle, and Prince Kang was often summoned to the capital Chang'an to accompany Emperor Cheng, who was sonless. Prince Kang was viewed by many as the potential heir – a status that was greatly despised by Emperor Cheng's uncles from the Wang clan. Consort Fu was happy to see this situation develop. In 24 BC, however, under pressure from his uncles, Emperor Cheng was forced to send Prince Kang back to his principality, and it would appear that the brothers did not see each other again alive, as Prince Kang died in September or October 23 BC and was succeeded by his son Prince Xin.

Princess Dowager Fu continued to carry her title after her son's death (as Han Dynasty did not appear to have a title such as "grand princess dowager"), and she maintained a strong grip on her grandson, the person now that she wants to eventually become emperor. As Emperor Cheng continued to be sonless, in 9 BC, he considered making either his younger brother Prince Liu Xing of Zhongshan or his nephew Prince Xin his heir. Emperor Cheng became convinced that Prince Xin was more capable, and at the same time, Princess Dowager Fu was endearing herself to Emperor Cheng's wife Empress Zhao Feiyan, her sister and favourite of Emperor Cheng Consort Zhao Hede, and Emperor Cheng's uncle Wang Gen with lavish gifts, and so the Zhaos and Wang Gen both praised Prince Xin as well. Emperor Cheng made Prince Xin crown prince in March 8 BC. Emperor Cheng, under the rationale that Prince Xin was now his adopted son and no longer Prince Kang's son, refused to allow Princess Dowager Fu and her daughter-in-law Consort Ding to see Prince Xin, but the more lenient Empress Dowager Wang allowed Princess Dowager Fu to see her grandson under the rationale that she was like a wet nurse.

Domination of her grandson's reign
Emperor Cheng died suddenly in April 7 BC, and Prince Xin ascended the throne as Emperor Ai. Almost immediately, the issue of the roles of Princess Dowager Fu and Consort Ding (and what honour, if any, to posthumously bestow on Emperor Ai's father Prince Kang) would quickly erupt into a major controversy. Initially, Grand Empress Dowager Wang decreed that Princess Dowager Fu and Consort Ding see him periodically, every 10 days. However, Princess Dowager Fu quickly began to visit her grandson everyday, and she insisted that two things be done: that she receive an Empress Dowager title, and that her relatives be granted titles, like the Wangs. Grand Empress Dowager Wang, sympathetic of the bind that Emperor Ai was in, first granted Prince Kang the unusual title of "Emperor Gong of Dingtao" (定陶共皇) and then, under the rationale of that title, granted Princess Dowager Fu the title "Empress Dowager Gong of Dingtao" (定陶共皇太后) and Consort Ding the title "Empress Gong of Dingtao" (定陶共皇后). Several members of the Fu and Ding clans were created marquesses. Grand Empress Dowager Wang also ordered her nephew Wang Mang, the commander of the armed forces, to resign and transfer power to the Fus and the Dings. Emperor Ai declined and begged Wang Mang to stay in his administration. Several months later, however, there would be another direct confrontation between Wang Mang and now-Empress Dowager Fu. At a major imperial banquet, the official in charge of seating placed Empress Dowager Fu's seat next to Grand Empress Dowager Wang's. When Wang Mang saw this, he rebuked the official and ordered that Empress Dowager Fu's seat be moved to the side, which drew great ire from Empress Dowager Fu, who refused to attend the banquet. To soothe her anger, Wang Mang resigned, and Emperor Ai approved his resignation. After Wang Mang's resignation, the Wangs gradually and inexorably began to lose their power. At Empress Dowager Fu's behest, the Fus and the Dings were installed in their place.

Empress Dowager Fu was not satisfied with what she saw was her inferior title (as only empress dowager, not grand empress dowager, and with the qualifier "of Dingtao"). Several key officials who opposed her move were reduced to commoner status without any other fault – including the prime minister Kong Guang (孔光) and the prime inspector Shi Dan (師丹) – two of the top three officials of the administration. The third one, Empress Dowager Fu's cousin Fu Xi (傅喜), who also opposed Empress Dowager Fu's actions notwithstanding his relationship with her, was removed from his position and sent back to his march.

In 6 BC, Empress Dowager Fu would further display her power and at the same time cause the people to be even more disappointed in Emperor Ai. Emperor Ai's cousin Liu Jizi (劉箕子), the Prince of Zhongshan (Prince Xing's son), had a congenital heart disorder, and his grandmother Feng Yuan, the princess dowager, cared for his treatment and often worshipped gods to pray for his healing. Emperor Ai, up on hearing his cousin's illness, sent imperial physicians along with his attendant Zhang You (張由) to go to Zhongshan (roughly modern Baoding, Hebei) to treat Prince Jizi. Zhang, however, was himself afflicted with a psychiatric condition (probably bipolar disorder), and when he got to Zhongshan, he suddenly, in a rage, left there and returned to Chang'an. Once he did and was ordered to explain his conduct, he made up a false reason – that he had discovered that Princess Dowager Feng was using witchcraft to curse Emperor Ai and Empress Dowager Fu. Empress Dowager Fu and Princess Dowager Feng were romantic rivals when they were both consorts to Emperor Yuan, and Empress Dowager Fu decided to use this opportunity to strike at Princess Dowager Feng. She sent a eunuch, Shi Li (史立), to serve as investigator, and Shi tortured a good number of Princess Dowager Feng's relations (including her sister Feng Xi (馮習) and her sister-in-law Junzhi (君之), some to death, but still could not build a solid case against Princess Dowager Feng. Shi Li decided to show Princess Dowager Feng who was actually behind the investigation, by referring to the wild bear incident. Princess Dowager Feng, realizing that Empress Dowager Fu was behind the investigation, went back to her palace and committed suicide. In total, 17 members of the Feng clan died as a result of the investigations. This was immediately viewed as a political case, and officials and the people all became disappointed in Emperor Ai.

In 5 BC, Empress Dowager Fu would finally get what she wanted. Emperor Ai removed the qualification "of Dingtao" from his father's posthumous title (thus making him simply "Emperor Gong"), and then gave his grandmother a variation of the grand empress dowager title (ditaitaihou (帝太太后), compared to Grand Empress Dowager Wang's title taihuangtaihou (太皇太后)) and his mother a variation of the empress dowager title (ditaihou (帝太后), compared to Empress Dowager Zhao's title huangtaihou (皇太后)), and there were therefore now four official empresses dowager in the capital, each with a full budget.

Grand Empress Dowager Fu died in February 2 BC, and was buried with Emperor Yuan – an act that drew even greater hatred from the Wangs, because now there would be no place for Grand Empress Dowager Wang, Emperor Yuan's wife, to be buried after her death. Further, Emperor Ai granted her a posthumous empress title, which was viewed to be inappropriate and disrespectful to Grand Empress Dowager Wang.

Posthumous developments
After Emperor Ai died in August 1 BC, Grand Empress Dowager Wang quickly seized power from Emperor Ai's male favourite and probable lover Dong Xian, and summoned Wang Mang back to Chang'an to serve as regent to the new Emperor Ping – the former Prince Jizi of Zhongshan. Wang Mang, bearing grudges against Grand Empress Dowager Fu and Emperor Ai, would immediately undertake several punitive actions against her posthumously. He had her title stripped, and further stripped her of her original title as princess dowager – and had her referred merely as "the mother of Prince Gong of Dingtao" (while her daughter-in-law suffered a similar posthumous demotion to "Consort Ding").

Against Grand Empress Dowager Wang's initial sentiment of stopping the punitive actions at this point, Wang Mang went further. In 5, he persuaded Grand Empress Dowager Wang to allow him to disinter Consorts Fu and Ding's caskets and strip their bodies of jade burial shells, and then returned to Dingtao to be buried there. Their tombs were then completely flattened and surrounded with thorns. According to legend, when their tombs were opened up, great fires started, damaging their bodies and the burial items.

Titles

 * Her Imperial Highness Fu Jieyu 傅婕妤
 * Her Imperial Highness Fu Zhaoyi 傅昭儀
 * Her Imperial Highness The Princess dowager of Dingtao 定陶王太后
 * Her Imperial Highness Princess Dowager Du of Dingtao 定陶傅王太后
 * Her Imperial Majesty Empress Dowager Gong 恭皇太后
 * Her Imperial Majesty The Empress Grandmother of the Emperor 帝太太后
 * Her Imperial Majesty The Grand Empress Dowager 皇太太后
 * Empress Xiaoyuan Fu 孝元傅皇后
 * Mother of Prince Gong of Dingtao 定陶恭王母

title: Consort Fu (Yuan)


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Defiance, Ohio (band)

Defiance, Ohio is a folk punk band from Columbus, Ohio, named after the town of Defiance, Ohio, in far northwest Ohio, near Toledo. The band features a violin, cello, and double bass. They have released four full-length LPs, as well as a handful of split recordings. Their music has typically been released through smaller independent labels. They are known for their association with Plan-It-X Records, as well as anti-capitalist themes.

History
Defiance, Ohio formed in 2002 as a three-piece in Columbus, Ohio with Geoff on guitar, Ryan on Bass, and Will on drums. By the time they recorded their first demo, BZ joined on violin and Sherri on cello. Their first album Share What Ya Got was self-released on CDr and on vinyl by Friends and Relatives. It was later reissued on vinyl by No Idea. The band moved to Bloomington, Indiana, released two split EPs with Ghost Mice, and One Reason and toured the US extensively, as well as Iceland. In 2004 Will moved to Eureka, California, and Theo Hilton filled in on drums. Will returned from California, and Theo stayed with the band, switching back and forth on drums and guitar.

The Great Depression was released in 2006 on No Idea. After making the decisions to work with No Idea, the band felt a need to explain their actions and maintain the sense of dialogue that they feel has been consistent in how they have conducted themselves as a band. They said that they would have preferred to release the album themselves, but that they did not have the capacity to do so while continuing to keep older releases in print.

In 2007 the band released a split EP with Environmental Youth Crunch, and a full-length The Fear, The Fear, The Fear on No Idea. Theo moved to Athens, Georgia, while the band continued to practice, tour and write together, and they toured Australia for the first time.

In 2009 the band released 4 songs as a benefit for the Icarus Project. These songs were later remastered and released as part of the full-length Midwestern Minutes in 2010. Geoff relocated to Chicago. Due to band members moving to different cities, and starting other musical projects, activity has slowed down in recent years compared to the consistent and extensive touring of earlier years. Their latest release is a 6-song EP, The Calling on No Idea. Their last tour was in 2015.

Most of the members of Defiance, Ohio also participate in other bands, such as Nana Grizol, Pretty Hot, Landlord, Pink Houses, Hymns, Disaster, High Dive, Memento Mori (now defunct), and other musical projects. Ryan Woods was also part-owner of the popular vegetarian restaurant The Owlery in Bloomington, Indiana. He has since sold off his ownership of the restaurant.

Members

 * Current
 * Ryan Woods – double bass, bass, vocals (2002–present)
 * Geoff Hing – guitar, vocals (2002–present)
 * Will Staler – drums, guitar, vocals (2002–present)
 * Elizabeth "BZ" Gibbs – violin (2002–present)
 * Sherri Miller – cello, banjo, vocals (2002–present)
 * Theo Hilton – drums, guitar, vocals (2005–present)

Discography

 * Albums
 * Share What Ya Got (2003)
 * The Great Depression (2006)
 * The Fear, The Fear, The Fear (2007)
 * Midwestern Minutes (2010)

 * Demos & EPs
 * Demo (2003)
 * 2003 Tour CDR (2003)
 * Songs for the Icarus Project (2009) (a benefit for the Icarus Project)
 * The Calling (2012)

 * Splits
 * Ghost Mice/Defiance Ohio Split CD (2004)
 * One Reason/Defiance, Ohio Split 7" (2004)
 * Environmental Youth Crunch/Defiance, Ohio Split 7" (2007)

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

title: Defiance, Ohio (band)


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Talk:The Power of Love (Frankie Goes to Hollywood song)

Who wrote the song?
Who wrote the song?

