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User:Bemoeial

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Typical set

In information theory, the typical set is a set of sequences whose probability is close to two raised to the negative power of the entropy of their source distribution. That this set has total probability close to one is a consequence of the asymptotic equipartition property (AEP) which is a kind of law of large numbers. The notion of typicality is only concerned with the probability of a sequence and not the actual sequence itself.

This has great use in compression theory as it provides a theoretical means for compressing data, allowing us to represent any sequence Xn using nH(X) bits on average, and, hence, justifying the use of entropy as a measure of information from a source.

The AEP can also be proven for a large class of stationary ergodic processes, allowing typical set to be defined in more general cases.

(Weakly) typical sequences (weak typicality, entropy typicality)
If a sequence x1, ..., xn is drawn from an independent identically-distributed random variable (IID) X defined over a finite alphabet $$\mathcal{X}$$, then the typical set, Aε(n)$$\in\mathcal{X}$$(n) is defined as those sequences which satisfy:

2^{-n( H(X)+\varepsilon)} \leqslant p(x_1, x_2, \dots, x_n) \leqslant 2^{-n( H(X)-\varepsilon)} $$

where

 * $$ H(X)  = - \sum_{x \isin \mathcal{X}}p(x)\log_2 p(x)  $$

is the information entropy of X. The probability above need only be within a factor of 2n ε. Taking the logarithm on all sides and dividing by -n, this definition can be equivalently stated as

 * $$ H(X) - \varepsilon \leq -\frac{1}{n}\log_2 p(x_1, x_2, \ldots, x_n) \leq H(X) + \varepsilon.$$

For i.i.d sequence, since

 * $$p(x_1, x_2, \ldots, x_n) = \prod_{i=1}^n p(x_i),$$

we further have

 * $$ H(X) - \varepsilon \leq -\frac{1}{n} \sum_{i=1}^n \log_2 p(x_i) \leq H(X) + \varepsilon.$$

By the law of large numbers, for sufficiently large n

 * $$-\frac{1}{n} \sum_{i=1}^n \log_2 p(x_i) \rightarrow H(X).$$

Properties
An essential characteristic of the typical set is that, if one draws a large number n of independent random samples from the distribution X, the resulting sequence (x1, x2, ..., xn) is very likely to be a member of the typical set, even though the typical set comprises only a small fraction of all the possible sequences. Formally, given any $$\varepsilon>0$$, one can choose n such that:
 * 1) The probability of a sequence from X(n) being drawn from Aε(n) is greater than 1 − ε, i.e. $$Pr[x^{(n)} \in A_\epsilon^{(n)}] \geq 1 - \varepsilon $$
 * 2) $$\left| {A_\varepsilon}^{(n)} \right| \leqslant 2^{n(H(X)+\varepsilon)}$$
 * 3) $$\left| {A_\varepsilon}^{(n)} \right| \geqslant (1-\varepsilon)2^{n(H(X)-\varepsilon)}$$
 * 4) If the distribution over $$\mathcal{X}$$ is not uniform, then the fraction of sequences that are typical is
 * $$\frac{|A_\epsilon^{(n)}|}{|\mathcal{X}^{(n)}|} \equiv \frac{2^{nH(X)}}{2^{n\log_2|\mathcal{X}|}} = 2^{-n(\log_2|\mathcal{X}|-H(X))} \rightarrow 0 $$

 * as n becomes very large, since $$H(X) < \log_2|\mathcal{X}|,$$ where $$|\mathcal{X}|$$ is the cardinality of $$\mathcal{X}$$.

For a general stochastic process {X(t)} with AEP, the (weakly) typical set can be defined similarly with p(x1, x2, ..., xn) replaced by p(x0τ) (i.e. the probability of the sample limited to the time interval [0, τ]), n being the degree of freedom of the process in the time interval and H(X) being the entropy rate. If the process is continuous valued, differential entropy is used instead.