Easy. Holly Johnson, Mark O'Toole and Brian Nash.
 * Despite the standard credits, it is widely believed that this song was largely written and performed by Anne Dudley and Trevor Horn. The entire Pleasuredome album didn't use any of the band performances apart from Holly Johnson's vocals, something that caused major grief at the time. --Ef80 (talk) 23:01, 25 June 2020 (UTC)

Fair use rationale for Image:Power of love single.jpg
Image:Power of love single.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.

Please go to the image description page and edit it to include a fair use rationale. Using one of the templates at Fair use rationale guideline is an easy way to insure that your image is in compliance with Wikipedia policy, but remember that you must complete the template. Do not simply insert a blank template on an image page.

If there is other other fair use media, consider checking that you have specified the fair use rationale on the other images used on this page. Note that any fair use images uploaded after 4 May, 2006, and lacking such an explanation will be deleted one week after they have been uploaded, as described on criteria for speedy deletion. If you have any questions please ask them at the Media copyright questions page. Thank you.BetacommandBot 08:03, 6 June 2007 (UTC)

Use in commercials
This song was used in an 80s commercial for Durex. I found this YouTube link, which obviously we can't use, but does anyone have a better source we can use? Paul MacDermott (talk) 21:02, 24 December 2013 (UTC)

External links modified
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Another cover
Hi all,

for the last section: Frank Dellé (from Seeed) covers the song in 2009 https://www.youtube.com/watch?v=pE0uRmETWKc — Preceding unsigned comment added by 89.134.118.153 (talk) 01:09, 21 February 2016 (UTC)

External links modified
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Critical Reception
Is not the critical reception section a little on the short side? As an aside, while almost everthing is considered a joke these days, should not songs with a straight/positive message be a little "straight-laced"? — Preceding unsigned comment added by 2.30.22.132 (talk) 15:45, 25 December 2023 (UTC)

Original camp ironic content?
Hi, what is "the track's original camp ironic content" mentioned in the article? Was the content changed? And how is the reference to the cartoon character "Hooded Claw" an example of camp and ironic? 2A0A:A541:8542:0:5C8F:642C:A5AB:ED7D (talk) 13:24, 3 June 2024 (UTC)


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Wikipedia:Articles for deletion/Chi-Chi (Scarface)

 This page is an archive of the proposed deletion of the article below. Further comments should be made on the appropriate discussion page (such as the article's talk page or on a Votes for Undeletion nomination). No further edits should be made to this page.

The result of the debate was delete. We have a close situation here (4 delete, 1 merge, 1 redirect), the entire content of the article is ""Chi-Chi" is a character from Scarface played by Angel Salazar. He is the short guy in the gang." Now, the Scarface itself is a disambiguation page, and so merging and redirecting would need to be to one of the actual movie articles, Scarface (1932 film) or Scarface (1983 film). Unfortunately, the article here does not say which one the character appears in. There is very little content here, and it is not easy to merge this with either of those articles anyway. With no good bearing of where or how to merge this, I will call this a delete. Sjakkalle (Check!)  08:44, 6 September 2005 (UTC)

Chi-Chi (Scarface)
One character in the movie -- more Wiki brah clean up. Paul Klenk 06:21, 28 August 2005 (UTC)
 * Author actually created a template to categorize Scarface-trivia.  Paul Klenk
 * Delete SaltyPig 12:08, 28 August 2005 (UTC)
 * Merge and redirect to Scarface. Pburka 15:39, August 28, 2005 (UTC)
 * Delete Minor character, expand Scarface if necessary. --12.42.48.204 15:54, 28 August 2005 (UTC)
 * redirect to scarface. __earth 04:57, August 29, 2005 (UTC)
 * Delete This user has now created a Scarface-related stub. If anything, combine it with the Scarface article. WATCH THIS USER -- he is deleting VfD tags, and posting a lot of nonsense. — Preceding unsigned comment added by Paul Klenk (talk • contribs) 08:00, 26 August 2005 (UTC)  –Comment merged here from Votes for deletion/Chi-Chi (Scarface)
 * This nom was never posted to a main VfD page; completing nomination. No vote. JYolkowski // talk 22:44, 3 September 2005 (UTC) –Comment merged here from Votes for deletion/Chi-Chi (Scarface)
 * Delete, hasn't this been deleted previously? -- Kjkolb 08:07, September 4, 2005 (UTC) –Comment merged here from Votes for deletion/Chi-Chi (Scarface)
 * Not this exact article, no. JYolkowski // talk 15:03, 4 September 2005 (UTC) –Comment merged here from Votes for deletion/Chi-Chi (Scarface)
 * There's already an open VFD page for this article at Pages for deletion/Chi-Chi (Scarface). Pburka 16:08, September 4, 2005 (UTC) –Comment merged here from Votes for deletion/Chi-Chi (Scarface)
 * Delete Possibly incorporate in Scarface. --Meiers Twins 14:59, 5 September 2005 (UTC)
 * The above discussion is preserved as an archive of the debate. Please do not modify it. Subsequent comments should be made on the appropriate discussion page (such as the article's talk page or in an undeletion request). No further edits should be made to this page.


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Talk:Detroit Safari

Delete an article?

 * The article Detroit Neons was put on vfd c. Mar 20, 2005; why would somebody delete an article that could just be redirected to Detroit Safari. Apparently the article Detroit Neon redirects there.

title: Talk:Detroit Safari


text:
Ruckman, West Virginia

Ruckman is an unincorporated community farming community in Hampshire County in the U.S. state of West Virginia. Ruckman is located at the intersections of Ash Ruckman Road (WV Secondary Route 7/4) with J.C. Ruckman (WV Secondary Route 12/6) and Edgar Loy (WV Secondary Route 7/7) Roads south of Augusta and northeast of Kirby. It is named for the Ruckman family that is still prevalent in the area. Ruckman's post office is no longer in operation.

title: Ruckman, West Virginia


text:
Leibniz integral rule

In calculus, the Leibniz integral rule for differentiation under the integral sign, named after Gottfried Wilhelm Leibniz, states that for an integral of the form $$\int_{a(x)}^{b(x)} f(x,t)\,dt,$$ where $$-\infty < a(x), b(x) < \infty$$ and the integrands are functions dependent on $$x,$$ the derivative of this integral is expressible as $$\begin{align} & \frac{d}{dx} \left (\int_{a(x)}^{b(x)} f(x,t)\,dt \right ) \\ &= f\big(x,b(x)\big)\cdot \frac{d}{dx} b(x) - f\big(x,a(x)\big)\cdot \frac{d}{dx} a(x) + \int_{a(x)}^{b(x)}\frac{\partial}{\partial x} f(x,t) \,dt \end{align}$$ where the partial derivative $$\tfrac{\partial}{\partial x}$$ indicates that inside the integral, only the variation of $$f(x, t)$$ with $$x$$ is considered in taking the derivative.

In the special case where the functions $$a(x)$$ and $$b(x)$$ are constants $$a(x) = a$$ and $$b(x) = b$$ with values that do not depend on $$x,$$ this simplifies to: $$\frac{d}{dx} \left(\int_a^b f(x,t)\,dt \right)= \int_a^b \frac{\partial}{\partial x} f(x,t) \,dt.$$

If $$a(x)=a$$ is constant and $$b(x)=x$$, which is another common situation (for example, in the proof of Cauchy's repeated integration formula), the Leibniz integral rule becomes: $$\frac{d}{dx} \left (\int_a^x f(x,t) \, dt \right )= f\big(x,x\big) + \int_a^x \frac{\partial}{\partial x} f(x,t) \, dt,$$

This important result may, under certain conditions, be used to interchange the integral and partial differential operators, and is particularly useful in the differentiation of integral transforms. An example of such is the moment generating function in probability theory, a variation of the Laplace transform, which can be differentiated to generate the moments of a random variable. Whether Leibniz's integral rule applies is essentially a question about the interchange of limits.

General form: differentiation under the integral sign
$$ The right hand side may also be written using Lagrange's notation as: $f(x, b(x)) \, b^\prime(x) - f(x, a(x)) \, a^\prime(x) + \displaystyle\int_{a(x)}^{b(x)} f_x(x, t) \,dt.$

Stronger versions of the theorem only require that the partial derivative exist almost everywhere, and not that it be continuous. This formula is the general form of the Leibniz integral rule and can be derived using the fundamental theorem of calculus. The (first) fundamental theorem of calculus is just the particular case of the above formula where $$a(x) = a \in \Reals$$ is constant, $$b(x) = x,$$ and $$f(x, t) = f(t)$$ does not depend on $$x.$$

If both upper and lower limits are taken as constants, then the formula takes the shape of an operator equation: $$\mathcal{I}_t \partial_x = \partial_x \mathcal{I}_t$$ where $$\partial_x$$ is the partial derivative with respect to $$x$$ and $$\mathcal{I}_t$$ is the integral operator with respect to $$t$$ over a fixed interval. That is, it is related to the symmetry of second derivatives, but involving integrals as well as derivatives. This case is also known as the Leibniz integral rule.

The following three basic theorems on the interchange of limits are essentially equivalent:
 * the interchange of a derivative and an integral (differentiation under the integral sign; i.e., Leibniz integral rule);
 * the change of order of partial derivatives;
 * the change of order of integration (integration under the integral sign; i.e., Fubini's theorem).

Three-dimensional, time-dependent case
A Leibniz integral rule for a two dimensional surface moving in three dimensional space is

$$\frac {d}{dt} \iint_{\Sigma (t)} \mathbf{F} (\mathbf{r}, t) \cdot d \mathbf{A} = \iint_{\Sigma (t)} \left(\mathbf{F}_t (\mathbf{r}, t) + \left[\nabla \cdot \mathbf{F} (\mathbf{r}, t) \right] \mathbf{v} \right) \cdot d \mathbf{A} - \oint_{\partial \Sigma (t)} \left[ \mathbf{v} \times \mathbf{F} ( \mathbf{r}, t) \right] \cdot d \mathbf{s},$$

where:
 * $F(r, t)$ is a vector field at the spatial position $r$ at time $t$,
 * $Σ$ is a surface bounded by the closed curve $∂Σ$,
 * $dA$ is a vector element of the surface $Σ$,
 * $ds$ is a vector element of the curve $∂Σ$,
 * $v$ is the velocity of movement of the region $Σ$,
 * $∇⋅$ is the vector divergence,
 * $×$ is the vector cross product,
 * The double integrals are surface integrals over the surface $Σ$, and the line integral is over the bounding curve $∂Σ$.

Higher dimensions
The Leibniz integral rule can be extended to multidimensional integrals. In two and three dimensions, this rule is better known from the field of fluid dynamics as the Reynolds transport theorem: $$\frac{d}{dt} \int_{D(t)} F(\mathbf x, t) \,dV = \int_{D(t)} \frac{\partial}{\partial t} F(\mathbf x, t)\,dV + \int_{\partial D(t)} F(\mathbf x, t) \mathbf v_b \cdot d\mathbf{\Sigma},$$

where $$F(\mathbf x, t)$$ is a scalar function, $D(t)$ and $∂D(t)$ denote a time-varying connected region of R3 and its boundary, respectively, $$\mathbf v_b$$ is the Eulerian velocity of the boundary (see Lagrangian and Eulerian coordinates) and $dΣ = n dS$ is the unit normal component of the surface element.

The general statement of the Leibniz integral rule requires concepts from differential geometry, specifically differential forms, exterior derivatives, wedge products and interior products. With those tools, the Leibniz integral rule in n dimensions is $$\frac{d}{dt}\int_{\Omega(t)}\omega=\int_{\Omega(t)} i_{\mathbf v}(d_x\omega)+\int_{\partial \Omega(t)} i_{\mathbf v} \omega + \int_{\Omega(t)} \dot{\omega},$$ where $Ω(t)$ is a time-varying domain of integration, ω is a p-form, $$\mathbf v=\frac{\partial \mathbf x}{\partial t}$$ is the vector field of the velocity, $$i_{\mathbf v}$$ denotes the interior product with $$\mathbf v$$, dxω is the exterior derivative of ω with respect to the space variables only and $$\dot{\omega}$$ is the time derivative of ω.

If we move into spacetime $$M = \mathbb{R} \times \mathbb{R}^3 $$ we can introduce the spacetime velocity field $$\Psi=\frac{\partial}{\partial t}+ \mathbf v$$ of the surface $$\Omega(t)$$ and the spacetime exterior derivative $$d\omega= dt\wedge \dot\omega + d_x\omega $$, so that the identity can be rewritten as a general statement about Lie derivatives: $$ \frac{d}{dt}\int_{\Omega(t)} \omega = \int_{\Omega(t)} \mathcal{L}_\Psi \omega.$$ Here, the differential form $$\omega$$ has only spatial components. Indeed, by Cartan's magic formula, the Lie derivative simplifies to $$ \mathcal{L}_\Psi \omega = \mathcal{L}_{\mathbf v} \omega + \mathcal{L}_{\frac{\partial}{\partial t}} \omega = i_{\mathbf v} d\omega + d i_{\mathbf v} \omega + i_{\frac{\partial}{\partial t}} d \omega = i_{\mathbf v} d_{x} \omega + d i_{\mathbf v} \omega + \dot\omega $$ which, after integrating over $$\Omega(t)$$ and using generalized Stokes' theorem on the second term, correspond to the three terms in the formula above.

Measure theory statement
Let $$X$$ be an open subset of $$\mathbf{R}$$, and $$\Omega$$ be a measure space. Suppose $$f\colon X \times \Omega \to \mathbf{R} $$ satisfies the following conditions:

 * 1) $$f(x,\omega)$$ is a Lebesgue-integrable function of $$\omega$$ for each $$x \in X$$.
 * 2) For almost all $$\omega \in \Omega$$, the partial derivative $$f_x$$ exists for all $$x \in X$$.
 * 3) There is an integrable function $$\theta \colon \Omega \to \mathbf{R}$$ such that $$|f_x(x,\omega)| \leq \theta ( \omega)$$ for all $$x \in X$$ and almost every $$\omega \in \Omega$$.

Then, for all $$x \in X$$, $$\frac{d}{dx} \int_\Omega f(x, \omega) \, d\omega = \int_{\Omega} f_x (x, \omega) \, d\omega.$$

The proof relies on the dominated convergence theorem and the mean value theorem (details below).

Proof of basic form
We first prove the case of constant limits of integration a and b.

We use Fubini's theorem to change the order of integration. For every $x$ and $h$, such that $h > 0$ and both $x$ and $x +h$ are within $[x_{0},x_{1}]$, we have: $$ \int_x^{x+h} \int_a^b f_x(x,t) \,dt \,dx = \int_a^b \int_x^{x+h} f_x(x,t) \,dx \,dt = \int_a^b \left(f(x+h,t) - f(x,t)\right) \,dt = \int_a^b f(x+h,t) \,dt - \int_a^b f(x,t) \,dt $$

Note that the integrals at hand are well defined since $$ f_x(x,t) $$ is continuous at the closed rectangle $$ [x_0, x_1] \times [a,b] $$ and thus also uniformly continuous there; thus its integrals by either dt or dx are continuous in the other variable and also integrable by it (essentially this is because for uniformly continuous functions, one may pass the limit through the integration sign, as elaborated below).

Therefore: $$\frac{\int_a^b f(x+h,t) \,dt - \int_a^b f(x,t) \,dt }{h} = \frac{1}{h}\int_x^{x+h} \int_a^b f_x(x,t) \,dt \,dx = \frac{F(x+h)-F(x)}{h}$$

Where we have defined: $$F(u) \equiv \int_{x_0}^{u} \int_a^b f_x(x,t) \,dt \,dx$$ (we may replace x0 here by any other point between x0 and x)

F is differentiable with derivative $\int_a^b f_x(x,t) \,dt $, so we can take the limit where $h$ approaches zero. For the left hand side this limit is: $$\frac{d}{dx}\int_a^b f(x,t) \, dt $$

For the right hand side, we get: $$ F'(x) = \int_a^b f_x(x,t) \, dt $$ And we thus prove the desired result: $$\frac{d}{dx}\int_a^b f(x,t) \, dt = \int_a^b f_x(x,t) \, dt $$

Another proof using the bounded convergence theorem
If the integrals at hand are Lebesgue integrals, we may use the bounded convergence theorem (valid for these integrals, but not for Riemann integrals) in order to show that the limit can be passed through the integral sign.

Note that this proof is weaker in the sense that it only shows that fx(x,t) is Lebesgue integrable, but not that it is Riemann integrable. In the former (stronger) proof, if f(x,t) is Riemann integrable, then so is fx(x,t) (and thus is obviously also Lebesgue integrable).

Let

By the definition of the derivative,

Substitute equation ($$) into equation ($$). The difference of two integrals equals the integral of the difference, and 1/h is a constant, so $$\begin{align} u'(x) &= \lim_{h \to 0} \frac{\int_a^bf(x + h, t)\,dt - \int_a^b f(x, t)\,dt}{h} \\ &= \lim_{h \to 0} \frac{\int_a^b\left( f(x + h, t) - f(x,t) \right)\,dt}{h} \\ &= \lim_{h \to 0} \int_a^b \frac{f(x + h, t) - f(x, t)}{h} \,dt. \end{align}$$

We now show that the limit can be passed through the integral sign.

We claim that the passage of the limit under the integral sign is valid by the bounded convergence theorem (a corollary of the dominated convergence theorem). For each δ > 0, consider the difference quotient $$f_\delta(x, t) = \frac{f(x + \delta, t) - f(x, t)}{\delta}.$$ For t fixed, the mean value theorem implies there exists z in the interval [x, x + δ] such that $$f_\delta(x, t) = f_x(z, t).$$ Continuity of fx(x, t) and compactness of the domain together imply that fx(x, t) is bounded. The above application of the mean value theorem therefore gives a uniform (independent of $$t$$) bound on $$f_\delta(x, t)$$. The difference quotients converge pointwise to the partial derivative fx by the assumption that the partial derivative exists.

The above argument shows that for every sequence {δn} → 0, the sequence $$\{f_{\delta_n}(x, t)\}$$ is uniformly bounded and converges pointwise to fx. The bounded convergence theorem states that if a sequence of functions on a set of finite measure is uniformly bounded and converges pointwise, then passage of the limit under the integral is valid. In particular, the limit and integral may be exchanged for every sequence {δn} → 0. Therefore, the limit as δ → 0 may be passed through the integral sign.

Variable limits form
For a continuous real valued function g of one real variable, and real valued differentiable functions $$ f_1 $$ and $$ f_2 $$ of one real variable, $$\frac{d}{dx} \left( \int_{f_1(x)}^{f_2(x)} g(t) \,dt \right )= g\left(f_2(x)\right) {f_2'(x)} - g\left(f_1(x)\right) {f_1'(x)}.$$

This follows from the chain rule and the First Fundamental Theorem of Calculus. Define $$ G(x) = \int_{f_1(x)}^{f_2(x)} g(t) \, dt, $$ and $$ \Gamma(x) = \int_{0}^{x} g(t) \, dt. $$ (The lower limit just has to be some number in the domain of $$ g $$)

Then, $$ G(x) $$ can be written as a composition: $$ G(x) = (\Gamma \circ f_2)(x) - (\Gamma \circ f_1)(x) $$. The Chain Rule then implies that $$ G'(x) = \Gamma'\left(f_2(x)\right) f_2'(x) - \Gamma'\left(f_1(x)\right) f_1'(x). $$ By the First Fundamental Theorem of Calculus, $$ \Gamma'(x) = g(x) $$. Therefore, substituting this result above, we get the desired equation: $$ G'(x) = g\left(f_2(x)\right) {f_2'(x)} - g\left(f_1(x)\right) {f_1'(x)}. $$

Note: This form can be particularly useful if the expression to be differentiated is of the form: $$\int_{f_1(x)}^{f_2(x)} h(x)g(t) \,dt$$ Because $$h(x)$$ does not depend on the limits of integration, it may be moved out from under the integral sign, and the above form may be used with the Product rule, i.e., $$\frac{d}{dx} \left( \int_{f_1(x)}^{f_2(x)} h(x)g(t) \,dt \right ) = \frac{d}{dx} \left(h(x) \int_{f_1(x)}^{f_2(x)} g(t) \,dt \right ) = h'(x)\int_{f_1(x)}^{f_2(x)} g(t) \,dt + h(x) \frac{d}{dx} \left(\int_{f_1(x)}^{f_2(x)} g(t) \,dt \right ) $$

General form with variable limits
Set $$\varphi(\alpha) = \int_a^b f(x,\alpha)\,dx,$$ where a and b are functions of α that exhibit increments Δa and Δb, respectively, when α is increased by Δα. Then, $$\begin{align} \Delta\varphi &= \varphi(\alpha + \Delta\alpha) - \varphi(\alpha) \\[4pt] &= \int_{a + \Delta a}^{b + \Delta b}f(x, \alpha + \Delta\alpha)\,dx - \int_a^b f(x, \alpha)\,dx \\[4pt] &= \int_{a + \Delta a}^af(x, \alpha + \Delta\alpha)\,dx + \int_a^bf(x, \alpha + \Delta\alpha)\,dx + \int_b^{b + \Delta b} f(x, \alpha+\Delta\alpha)\,dx - \int_a^b f(x, \alpha)\,dx \\[4pt] &= -\int_a^{a + \Delta a} f(x, \alpha + \Delta\alpha)\,dx + \int_a^b [f(x, \alpha + \Delta\alpha) - f(x,\alpha)]\,dx + \int_b^{b + \Delta b} f(x, \alpha + \Delta\alpha)\,dx. \end{align}$$

A form of the mean value theorem, $\int_a^b f(x)\,dx = (b - a)f(\xi)$, where a < ξ < b, may be applied to the first and last integrals of the formula for Δφ above, resulting in $$\Delta\varphi = -\Delta a f(\xi_1, \alpha + \Delta\alpha) + \int_a^b [f(x, \alpha + \Delta\alpha) - f(x,\alpha)]\,dx + \Delta b f(\xi_2, \alpha + \Delta\alpha).$$

Divide by Δα and let Δα → 0. Notice ξ1 → a and ξ2 → b. We may pass the limit through the integral sign: $$\lim_{\Delta\alpha\to 0}\int_a^b \frac{f(x,\alpha + \Delta\alpha) - f(x,\alpha)}{\Delta\alpha}\,dx = \int_a^b \frac{\partial}{\partial\alpha}f(x, \alpha)\,dx,$$ again by the bounded convergence theorem. This yields the general form of the Leibniz integral rule, $$\frac{d\varphi}{d\alpha} = \int_a^b \frac{\partial}{\partial\alpha}f(x, \alpha)\,dx + f(b, \alpha) \frac{db}{d\alpha} - f(a, \alpha)\frac{da}{d\alpha}.$$

Alternative proof of the general form with variable limits, using the chain rule
The general form of Leibniz's Integral Rule with variable limits can be derived as a consequence of the basic form of Leibniz's Integral Rule, the multivariable chain rule, and the first fundamental theorem of calculus. Suppose $$ f $$ is defined in a rectangle in the $$ x-t $$ plane, for $$ x \in [x_1, x_2] $$ and $$ t \in [t_1, t_2] $$. Also, assume $$ f $$ and the partial derivative $ \frac{\partial f}{\partial x} $ are both continuous functions on this rectangle. Suppose $$ a, b$$ are differentiable real valued functions defined on $$ [x_1, x_2]$$, with values in $$ [t_1, t_2] $$ (i.e. for every $$ x \in [x_1, x_2], a(x), b(x) \in [t_1, t_2] $$). Now, set $$ F(x,y) = \int_{t_1}^{y} f(x,t)\,dt, \qquad \text{for} ~ x \in [x_1, x_2] ~\text{and}~ y \in [t_1, t_2] $$ and $$ G(x) = \int_{a(x)}^{b(x)} f(x,t)\,dt, \quad \text{for} ~ x \in [x_1, x_2] $$

Then, by properties of definite Integrals, we can write $$ G(x) = \int_{t_1}^{b(x)} f(x,t)\,dt - \int_{t_1}^{a(x)} f(x,t)\,dt = F(x, b(x)) - F(x, a(x))$$

Since the functions $$ F, a, b $$ are all differentiable (see the remark at the end of the proof), by the multivariable chain rule, it follows that $$ G $$ is differentiable, and its derivative is given by the formula: $$ G'(x) = \left(\frac{\partial F}{\partial x} (x, b(x)) + \frac{\partial F}{\partial y} (x, b(x) ) b'(x) \right) - \left(\frac{\partial F}{\partial x} (x, a(x)) + \frac{\partial F}{\partial y} (x, a(x)) a'(x) \right) $$Now, note that for every $$ x \in [x_1, x_2] $$, and for every $$ y \in [t_1, t_2] $$, we have that $ \frac{\partial F}{\partial x}(x, y) = \int_{t_1}^y \frac{\partial f}{\partial x}(x,t) \, dt $, because when taking the partial derivative with respect to $$ x $$ of $$ F $$, we are keeping $$ y $$ fixed in the expression $ \int_{t_1}^{y} f(x,t)\,dt $ ; thus the basic form of Leibniz's Integral Rule with constant limits of integration applies. Next, by the first fundamental theorem of calculus, we have that $ \frac{\partial F}{\partial y}(x, y) = f(x,y) $ ; because when taking the partial derivative with respect to $$ y $$ of $$ F $$, the first variable $$ x $$ is fixed, so the fundamental theorem can indeed be applied.