Example
Counter-intuitively, the most likely sequence is often not a member of the typical set. For example, suppose that X is an i.i.d Bernoulli random variable with p(0)=0.1 and p(1)=0.9. In n independent trials, since p(1)>p(0), the most likely sequence of outcome is the sequence of all 1's, (1,1,...,1). Here the entropy of X is H(X)=0.469, while

 * $$ -\frac{1}{n}\log_2 p\left(x^{(n)}=(1,1,\ldots,1)\right) = -\frac{1}{n}\log_2 (0.9^n) = 0.152$$

So this sequence is not in the typical set because its average logarithmic probability cannot come arbitrarily close to the entropy of the random variable X no matter how large we take the value of n.

For Bernoulli random variables, the typical set consists of sequences with average numbers of 0s and 1s in n independent trials. This is easily demonstrated: If p(1) = p and p(0) = 1-p, then for n trials with m 1's, we have

 * $$ -\frac{1}{n} \log_2 p(x^{(n)}) = -\frac{1}{n} \log_2 p^m (1-p)^{n-m} = -\frac{m}{n} \log_2 p - \left( \frac{n-m}{n} \right) \log_2 (1-p).$$

The average number of 1's in a sequence of Bernoulli trials is m = np. Thus, we have

 * $$ -\frac{1}{n} \log_2 p(x^{(n)}) = - p \log_2 p - (1-p) \log_2 (1-p) = H(X).$$

For this example, if n=10, then the typical set consist of all sequences that have a single 0 in the entire sequence. In case p(0)=p(1)=0.5, then every possible binary sequences belong to the typical set.

Strongly typical sequences (strong typicality, letter typicality)
If a sequence x1, ..., xn is drawn from some specified joint distribution defined over a finite or an infinite alphabet $$\mathcal{X}$$, then the strongly typical set, Aε,strong(n)$$\in\mathcal{X}$$ is defined as the set of sequences which satisfy

\left|\frac{N(x_i)}{n}-p(x_i)\right| < \frac{\varepsilon}{\|\mathcal{X}\|}. $$

where $${N(x_i)}$$ is the number of occurrences of a specific symbol in the sequence.

It can be shown that strongly typical sequences are also weakly typical (with a different constant ε), and hence the name. The two forms, however, are not equivalent. Strong typicality is often easier to work with in proving theorems for memoryless channels. However, as is apparent from the definition, this form of typicality is only defined for random variables having finite support.

Jointly typical sequences
Two sequences $$x^n$$ and $$y^n$$ are jointly ε-typical if the pair $$(x^n,y^n)$$ is ε-typical with respect to the joint distribution $$p(x^n,y^n)=\prod_{i=1}^n p(x_i,y_i)$$ and both $$x^n$$ and $$y^n$$ are ε-typical with respect to their marginal distributions $$p(x^n)$$ and $$p(y^n)$$. The set of all such pairs of sequences $$(x^n,y^n)$$ is denoted by $$A_{\varepsilon}^n(X,Y)$$. Jointly ε-typical n-tuple sequences are defined similarly.

Let $$\tilde{X}^n$$ and $$\tilde{Y}^n$$ be two independent sequences of random variables with the same marginal distributions $$p(x^n)$$ and $$p(y^n)$$. Then for any ε>0, for sufficiently large n, jointly typical sequences satisfy the following properties:
 * 1) $$ P\left[ (X^n,Y^n) \in A_{\varepsilon}^n(X,Y) \right] \geqslant 1 - \epsilon $$
 * 2) $$ \left| A_{\varepsilon}^n(X,Y) \right| \leqslant 2^{n (H(X,Y) + \epsilon)} $$
 * 3) $$ \left| A_{\varepsilon}^n(X,Y) \right| \geqslant (1 - \epsilon) 2^{n (H(X,Y) - \epsilon)} $$
 * 4) $$ P\left[ (\tilde{X}^n,\tilde{Y}^n) \in A_{\varepsilon}^n(X,Y) \right] \leqslant 2^{-n (I(X;Y) - 3 \epsilon)} $$
 * 5) $$ P\left[ (\tilde{X}^n,\tilde{Y}^n) \in A_{\varepsilon}^n(X,Y) \right] \geqslant (1 - \epsilon) 2^{-n (I(X;Y) + 3 \epsilon)}$$