Substituting these results into the equation for $$ G'(x) $$ above gives: $$ \begin{align} G'(x) &= \left(\int_{t_1}^{b(x)} \frac{\partial f}{\partial x}(x,t) \, dt + f(x, b(x)) b'(x) \right) - \left(\int_{t_1}^{a(x)} \dfrac{\partial f}{\partial x}(x,t) \, dt + f(x, a(x)) a'(x) \right) \\[2pt] &= f(x,b(x)) b'(x) - f(x,a(x)) a'(x) + \int_{a(x)}^{b(x)} \frac{\partial f}{\partial x}(x,t) \, dt, \end{align}$$ as desired.

There is a technical point in the proof above which is worth noting: applying the Chain Rule to $$ G $$ requires that $$ F $$ already be differentiable. This is where we use our assumptions about $$ f $$. As mentioned above, the partial derivatives of $$ F $$ are given by the formulas $ \frac{\partial F}{\partial x}(x, y) = \int_{t_1}^y \frac{\partial f}{\partial x}(x,t) \, dt $  and $ \frac{\partial F}{\partial y}(x, y) = f(x,y) $. Since $ \dfrac{\partial f}{\partial x}$ is continuous, its integral is also a continuous function, and since $$ f $$ is also continuous, these two results show that both the partial derivatives of $$ F $$ are continuous. Since continuity of partial derivatives implies differentiability of the function, $$ F $$ is indeed differentiable.

Three-dimensional, time-dependent form
At time t the surface Σ in Figure 1 contains a set of points arranged about a centroid $$\mathbf{C}(t)$$. The function $$\mathbf{F}(\mathbf{r}, t)$$ can be written as $$\mathbf{F}(\mathbf{C}(t) + \mathbf{r} - \mathbf{C}(t), t) = \mathbf{F}(\mathbf{C}(t) + \mathbf{I}, t),$$ with $$\mathbf{I}$$ independent of time. Variables are shifted to a new frame of reference attached to the moving surface, with origin at $$\mathbf{C}(t)$$. For a rigidly translating surface, the limits of integration are then independent of time, so: $$\frac {d}{dt} \left (\iint_{\Sigma (t)} d \mathbf{A}_{\mathbf{r}}\cdot \mathbf{F}(\mathbf{r}, t) \right) = \iint_\Sigma d \mathbf{A}_{\mathbf{I}} \cdot \frac {d}{dt}\mathbf{F}(\mathbf{C}(t) + \mathbf{I}, t),$$ where the limits of integration confining the integral to the region Σ no longer are time dependent so differentiation passes through the integration to act on the integrand only: $$\frac {d}{dt}\mathbf{F}( \mathbf{C}(t) + \mathbf{I}, t) = \mathbf{F}_t(\mathbf{C}(t) + \mathbf{I}, t) + \mathbf{v \cdot \nabla F}(\mathbf{C}(t) + \mathbf{I}, t) = \mathbf{F}_t(\mathbf{r}, t) + \mathbf{v} \cdot \nabla \mathbf{F}(\mathbf{r}, t),$$ with the velocity of motion of the surface defined by $$\mathbf{v} = \frac {d}{dt} \mathbf{C} (t).$$

This equation expresses the material derivative of the field, that is, the derivative with respect to a coordinate system attached to the moving surface. Having found the derivative, variables can be switched back to the original frame of reference. We notice that (see article on curl) $$\nabla \times \left(\mathbf{v} \times \mathbf{F}\right) = (\nabla \cdot \mathbf{F} + \mathbf{F} \cdot \nabla) \mathbf{v}- (\nabla \cdot \mathbf{v} + \mathbf{v} \cdot \nabla) \mathbf{F},$$ and that Stokes theorem equates the surface integral of the curl over Σ with a line integral over $∂Σ$: $$\frac{d}{dt} \left(\iint_{\Sigma (t)} \mathbf{F} (\mathbf{r}, t) \cdot d \mathbf{A}\right) = \iint_{\Sigma (t)} \big(\mathbf{F}_t (\mathbf{r}, t) + \left(\mathbf{F \cdot \nabla} \right)\mathbf{v} + \left(\nabla \cdot \mathbf{F} \right) \mathbf{v} - (\nabla \cdot \mathbf{v})\mathbf{F}\big)\cdot d\mathbf{A} - \oint_{\partial \Sigma (t)}\left(\mathbf{v} \times \mathbf{F}\right)\cdot d\mathbf{s}. $$

The sign of the line integral is based on the right-hand rule for the choice of direction of line element ds. To establish this sign, for example, suppose the field F points in the positive z-direction, and the surface Σ is a portion of the xy-plane with perimeter ∂Σ. We adopt the normal to Σ to be in the positive z-direction. Positive traversal of ∂Σ is then counterclockwise (right-hand rule with thumb along z-axis). Then the integral on the left-hand side determines a positive flux of F through Σ. Suppose Σ translates in the positive x-direction at velocity v. An element of the boundary of Σ parallel to the y-axis, say ds, sweeps out an area vt × ds in time t. If we integrate around the boundary ∂Σ in a counterclockwise sense, vt × ds points in the negative z-direction on the left side of ∂Σ (where ds points downward), and in the positive z-direction on the right side of ∂Σ (where ds points upward), which makes sense because Σ is moving to the right, adding area on the right and losing it on the left. On that basis, the flux of F is increasing on the right of ∂Σ and decreasing on the left. However, the dot product $v × F ⋅ ds = −F × v ⋅ ds = −F ⋅ v × ds$. Consequently, the sign of the line integral is taken as negative.

If v is a constant, $$\frac {d}{dt} \iint_{\Sigma (t)} \mathbf{F} (\mathbf{r}, t) \cdot d \mathbf{A} = \iint_{\Sigma (t)} \big(\mathbf{F}_t (\mathbf{r}, t) + \left(\nabla \cdot \mathbf{F} \right)  \mathbf{v}\big) \cdot d \mathbf{A} - \oint_{\partial \Sigma (t)}\left(\mathbf{v} \times \mathbf{F}\right) \cdot \,d\mathbf{s},$$ which is the quoted result. This proof does not consider the possibility of the surface deforming as it moves.

Alternative derivation
Lemma. One has: $$\frac{\partial}{\partial b} \left (\int_a^b f(x) \,dx \right ) = f(b), \qquad \frac{\partial}{\partial a} \left (\int_a^b f(x) \,dx \right )= -f(a).$$

Proof. From the proof of the fundamental theorem of calculus,

$$\begin{align} \frac{\partial}{\partial b} \left(\int_a^b f(x) \,dx\right) &= \lim_{\Delta b \to 0} \frac{1}{\Delta b} \left( \int_a^{b+\Delta b} f(x)\,dx - \int_a^b f(x)\,dx \right) \\[1ex] &= \lim_{\Delta b \to 0} \frac{1}{\Delta b} \left( \int_a^b f(x) \, dx + \int_b^{b+\Delta b} f(x)\,dx - \int_a^b f(x)\,dx \right) \\[1ex] &= \lim_{\Delta b \to 0} \frac{1}{\Delta b} \int_b^{b+\Delta b} f(x)\,dx \\[1ex] &= \lim_{\Delta b \to 0} \frac{1}{\Delta b} \left[ f(b) \Delta b + O\left(\Delta b^2\right) \right] \\[1ex] &= f(b), \end{align}$$ and $$\begin{align} \frac{\partial}{\partial a} \left (\int_a^b f(x) \,dx \right )&= \lim_{\Delta a \to 0} \frac{1}{\Delta a} \left[ \int_{a+\Delta a}^b f(x)\,dx - \int_a^b f(x)\,dx \right] \\[6pt] &= \lim_{\Delta a \to 0} \frac{1}{\Delta a} \int_{a+\Delta a}^a f(x)\,dx \\[6pt] &= \lim_{\Delta a \to 0} \frac{1}{\Delta a} \left[ -f(a) \Delta a + O\left(\Delta a^2\right) \right]\\[6pt] &= -f(a). \end{align}$$

Suppose a and b are constant, and that f(x) involves a parameter α which is constant in the integration but may vary to form different integrals. Assume that f(x, α) is a continuous function of x and α in the compact set {(x, α) : α0 ≤ α ≤ α1 and a ≤ x ≤ b}, and that the partial derivative fα(x, α) exists and is continuous. If one defines: $$\varphi(\alpha) = \int_a^b f(x,\alpha)\,dx,$$ then $$\varphi$$ may be differentiated with respect to α by differentiating under the integral sign, i.e., $$\frac{d\varphi}{d\alpha}=\int_a^b\frac{\partial}{\partial\alpha}f(x,\alpha)\,dx.$$

By the Heine–Cantor theorem it is uniformly continuous in that set. In other words, for any ε > 0 there exists Δα such that for all values of x in [a, b], $$|f(x,\alpha+\Delta \alpha)-f(x,\alpha)|<\varepsilon.$$

On the other hand, $$\begin{align} \Delta\varphi &=\varphi(\alpha+\Delta \alpha)-\varphi(\alpha) \\[6pt] &=\int_a^b f(x,\alpha+\Delta\alpha)\,dx - \int_a^b f(x,\alpha)\, dx \\[6pt] &=\int_a^b \left (f(x,\alpha+\Delta\alpha)-f(x,\alpha) \right )\,dx \\[6pt] &\leq \varepsilon (b-a). \end{align}$$

Hence φ(α) is a continuous function.

Similarly if $$\frac{\partial}{\partial\alpha} f(x,\alpha)$$ exists and is continuous, then for all ε > 0 there exists Δα such that: $$\forall x \in [a, b], \quad \left|\frac{f(x,\alpha+\Delta \alpha)-f(x,\alpha)}{\Delta \alpha} - \frac{\partial f}{\partial\alpha}\right|<\varepsilon.$$

Therefore, $$\frac{\Delta \varphi}{\Delta \alpha}=\int_a^b\frac{f(x,\alpha+\Delta\alpha)-f(x,\alpha)}{\Delta \alpha}\,dx = \int_a^b \frac{\partial f(x,\alpha)}{\partial \alpha}\,dx + R,$$ where $$|R| < \int_a^b \varepsilon\, dx = \varepsilon(b-a).$$

Now, ε → 0 as Δα → 0, so $$\lim_{{\Delta \alpha} \to 0}\frac{\Delta\varphi}{\Delta \alpha}= \frac{d\varphi}{d\alpha} = \int_a^b \frac{\partial}{\partial \alpha} f(x,\alpha)\,dx.$$

This is the formula we set out to prove.

Now, suppose $$\int_a^b f(x,\alpha)\,dx=\varphi(\alpha),$$ where a and b are functions of α which take increments Δa and Δb, respectively, when α is increased by Δα. Then, $$\begin{align} \Delta\varphi &=\varphi(\alpha+\Delta\alpha)-\varphi(\alpha) \\[6pt] &=\int_{a+\Delta a}^{b+\Delta b}f(x,\alpha+\Delta\alpha)\,dx -\int_a^b f(x,\alpha)\,dx \\[6pt] &=\int_{a+\Delta a}^af(x,\alpha+\Delta\alpha)\,dx+\int_a^bf(x,\alpha+\Delta\alpha)\,dx+\int_b^{b+\Delta b}f(x,\alpha+\Delta\alpha)\,dx -\int_a^b f(x,\alpha)\,dx \\[6pt] &=-\int_a^{a+\Delta a} f(x,\alpha+\Delta\alpha)\,dx+\int_a^b[f(x,\alpha+\Delta\alpha)-f(x,\alpha)]\,dx+\int_b^{b+\Delta b} f(x,\alpha+\Delta\alpha)\,dx. \end{align}$$

A form of the mean value theorem, $\int_a^b f(x)\,dx=(b-a)f(\xi),$ where a < ξ < b, can be applied to the first and last integrals of the formula for Δφ above, resulting in $$\Delta\varphi=-\Delta a\,f(\xi_1,\alpha+\Delta\alpha)+\int_a^b[f(x,\alpha+\Delta\alpha)-f(x,\alpha)]\,dx+\Delta b\,f(\xi_2,\alpha+\Delta\alpha).$$

Dividing by Δα, letting Δα → 0, noticing ξ1 → a and ξ2 → b and using the above derivation for $$\frac{d\varphi}{d\alpha} = \int_a^b\frac{\partial}{\partial \alpha} f(x,\alpha)\,dx$$ yields $$\frac{d\varphi}{d\alpha} = \int_a^b\frac{\partial}{\partial \alpha} f(x,\alpha)\,dx+f(b,\alpha)\frac{\partial b}{\partial \alpha}-f(a,\alpha)\frac{\partial a}{\partial \alpha}. $$

This is the general form of the Leibniz integral rule.

Example 1: Fixed limits
Consider the function $$\varphi(\alpha)=\int_0^1\frac{\alpha}{x^2+\alpha^2}\,dx.$$

The function under the integral sign is not continuous at the point (x, α) = (0, 0), and the function φ(α) has a discontinuity at α = 0 because φ(α) approaches ±π/2 as α → 0±.

If we differentiate φ(α) with respect to α under the integral sign, we get $$\frac{d}{d\alpha} \varphi(\alpha)=\int_0^1\frac{\partial}{\partial\alpha}\left(\frac{\alpha}{x^2+\alpha^2}\right)\,dx=\int_0^1\frac{x^2-\alpha^2}{(x^2+\alpha^2)^2} dx=\left.-\frac{x}{x^2+\alpha^2}\right|_0^1=-\frac{1}{1+\alpha^2},$$ which is, of course, true for all values of α except α = 0. This may be integrated (with respect to α) to find $$\varphi(\alpha) = \begin{cases} 0, & \alpha = 0, \\ -\arctan({\alpha})+\frac{\pi}{2}, & \alpha \neq 0. \end{cases}$$

Example 2: Variable limits
An example with variable limits: $$\begin{align} \frac{d}{dx} \int_{\sin x}^{\cos x} \cosh t^2\,dt &= \cosh\left(\cos^2 x\right) \frac{d}{dx}(\cos x) - \cosh\left(\sin^2 x\right) \frac{d}{dx} (\sin x) + \int_{\sin x}^{\cos x} \frac{\partial}{\partial x} (\cosh t^2) \, dt \\[6pt] &= \cosh(\cos^2 x) (-\sin x) - \cosh(\sin^2 x) (\cos x) + 0 \\[6pt] &= - \cosh(\cos^2 x) \sin x - \cosh(\sin^2 x) \cos x. \end{align}$$

Evaluating definite integrals
The formula $$\frac{d}{dx} \left (\int_{a(x)}^{b(x)}f(x,t) \, dt \right) = f\big(x,b(x)\big)\cdot \frac{d}{dx} b(x) - f\big(x,a(x)\big)\cdot \frac{d}{dx} a(x) + \int_{a(x)}^{b(x)}\frac{\partial}{\partial x} f(x,t) \, dt$$ can be of use when evaluating certain definite integrals. When used in this context, the Leibniz integral rule for differentiating under the integral sign is also known as Feynman's trick for integration.

Example 3
Consider $$\varphi(\alpha)=\int_0^\pi \ln \left (1-2\alpha\cos(x)+\alpha^2 \right )\,dx, \qquad |\alpha| \neq 1.$$

Now, $$\begin{align} \frac{d}{d\alpha} \varphi(\alpha) &=\int_0^\pi \frac{-2\cos(x)+2\alpha }{1-2\alpha \cos(x)+\alpha^2} dx \\[6pt] &=\frac{1}{\alpha}\int_0^\pi \left(1-\frac{1-\alpha^2}{1-2\alpha \cos(x)+\alpha^2} \right) dx  \\[6pt] &=\left. \frac{\pi}{\alpha}-\frac{2}{\alpha}\left\{\arctan\left(\frac{1+\alpha}{1-\alpha} \tan\left(\frac{x}{2}\right)\right) \right\} \right|_0^\pi. \end{align}$$

As $$x$$ varies from $$0$$ to $$\pi$$, we have $$\begin{cases} \frac{1+\alpha}{1-\alpha} \tan\left(\frac{x}{2}\right) \geq 0, & |\alpha| < 1, \\ \frac{1+\alpha}{1-\alpha} \tan \left( \frac{x}{2}\right) \leq 0, & |\alpha| > 1. \end{cases}$$

Hence, $$\left. \arctan\left(\frac{1+\alpha}{1-\alpha}\tan\left(\frac{x}{2}\right)\right)\right|_0^\pi= \begin{cases} \frac{\pi}{2}, & |\alpha| < 1, \\ -\frac{\pi}{2}, & |\alpha| > 1. \end{cases}$$

Therefore,

$$\frac{d}{d\alpha} \varphi(\alpha)= \begin{cases} 0, & |\alpha| < 1, \\ \frac{2\pi}{\alpha}, & |\alpha| > 1. \end{cases}$$

Integrating both sides with respect to $$\alpha$$, we get: $$\varphi (\alpha) = \begin{cases} C_1, & |\alpha| < 1, \\ 2\pi \ln |\alpha| + C_2, & |\alpha| > 1. \end{cases}$$

$$C_{1} = 0$$ follows from evaluating $$\varphi (0)$$: $$\varphi(0) =\int_0^\pi \ln(1)\,dx =\int_0^\pi 0\,dx=0.$$

To determine $$C_2$$ in the same manner, we should need to substitute in a value of $$\alpha$$ greater than 1 in $$\varphi (\alpha)$$. This is somewhat inconvenient. Instead, we substitute $\alpha = \frac{1}{\beta}$, where $$|\beta| < 1$$. Then, $$\begin{align} \varphi(\alpha)   &=\int_0^\pi\left(\ln \left (1-2\beta \cos(x)+\beta^2 \right )-2\ln|\beta|\right) dx \\[6pt] &= \int_0^\pi \ln \left (1-2\beta \cos(x)+\beta^2 \right )\,dx -\int_0^\pi 2\ln|\beta| dx \\[6pt] &=0-2\pi\ln|\beta|  \\[6pt] &=2\pi\ln|\alpha|. \end{align}$$

Therefore, $$C_{2} = 0$$

The definition of $$\varphi (\alpha)$$ is now complete: $$\varphi (\alpha) = \begin{cases} 0, & |\alpha| < 1, \\ 2\pi \ln |\alpha|, & |\alpha| > 1. \end{cases}$$

The foregoing discussion, of course, does not apply when $$\alpha = \pm 1$$, since the conditions for differentiability are not met.

Example 4
$$I = \int_0^{\pi/2} \frac{1}{\left(a\cos^2 x +b\sin^2 x\right)^2}\,dx,\qquad a,b > 0.$$

First we calculate: $$\begin{align} J &= \int_0^{\pi/2} \frac{1}{a\cos^2 x + b \sin^2 x} dx \\[6pt] &= \int_0^{\pi/2} \frac{\frac{1}{\cos^2 x}}{a + b \frac{\sin^2 x}{\cos^2 x}} dx \\[6pt] &= \int_0^{\pi/2} \frac{\sec^2 x}{a +b \tan^2 x} dx \\[6pt] &= \frac{1}{b} \int_0^{\pi/2} \frac{1}{\left(\sqrt{\frac{a}{b}}\right)^2+\tan^2 x}\,d(\tan x) \\[6pt] &= \left.\frac{1}{\sqrt{ab}}\arctan \left(\sqrt{\frac{b}{a}}\tan x\right) \right|_0^{\pi/2} \\[6pt] &= \frac{\pi}{2\sqrt{ab}}. \end{align}$$

The limits of integration being independent of $$a$$, we have: $$\frac{\partial J}{\partial a}=-\int_0^{\pi/2} \frac{\cos^2 x}{\left(a\cos^2 x+b \sin^2 x\right)^2}\,dx$$

On the other hand: $$\frac{\partial J}{\partial a}= \frac{\partial}{\partial a} \left(\frac{\pi}{2\sqrt{ab}}\right) =-\frac{\pi}{4\sqrt{a^3b}}.$$

Equating these two relations then yields $$\int_0^{\pi/2} \frac{\cos^2 x}{\left(a \cos^2 x+b \sin^2 x\right)^2}\,dx=\frac{\pi}{4\sqrt{a^3b}}.$$

In a similar fashion, pursuing $$\frac{\partial J}{\partial b}$$ yields $$\int_0^{\pi/2}\frac{\sin^2 x}{\left(a\cos^2 x+b\sin^2 x\right)^2}\,dx = \frac{\pi}{4\sqrt{ab^3}}.$$

Adding the two results then produces $$I = \int_0^{\pi/2}\frac{1}{\left(a\cos^2x+b\sin^2x\right)^2}\,dx=\frac{\pi}{4\sqrt{ab}}\left(\frac{1}{a}+\frac{1}{b}\right),$$ which computes $$I$$ as desired.

This derivation may be generalized. Note that if we define $$I_n = \int_0^{\pi/2} \frac{1}{\left(a\cos^2 x+b\sin^2 x\right)^n}\,dx,$$ it can easily be shown that $$(1-n) I_n = \frac{\partial I_{n-1}}{\partial a} + \frac{\partial I_{n-1}}{\partial b}$$

Given $$I_1$$, this integral reduction formula can be used to compute all of the values of $$I_n$$ for $$n > 1$$. Integrals like $$I$$ and $$J$$ may also be handled using the Weierstrass substitution.

Example 5
Here, we consider the integral $$I(\alpha) = \int_0^{\pi/2} \frac{\ln (1+\cos\alpha \cos x)}{\cos x}\,dx, \qquad 0 < \alpha < \pi.$$

Differentiating under the integral with respect to $$\alpha$$, we have $$\begin{align} \frac{d}{d\alpha} I(\alpha) &= \int_0^{\pi/2} \frac{\partial}{\partial\alpha} \left(\frac{\ln(1 + \cos\alpha \cos x)}{\cos x}\right) \, dx \\[6pt] &=-\int_0^{\pi/2}\frac{\sin \alpha}{1+\cos \alpha \cos x}\,dx \\ &=-\int_0^{\pi/2}\frac{\sin \alpha}{\left(\cos^2 \frac{x}{2}+\sin^2 \frac{x}{2}\right)+\cos \alpha \left(\cos^2 \frac{x}{2}-\sin^2 \frac{x}{2}\right)} \, dx \\[6pt] &=-\frac{\sin\alpha}{1-\cos\alpha} \int_0^{\pi/2} \frac{1}{\cos^2\frac{x}{2}} \frac{1}{\frac{1+\cos \alpha}{1-\cos \alpha} +\tan^2 \frac{x}{2}} \, dx \\[6pt] &=-\frac{2\sin\alpha}{1-\cos\alpha} \int_0^{\pi/2} \frac{\frac{1}{2} \sec^2 \frac{x}{2}}{\frac{2 \cos^2 \frac{\alpha}{2}}{2 \sin^2\frac{\alpha}{2}} + \tan^2 \frac{x}{2}} \, dx \\[6pt] &=-\frac{2\left(2 \sin \frac{\alpha}{2} \cos \frac{\alpha}{2}\right)}{2 \sin^2 \frac{\alpha}{2}} \int_0^{\pi/2} \frac{1}{\cot^2\frac{\alpha}{2} + \tan^2 \frac{x}{2}} \, d\left(\tan \frac{x}{2}\right)\\[6pt] &=-2\cot \frac{\alpha}{2}\int_0^{\pi/2} \frac{1}{\cot^2\frac{\alpha}{2} + \tan^2\frac{x}{2}}\,d\left(\tan \frac{x}{2}\right)\\[6pt] &=-2\arctan \left(\tan \frac{\alpha}{2} \tan \frac{x}{2} \right) \bigg|_0^{\pi/2}\\[6pt] &=-\alpha. \end{align}$$

Therefore: $$I(\alpha) = C - \frac{\alpha^2}{2}.$$

But $I{\left(\frac{\pi}{2} \right)} = 0$ by definition so $C = \frac{\pi^2}{8}$  and $$ I(\alpha) = \frac{\pi^2}{8}-\frac{\alpha^2}{2}.$$

Example 6
Here, we consider the integral $$\int_0^{2\pi} e^{\cos\theta} \cos(\sin\theta) \, d\theta.$$

We introduce a new variable φ and rewrite the integral as $$f(\varphi) = \int_0^{2\pi} e^{\varphi\cos\theta} \cos(\varphi\sin\theta)\,d\theta.$$

When φ = 1 this equals the original integral. However, this more general integral may be differentiated with respect to $$\varphi$$: $$\frac{df}{d\varphi} = \int_0^{2\pi} \frac{\partial}{\partial\varphi}\left[e^{\varphi\cos\theta} \cos(\varphi\sin\theta)\right] d\theta = \int_0^{2\pi} e^{\varphi\cos\theta} \left[ \cos\theta\cos(\varphi\sin\theta)-\sin\theta\sin(\varphi\sin\theta) \right] d\theta.$$

Now, fix φ, and consider the vector field on $$ \Reals^2 $$ defined by $$ \mathbf{F}(x,y) = (F_1(x,y), F_2(x,y)) := (e^{\varphi x} \sin (\varphi y), e^{\varphi x} \cos (\varphi y)) $$. Further, choose the positive oriented parameterization of the unit circle $$ S^1 $$ given by $$ \mathbf{r} \colon [0, 2\pi) \to \Reals^2 $$, $$\mathbf{r}(\theta) := (\cos \theta, \sin \theta) $$, so that $$ \mathbf{r}'(t) = (-\sin \theta, \cos \theta) $$. Then the final integral above is precisely $$\begin{align} & \int_0^{2\pi} e^{\varphi\cos\theta} \left[ \cos\theta\cos(\varphi\sin\theta)-\sin\theta\sin(\varphi\sin\theta) \right] d\theta \\[6pt] = {} & \int_0^{2\pi} \begin{bmatrix} e^{\varphi \cos \theta} \sin (\varphi \sin \theta) \\ e^{\varphi \cos \theta} \cos (\varphi \sin \theta) \end{bmatrix} \cdot \begin{bmatrix} -\sin \theta \\ \hphantom{-} \cos \theta\end{bmatrix} \, d\theta\\[6pt] = {} & \int_0^{2\pi} \mathbf{F}(\mathbf{r}(\theta)) \cdot \mathbf{r}'(\theta) \, d\theta\\[6pt] = {} & \oint_{S^1} \mathbf{F}(\mathbf{r}) \cdot d\mathbf{r} = \oint_{S^1} F_1 \, dx + F_2 \, dy, \end{align}$$ the line integral of $$\mathbf{F}$$ over $$S^1$$. By Green's Theorem, this equals the double integral $$\iint_D \frac{\partial F_2}{\partial x} - \frac{\partial F_1}{\partial y} \, dA,$$ where $$D$$ is the closed unit disc. Its integrand is identically 0, so $$df/d\varphi$$ is likewise identically zero. This implies that f(φ) is constant. The constant may be determined by evaluating $$f$$ at $$\varphi = 0$$: $$f(0) = \int_0^{2\pi} 1\,d\theta = 2\pi.$$

Therefore, the original integral also equals $$2\pi$$.

Other problems to solve
There are innumerable other integrals that can be solved using the technique of differentiation under the integral sign. For example, in each of the following cases, the original integral may be replaced by a similar integral having a new parameter $$\alpha$$: $$\begin{align} \int_0^\infty \frac{\sin x}{x}\,dx &\to \int_0^\infty e^{-\alpha x} \frac{\sin x}{x} dx,\\[6pt] \int_0^{\pi/2} \frac{x}{\tan x}\,dx &\to\int_0^{\pi/2} \frac{\tan^{-1}(\alpha \tan x)}{\tan x} dx,\\[6pt] \int_0^\infty \frac{\ln (1+x^2)}{1+x^2}\,dx &\to\int_0^\infty \frac{\ln (1+\alpha^2 x^2)}{1+x^2} dx \\[6pt] \int_0^1 \frac{x-1}{\ln x}\,dx &\to \int_0^1 \frac{x^\alpha-1}{\ln x} dx. \end{align}$$

The first integral, the Dirichlet integral, is absolutely convergent for positive α but only conditionally convergent when $$\alpha = 0$$. Therefore, differentiation under the integral sign is easy to justify when $$\alpha > 0$$, but proving that the resulting formula remains valid when $$\alpha = 0$$ requires some careful work.

Infinite series
The measure-theoretic version of differentiation under the integral sign also applies to summation (finite or infinite) by interpreting summation as counting measure. An example of an application is the fact that power series are differentiable in their radius of convergence.

Euler-Lagrange equations
The Leibniz integral rule is used in the derivation of the Euler-Lagrange equation in variational calculus.

In popular culture
Differentiation under the integral sign is mentioned in the late physicist Richard Feynman's best-selling memoir Surely You're Joking, Mr. Feynman! in the chapter "A Different Box of Tools". He describes learning it, while in high school, from an old text, Advanced Calculus (1926), by Frederick S. Woods (who was a professor of mathematics in the Massachusetts Institute of Technology). The technique was not often taught when Feynman later received his formal education in calculus, but using this technique, Feynman was able to solve otherwise difficult integration problems upon his arrival at graduate school at Princeton University:

One thing I never did learn was contour integration. I had learned to do integrals by various methods shown in a book that my high school physics teacher Mr. Bader had given me. One day he told me to stay after class. "Feynman," he said, "you talk too much and you make too much noise. I know why. You're bored. So I'm going to give you a book. You go up there in the back, in the corner, and study this book, and when you know everything that's in this book, you can talk again." So every physics class, I paid no attention to what was going on with Pascal's Law, or whatever they were doing. I was up in the back with this book: "Advanced Calculus", by Woods. Bader knew I had studied "Calculus for the Practical Man" a little bit, so he gave me the real works—it was for a junior or senior course in college. It had Fourier series, Bessel functions, determinants, elliptic functions—all kinds of wonderful stuff that I didn't know anything about. That book also showed how to differentiate parameters under the integral sign—it's a certain operation. It turns out that's not taught very much in the universities; they don't emphasize it. But I caught on how to use that method, and I used that one damn tool again and again. So because I was self-taught using that book, I had peculiar methods of doing integrals. The result was, when guys at MIT or Princeton had trouble doing a certain integral, it was because they couldn't do it with the standard methods they had learned in school. If it was contour integration, they would have found it; if it was a simple series expansion, they would have found it. Then I come along and try differentiating under the integral sign, and often it worked. So I got a great reputation for doing integrals, only because my box of tools was different from everybody else's, and they had tried all their tools on it before giving the problem to me.

title: Leibniz integral rule


text: Category:Hormones of the pineal gland

title: Category:Hormones of the pineal gland


text:
Category:Parapsychology

Parapsychology is the use of scientific methods to study paranormal psychological phenomena, such as extra-sensory perception, psychokinesis, and survival of consciousness after death. This category is limited to subjects involved with the field of parapsychology, which does not study all paranormal phenomena.

title: Category:Parapsychology


text:
Nino Khurtsidze

Nino Khurtsidze (28 September 1975 – 22 April 2018) was a Georgian chess player. She was awarded the FIDE titles of Woman Grandmaster (WGM) in 1993 and International Master (IM) in 1999. She won the World Girls U-20 Chess Championship in 1993 and 1995. Khurtsidze also won the World Girls U-16 Chess Championship of 1991 in Guarapuava, Brazil, the European U-20 Girls Championship in 1992, the Men's Georgian Chess Championship in 1998 (the only Georgian female player to do so) and the women's Georgian championship five times (in 1989, 1993, 2005, 2006 and 2013).

She took part in the Women's Interzonal Tournament in 1993 in Jakarta, Indonesia and 1995 in Chişinău, Moldova. These events were part of the Women's World Chess Championship cycle of 1996 and 1999 respectively.

In 1998 she won in Rotterdam the Women's championship for Universities, organised by the FISU.

Khurtsidze competed in the Women's World Championship held with the knockout format in 2000, 2001, 2004, 2006, 2008, 2012, 2015 and 2017. Her best results occurred at Moscow 2001 (reaching the final 8, where she was defeated by the eventual winner, Zhu Chen), at Yekaterinburg 2006 (final 8 - defeated by Alisa Galliamova), and at Tehran 2017 (final 16 - defeated by Antoaneta Stefanova).

In team events, Khurtsidze played for Georgia in the Women's Chess Olympiad in 1998, 2000, 2002, 2006 and 2012, the Women's World Team Chess Championship in 2007, 2011, the team winning the bronze medal, and 2013, and the Women's European Team Chess Championship in 1999, 2003, 2005, 2007, 2009, 2011 and 2013. She was on the Georgian women's team in the 4th World Team Chess Championship, held in Lucerne, Switzerland in 1997. Khurtsidze also competed in the European Club Cup for Women with five different teams in 1996, 1999, 2001, 2005, 2006, 2007, 2008, 2009 and 2010.

In 2004, she came second behind Yevgeny Shaposhnikov in the Essen Open tournament. In 2005, she was the number two rated female chess player in Georgia with a FIDE rating of 2420. In 2012 Khurtsidze took part in the FIDE Women's Grand Prix tournament in Jermuk, Armenia.

She resided in Tbilisi and was coached for many years by GM Konstantin Aseev.

Khurtsidze died of cancer in April 2018.

title: Nino Khurtsidze


text:
NCAA Season 81

NCAA Season 81 was the 2005–06 season of the National Collegiate Athletic Association (Philippines). Colegio de San Juan de Letran was the season's host.

To usher in the new season, Letran and the other NCAA member schools performed the opening ceremonies at the Araneta Coliseum, followed by four basketball games on June 25, 2005. Other disciplines started their own tournaments within the following months, with beach volleyball as the last tournament on mid-March.

De La Salle-College of Saint Benilde won its first General Championship honors for the Seniors division, while the San Sebastian Recoletos High School won its second Juniors General Championship, via tiebreakers, when La Salle Green Hills emerged with an equal number of points.

Individual awards

 * Most Valuable Player: Ernie Jay Sagad (CSB)
 * Rookie of the Year: Kelvin dela Peña (Mapua)

Individual awards

 * Most Valuable Player: Allan Mangahas (PCU-Union)
 * Rookie of the Year: Darell Green (Letran)

Championship results
Arranged by order of conclusion:

Seniors division
¹ Not a part in the tabulation of the General Championship

Juniors division
¹ Since San Sebastian had more championships (4) than LSGH (3), San Sebastian won the General Championship, despite being tied for first place. ² Mapua did not send representatives in the Juniors division

title: NCAA Season 81


text:
Red-capped cardinal

The red-capped cardinal (Paroaria gularis) is a small species of bird in the tanager family Thraupidae. It is found in South America.

Taxonomy
In 1760 the French zoologist Mathurin Jacques Brisson included a description of the red-capped cardinal in the supplement to his Ornithologie based on a specimen collected in "America". He used the French name Le cardinal d'Amérique and the Latin name Cardinalis americanis. Although Brisson coined Latin names, these do not conform to the binomial system and are not recognised by the International Commission on Zoological Nomenclature. When in 1766 the Swedish naturalist Carl Linnaeus updated his Systema Naturae for the twelfth edition he added 240 species that had been previously described by Brisson in his Ornithologie. One of these was the red-capped cardinal. Linnaeus included a terse description, coined the binomial name Tanagra gularis and cited Brisson's work. The specific name gularis is Latin for "of the throat". The type locality has been restricted to French Guiana. This species is now placed in the genus Paroaria that was introduced by the French ornithologist Charles Lucien Bonaparte in 1832.

The genus Paroaria is classified in the family Thraupidae, unlike the cardinals proper which are in the Cardinalidae.

The masked cardinal was formerly considered conspecific with the red-capped cardinal. The masked cardinal markedly differs from the red-capped cardinal by the black ear-coverts and the red (not black) lower throat and upper chest. Additionally, its lower mandible is typically whitish, but this is not entirely consistent, as it occasionally is pale flesh-coloured. This distinctive taxon was determined a different species, as suggested by the lack of hybridization with P. g. gularis in the limited area of overlap in southern Venezuela.

There are two subspecies:
 * P. g. gularis (Linnaeus, 1766) (eastern Colombia to eastern Peru and through the Guianas to central Brazil)
 * P. g. cervicalis Sclater, PL, 1862 (north east Bolivia and south west Brazil

Description
The adult red-capped cardinal is 16.5 cm long and weighs about 22 g. The nominate subspecies has a crimson head, blackish lores and ocular region, and shiny black upperparts, apart from a white partial collar extending up the neck sides from the white underparts. The throat is black, extending to a point on the upper chest. The upper mandible is black, while the lower is pale flesh-coloured. The legs are dark grey (almost black) and the iris is brownish-orange. In pattern the juvenile resemble the adults, but the upperparts are dusky-brown, the head is deep brownish-buff (darker on the cap), the bill is entirely black and the iris is pale, dull creamy-yellow.

The song is a variable, often repeated series of suweet-chu notes, and the call is a sharp chep.

Distribution and habitat
It occurs in lowlands of the Guianas, Venezuela, eastern Colombia, eastern Ecuador, eastern Peru, northern and eastern Bolivia and the Amazon basin in Brazil. In Brazil it is, except for populations in the relatively open lowlands of north-eastern Roraima and along the Branco River and lower Rio Negro, apparently largely absent from the regions north of the northern bank of the Amazon River, but these regions are generally very poorly known and its presence cannot be discounted, especially in the section between the lower Rio Negro and Rio Jari (the river on the border between Pará and Amapá).

This is a bird of swamps, mangrove, várzea and other semi-open areas near water. It is generally common, and even occurs in lightly wooded wet habitats in towns or cities (e.g. Manaus and Puerto Maldonado).

Behaviour
The red-capped cardinal feeds on insects, rice and fruit. It is usually found in conspicuous pairs or family groups. If there is a well-marked breeding season, it extends over much of the year. The red-capped cardinal has been recorded to breed at least from June to September in northern South America but at Sacha Lodge at the Napo River (Ecuador) an adult was seen feeding a fledgling shiny cowbird (a brood parasite) in late March. This suggests that breeding can also occur as early as February/March. There is no indication for a difference in breeding seasons between the red-capped and the masked cardinals.

They build a shallow open cup nest, some 9 – wide and 7 cm high outside with a 7 cm wide by 4 cm deep cup, in a tree or some other secure spot. In Ecuador, a nest in the Cuyabeno Wildlife Reserve was built in the roots of a Macrolobium sp. standing in water, while one at Sacha Lodge was 4 m above ground in a tree. For nest material, it uses rootlets, thin twigs, and ferns. The clutch is two or three eggs. These have a background colour varying between whitish and dull olive and have dense dark brown flecks and blotches, heavier at the blunt end. They measure about 21.5 x. This species is parasitised by the shiny cowbird (Molothrus bonariensis).

title: Red-capped cardinal


text:
Kenny Hotz

Kenneth Joel Hotz (born May 3, 1967) is a Canadian comedy writer, filmmaker, entertainer and television personality. He is best known as the star of the reality comedy show Kenny vs. Spenny alongside Spencer Rice. Hotz is the creator of the FX series Testees, and Kenny Hotz's Triumph of the Will. Between 2004 and 2006, he served as a staff writer (and later occasional writer) for South Park. Hotz has directed a number of films, including Pitch, It Don't Cost Nothin' to Say Good Morning, The Papal Chase and Subscribe. He also co-directed the music video for the song "Monophobia" by Deadmau5. Hotz is a regular contributor for Vice Media and began his career as a war correspondent and photojournalist during the Gulf War.

Early life and education
Hotz was born to a Jewish family and graduated from the Forest Hill Collegiate Institute. In 1992, Hotz graduated from the Media Arts program at the Ryerson Polytechnical Institute. Hotz met Spencer Rice in high school, and the pair later went on to direct films together and create Kenny vs Spenny. As a teenager and young adult, Hotz dealt cannabis.

Photo-journalism
Hotz began as a documentary filmmaker and news photographer, creating photo essays on subjects including Auschwitz, Dachau, and the Gulf War. Hotz was the only registered Canadian photo-journalist to cover the Gulf War. His additional photo essays also include Needle Park in Zürich, Switzerland, David Koresh, Mount Carmel, Romanian orphanages after Nicolae Ceauşescu and New Year's Eve 2000 in Times Square. Both national and provincial archives have purchased his work for their collections.

Early film work
Hotz and Spencer Rice created the 18-minute short film It Don't Cost Nothin' to Say Good Morning.

In 1996, Hotz created and starred (with Rice) in the documentary Pitch which premiered at the Toronto International Film Festival in 1997. The film had numerous cameos including Al Pacino, Samuel Z. Arkoff and Neil Simon.

In 1997, he co-wrote an episode of Ninja Turtles: The Next Mutation with Rice, titled Truce or Consequences.

In 1999, Hotz moved with his long-time writing partner Rice to Los Angeles, where they worked on the development of the show Kenny vs. Spenny.

The Papal Chase
Hotz starred in his second feature film titled The Papal Chase, a documentary about Hotz trying to meet the Pope John Paul II, released in 2004. The film received the Phillip Borsos Award for Best Canadian Feature Film at the 2004 Whistler Film Festival, and in 2005 won the People's Choice Award for Best Canadian Feature at the Canadian Filmmakers Festival and the Audience Award for Best Documentary at the Brooklyn International Film Festival.

Kenny vs. Spenny
Kenny vs. Spenny was first aired on CBC Television in 2003 and went on to be nominated for eight Gemini awards. The original pilot had been commissioned by USA Network, but never aired before the show was picked up by the CBC for 26 episodes. The series centred on Hotz and friend Spencer Rice as they competed against each other in competitions, the loser of which would perform a "humiliation". The show was embroiled in a deliberate controversy with the British Columbia Human Rights Commission for Hotz's actions in the episode Who Can Piss Off More People, where he paid to have a plane fly over Toronto dragging a banner that reads "Jesus Sucks". Though he claimed his friend Jesus was getting married and it was not directed towards a religion. The show moved from CBC to Showcase television in Canada after the nationally run CBC censored two episodes of Kenny vs. Spenny, refusing to air them on their network.

In 2004, Hotz also created the concept for the videogame Versusville to accompany the television show, which was nominated for a Canadian New Media Award for Excellence in Gaming.

In 2009, Maclean's magazine named Kenny vs. Spenny the number eight pick in its list of the top ten Canadian television shows of the previous decade. The magazine said of the show and Kenny's on air role that, "Many episodes feature the evil Kenny destroying his supposed friend through deceit, trickery and blatant cheating." Hotz produced the German spinoff series Elton vs. Simon that aired between 2004 and 2006. Kenny vs. Spenny show was broadcast in over 25 countries and licensed globally, with remakes including Cenk vs. Erdem in Turkey, Juan vs. Roman in Colombia, Katja vs. Bridget in the Netherlands, and Ed vs. Spencer in the UK. This resulted in the show becoming the largest selling comedy format in Canadian television history. The show was named #8 among the top 10 Canadian television shows of the 21st century. In July 2013, the show was ranked as the highest rated reality television show by IMDb.

The last episode, ("The Kenny vs. Spenny Christmas Special"), was aired in 2010. The series produced 89 original episodes, over six seasons. Each of the main characters of the show would go on to have their own shows the following year.

In 2014, a fan campaign to have the show return to production on Netflix began, following Hotz's announcement that he and Spenny were planning the live comedy tour Kenny vs. Spenny vs. Canada. Also that year, Hotz and Rice designed a travelling stage show based on the show with a world tour of approximately 100 dates. Part of this tour included the Canadian national Kenny vs. Spenny vs. Canada tour.

In November 2020, along with Spencer Rice, Hotz directed, wrote and produced Kenny & Spenny: Paldemic, a CBC Gem special that focused on the pair's friendship and careers since Kenny vs. Spenny ended in 2010.

South Park and Comedy Central
Hotz worked as a consultant and writer on South Park, providing material for episodes including "Follow That Egg!" and "Two Days Before the Day After Tomorrow". Between 2004 and 2006 Hotz was a consultant writer, and sometimes staff writer, for 42 different episodes of the show, returning again in 2012 to write for the episode "Cartman Finds Love". Trey Parker and Matt Stone, the creators of South Park, joined to help produce future episodes of Kenny vs. Spenny in the fall of 2007, when the show began airing on Comedy Central, becoming executive producers. They stated that they first came across the show while staying up late finishing their movie Team America: World Police, when they viewed the episode "Who Can Stay Awake the Longest?" from the show's first season. The Writers Guild of America named him and the rest of the South Park writing staff as the 63rd best writing team in television history.

Testees
In 2008, Hotz created the sitcom Testees for FX Networks, producing 13 shows centering on two roommates who work as guinea pigs for a mediocre product testing facility. The show produced 13 episodes and started airing in October 2008 on FX. Hotz stated that the inspiration of the series was Woody Allen's character Fielding Mellish in the film Bananas. The show met with positive reviews. In a review for The Washington Post, Tom Shales referred to the show as "wall-to-wall uproarious."

Kenny Hotz's Triumph of the Will
In 2011, Hotz starred and produced a new series titled Kenny Hotz's Triumph of the Will, which premiered on Canada's Action network on July 22, 2011. The series produced six episodes and showed Hotz attempting to complete major tasks that challenged the social assumptions of both his audience and himself, including the building of a Jewish funded mosque for Muslims, and finding his widowed mother a new boyfriend. The show was shot on location in countries including Canada, the US, and France.

Vice
Hotz is an executive producer, host and columnist for Vice, and won the publication's best fiction article of the year award in 2006. In 2015, he directed and starred in the Vice talk show Me on You. Then in 2016 Vice Media aired a TV special starring Hotz, entitled Getting an Election executive produced by Spike Jonze. It was nominated for the Academy of Canadian Television and Cinema award for Best Original Series produced for Digital Media – Non-Fiction. Hotz has also been involved as a spokesperson and on-stage presence at Vice Media project and network launches.

Other work
In 2013, Hotz signed a deal with Sirius Radio in the U.S. for a weekly radio show that airs in Canada and the United States. In 2015, he had a role in the film The Chickening, which premiered at the Toronto International Film Festival. He also served as the film's producer. In 2018, Hotz co-directed a music video for the track "Monophobia" by Deadmau5. The video was later nominated for the Prism Prize.

Business ventures and involvement in restaurants
Hotz owned The Hoxton, a bar and music venue in Toronto which has since closed. In December 2013, The Hoxton was the site of an impromptu post-concert performance by Kanye West and Drake.

Hotz owns the Dog & Bear restaurant and bar in Toronto.

In 2018, Hotz was a judge for the Canada's Top 100 Restaurants competition.

Personal life
Hotz identifies as an atheist. He can speak Hebrew.

Hotz is the godson of folk singer Joni Mitchell.

In 2007, Hotz married former model Audrey Gair; the couple have three children.

Honours and accolades
In November 2013, Hotz was nominated for the "Digital Personality of the Year" at the Canadian Digi Awards.

In 2013, he received the "Sovereign's Medal for Volunteers" from the Governor General of Canada for his work supporting childhood survivors of sexual abuse.

title: Kenny Hotz


text:
Wollongong Wolves FC

Wollongong Wolves Football Club is an Australian semi-professional soccer club based in Wollongong, in the Illawarra region of New South Wales. The club currently competes in the National Premier Leagues NSW, the second-tier of soccer in Australia. The club plays its home games at WIN Stadium.

The club has been successful throughout their history winning one continental championship, two NSL championships, three state championships and a National Premier Leagues title (2019)

The club was formed in 1980 and competed in the National Soccer League from 1981 until the competition ceased in 2004, with the exception of a brief stint in the NSW State League in 1987. The club's most successful period in the national tournament came at the turn of the century when they won two consecutive championships in 1999–2000 and 2000–01. As a result, the club qualified for the region's continental championship in 2001 which it also won. By doing so, the Wolves became only the second Australian club to win a continental double after South Melbourne FC achieved the same feat two years prior. Their success at this particular event qualified the club for the 2001 FIFA Club World Championship, however, this tournament would ultimately be cancelled.

The club then joined the state league in the 2005 as it failed to join the newly formed national competition, the A-League after their owners decided not to apply. In 2009, the club was in financial crisis and could only survive through community support. The club was re-branded as a not for profit organisation, run and owned by the community.

The Wollongong Wolves have traditionally competed in a red and white kit, like many other sporting teams in the region. The club has operated under the names Wollongong City (1981–1996), Wollongong Wolves (1996–2006), Wollongong FC (2007–2008), Wollongong Community FC (2009), South Coast Wolves FC (2010–2015) and will once again use the moniker Wollongong Wolves FC (from 2016). In April 2016, the club president declared a desire to enter the A-League as a Wollongong Franchise in the next 3–5 years.

Foundation
The club was founded in 1980 as Wollongong City specifically to compete in the National Soccer League (NSL). The club joined the competition in the 1981 season, to which they finished in 11th position. The club was relegated from the NSL after the 1986 season, as the league reduced in size for the 1987 season. During 1987, Wollongong competed in the NSW First Division, where they were crowned both premiers and champions. The club returned to the NSL in the 1988 season where they were crowned NSL premiers. However, the honour would be he last major title the club would receive for 11 years.

Championship years: 1999–2001
In 1996, the club renamed to Wollongong Wolves. In the 1999–2000 season the team finished the league in second position and after winning their semi-final match 2–1 against Carlton, Wolves were matched-up against Perth Glory in the 2000 NSL Grand Final. The grand final match went down as one of the best comebacks ever seen in Australian football. At half time Perth Glory were leading 3–0, but at 56 minutes the Wolves found a goal through Scott Chipperfield to make it 3–1. Matt Horsley gave the Wolves a sniff at a comeback when he put the ball in the back of the net in the 69th minute, but it wasn't until Paul Reid pulled the game right back when his strike in the 89th minute made the score line 3–3. After no goals were scored in extra time, the game was forced into a penalty shoot out. Wolves goalkeeper, Les Pogliacomi, pulled off two saves to win the match 7–6 on penalties and give the club's first nation championship title.

The Wolves followed up their past season win by again placing second in the 2000–01 NSL season. After downing South Melbourne 2–1 in a semi-final match, the Wolves found themselves in a second consecutive NSL Grand Final. The Wolves earned the right to host the grand final for 2001. However, it was decided by Soccer Australia that the final would be moved from Wollongong and instead be played at Parramatta Stadium in Sydney. The reason cited was that the 20,000 capacity at WIN Stadium was not large enough to host such an event. Ironically, the crowd attendance for the match was 13,400. The Wolves went on to claim their second NSL title and qualify for the 2001 Oceania Club Championship, defeating their semi-final opponents, South Melbourne, 2–1. Stuart Young and Sasho Petrovski were among the goal-scorers for the Wolves, scoring on the 56th and 57th minute marks.

In the 2001 Oceania Club Championship, the Wolves beating Tafea of Vanuatu 1–0 in the final. The Oceania Club Championship title, however, did not lead to them playing in the 2001 FIFA Club World Championship which was scheduled to be played in Spain, as FIFA had cancelled the Club World Championship due to financial reasons.

Demise of NSL, return to state league
After the demise of the national competition, the NSL, the Wolves moved into the NSW Premier League in the 2004–05 season. In 2007, Wollongong Wolves changed its name to Wollongong FC and won the Waratah Cup. The club secured a grand final win in the 2008 season. However, the win didn't prevent a financial crisis later in the year. The club had shown many financial losses including A$240,000 in debts and losses of players and coaches. It was later announced that the club would cease to exist and a new community organisation would be formed to ensure money problems would not plague the club in the future. The club was re-branded as Wollongong Community FC, a not for profit organisation, run and owned by the community.

After a dismal 2009 season, Wollongong Community FC requested a name change back to the former "Wolves" moniker, with the formal request being approved in November 2009, and the name South Coast Wolves adopted in 2010.

The Wolves competed in the inaugural 2014 FFA Cup, where they were eliminated from the round of 32 by Central Coast Mariners 1–0 in front of a crowd of 5,238 at WIN Stadium.

In July 2015, as part of the club's 35th anniversary celebrations, it was announced that along with a brand new club logo, the name of the club would also change from South Coast Wolves back to Wollongong Wolves. These changes will be brought in for the 2016 season onwards.

Wollongong qualified for the 2016 FFA Cup and drew A-League powerhouse Sydney FC in the Round of 32. The match, played on 10 August 2016 at WIN Stadium in front of 8,029 fans, saw the Sky Blues run out 3–0 victors, with two goals to Matt Simon and one to David Carney.

Move to National Second Division (2025–)
Wollongong Wolves FC announcement officially play in National Second Division from March or April 2025 after submitting in Australian Professional League and club founder from eight teams on 20 November 2023. They will continue to play in the NPL NSW for the 2024 season, before transitioning to the new league following season.

Colours, kit and emblem

Like many other teams from the Illawarra region, the Wolves have traditionally played in red and white. These colours are taken from the local floral emblem – the Illawarra Flametree – which is used by representative teams and sporting associations throughout the area. Variations of these colours have been used throughout the years. For the 1996–97 season, the first season with the official Wolves moniker, the club played in a predominantly red kit. The shirt was bright red with maroon paneling on the lower torso and collar. Both shorts and socks were also bright red. For the 2000–01 championship season the club had returned to bright red shirt with a white collar and black detailing on the shoulders. The shorts and socks were both white. In 2002–03 the club changed to black collar, shorts and socks, and the shirt bright red with white paneling down the sides. Generally the away kits have been all-white throughout much of the Wolves history. After the club's financial troubles of the 2009 season and the move to the new moniker South Coast Wolves, both the home and away kit evolved also. The club's home kit was an all-white shirt, red shorts and socks. The away kit was a yellow with blue collar and side paneling, blue shorts and socks. For the 2014 FFA Cup, the club played in an all-red kit with white detailing. As of the 2015 season the club are back playing in a red shirt, black shorts and white socks. The away kit is also predominantly all-white with black socks.

The club's emblem has evolved throughout the years also. The original crest featured a wolf facing forward standing on its forefeet with a football in front. "Wollongong Wolves" was titled in white against a red background above the wolf. From 1996 the emblem included only the head of a wolf looking head-on, framed with large text of "Wolves" with smaller text "Wollongong" above it. Above this sat a football. With the name change to South Coast Wolves, the wolf also changed orientation to face east. There was no framing, just text "South Coast" above and "Wolves" underneath the wolf. During 2013, a company set out to produce a new club crest, citing the crest lacked traditional themes present in clubs such as Manchester United, Inter Milan and Bayern Munich. They produced a simple logo of a wolf looking face on and created a simple crest to frame the wolf. The club announced it would adopt this design from the 2016 season with the addition of two gold stars and two silver stars to represent the national and state championships won.

The Wolves kits are currently manufactured by Puma and the kit main sponsor is MCR Multi Civil and Rail. The back of the kit sponsor for season 2018 is Central Data Networks (CDN).

Stadium

When Wollongong City was admitted into the NSL in 1981, the club played out of the Wollongong Showground and then in Corrimal before it made a home at Brandon Park, Fairy Meadow in 1988. The ground had a capacity of 15,000. However, this site was demolished in 2003 in order for the University of Wollongong to build their new Wollongong Innovation Campus. At the time, the Wolves had six years left on their lease to the ground, but had been promised to play out of WIN Stadium as co-tenants with NRL team, the St. George-Illawarra Dragons. The switch to WIN Stadium occurred in 2000 following the club's title-winning season. However, due to the demise of the NSL and the Lysaghts Oval fiasco followed by the former owners handing the club to the football community has left the Wolves homeless. The Wolves moved through various homegrounds, including Macedonia Park (Berkeley) and Wetherall Park (Primbee) before settling on John Crehan Park in Cringila as its base, which has a capacity of 7,500. The Wolves moved back to WIN Stadium on a part-time contract in 2014 before announcing a permanent move in the PS4 National Premier League NSW Men's 1 for the 2015 season. The club also announced it would move their training facilities to The Thomas Dalton Park Regional Sports Centre in Fairy Meadow.

Current squad
As of 17 February 2024

Management

 * Chairman: Tory Lavalle
 * CEO: Strebre Delovski
 * Board Members: Luke Wilkshire Laura Davis Courtney Rawson

Regional

 * National Premier Leagues NSW
 * Champions (2): 1987, 2008
 * Premiers (2): 1987, 2019
 * Waratah Cup
 * Champions (2): 1997, 2007

National

 * National Soccer League
 * Champions (2): 1999–2000, 2000–01
 * Premiers (1): 1988
 * National Premier Leagues
 * Champions (1): 2019

Continental

 * OFC Champions League
 * Champions (1): 2001

title: Wollongong Wolves FC


text: Category:Anterior pituitary hormones

title: Category:Anterior pituitary hormones


text:
Mortal Coils

Mortal Coils is a collection of five short fictional pieces written by Aldous Huxley, published in 1921. The book consists of three short stories, a novelette and a play.

The title uses a phrase from Hamlet, Act 3, Scene 1: [...] To die, to sleep,

To sleep, perchance to dream; aye, there's the rub,

For in that sleep of death, what dreams may come,

When we have shuffled off this mortal coil,

Must give us pause [...]

The stories all concern themselves with some sort of trouble, normally of an amorous nature, and often ending with disappointment.

Content

 * "The Gioconda Smile", novelette:
 * is a mixture of social satire and murder story, which Huxley later adapted into a film called A Woman's Vengeance (1948).
 * "Permutations Among the Nightingales", play:
 * is a play concerning the amorous problems encountered by various patrons of a hotel.
 * "The Tillotson Banquet":
 * tells of an old artist who was thought to be dead, and is "rediscovered"; a not entirely successful honorary dinner is organised for him.
 * "Green Tunnels":
 * is about the boredom of a young girl on holiday with her family. She develops a romantic fantasy, and is ultimately disillusioned.
 * "Nuns at Luncheon":
 * is a second-hand story told of a nun falling in love. The story mocks the writer's process, a concept Huxley used in his novel Crome Yellow.

Adaptations
Based on novelette "The Gioconda Smile":
 * A Woman's Vengeance (1948), film directed by Zoltan Korda
 * Mortal Coils: Play (1948), play by Aldous Huxley
 * Das Lächeln der Gioconda (1958), TV movie directed by Michael Kehlmann
 * The Gioconda Smile (1963), TV movie directed by Patrick Barton
 * Das Lächeln der Gioconda (1966), TV movie directed by Ilo von Jankó
 * Mona Lisan hymy (1966), TV movie directed by Jukka Sipilä
 * Úsmev Mony Lízy (1968), TV movie directed by Bedřich Kramosil

Based on play Mortal Coils: Play:
 * Das Lächeln der Gioconda (1953), TV movie directed by Werner Völger
 * Il sorriso della Gioconda (1969), TV movie directed by Enrico Colosimo

title: Mortal Coils