Typical set encoding
In information theory, typical set encoding encodes only the sequences in the typical set of a stochastic source with fixed length block codes. Since the size of the typical set is about 2nH(X), only nH(X) bits are required for the coding, while at the same time ensuring that the chances of encoding error is limited to ε. Asymptotically, it is, by the AEP, lossless and achieves the minimum rate equal to the entropy rate of the source.

Typical set decoding
In information theory, typical set decoding is used in conjunction with random coding to estimate the transmitted message as the one with a codeword that is jointly ε-typical with the observation. i.e.
 * $$\hat{w}=w \iff (\exists w)( (x_1^n(w),y_1^n)\in A_{\varepsilon}^n(X,Y)) $$

where $$\hat{w},x_1^n(w),y_1^n$$ are the message estimate, codeword of message $$w$$ and the observation respectively. $$A_{\varepsilon}^n(X,Y)$$ is defined with respect to the joint distribution $$p(x_1^n)p(y_1^n|x_1^n)$$ where $$p(y_1^n|x_1^n)$$ is the transition probability that characterizes the channel statistics, and $$p(x_1^n)$$ is some input distribution used to generate the codewords in the random codebook.

title: Typical set


text:
Ústí nad Labem

Ústí nad Labem (Aussig) is a city in the Czech Republic. It has about 91,000 inhabitants and is the capital of the Ústí nad Labem Region. It is a major industrial centre and, besides being an active river port, is an important railway junction.

Administrative division
Ústí nad Labem is divided into four boroughs, which are further divided into 22 administrative parts (in brackets):
 * Ústí nad Labem-město (Ústí nad Labem-centrum, Božtěšice, Bukov, Habrovice, Hostovice, Klíše, Předlice, Skorotice, Strážky, Vaňov and Všebořice);
 * Ústí nad Labem-Neštěmice (Krásné Březno, Mojžíř and Neštěmice);
 * Ústí nad Labem-Severní Terasa (Severní Terasa);
 * Ústí nad Labem-Střekov (Brná, Církvice, Kojetice, Olešnice, Sebuzín, Střekov and Svádov).

Etymology
The name of Ústí nad Labem is formed from the Old Czech ustie ("river mouth") and Labe (the Elbe River). It thus literally means "Mouth-upon-the-Elbe", in reference to its location at the Bílina's confluence with the Elbe.

The Czech name was Latinized as Usk super Albium and Germanized as Aussig or Außig. Before Czechoslovak independence amid the dissolution of Austria-Hungary following World War I, the town was usually known in English as Aussig.

Geography
Ústí nad Labem is located about 65 km north of Prague and 45 km south of Dresden in Germany. It lies mostly in a hilly landscape of the Central Bohemian Uplands, but it also extends into the Most Basin in the northwest. The highest point is the hill Široký vrch at 659 m above sea level. The city is situated at the confluence of the Elbe and Bílina rivers. Half of Lake Milada lies in the municipal territory. The southern part of the territory lies in the České středohoří Protected Landscape Area.

History
For years, the charter of the Prague Benedictine monastery from 993 was considered to be the first written mention of Ústí nad Labem, but it has been proven to be a hoax. The first verified written mention is in the charter of the chapter at the Church of St. Stephen in Litoměřice, dated to 1056 or 1057. In 1249, it was first mentioned with the title of royal town.

In the second half of the 13th century, King Ottokar II of Bohemia invited German settlers into the country and granted them a German form of municipal incorporation, thereby founding the city proper. In 1423, as King of Bohemia, Sigismund pledged the town to Elector Frederick I of Meissen, who occupied it with a Saxon garrison. It was besieged by the Hussites in 1426: a German army of 70,000 was sent to its relief but the 25,000 besiegers defeated them amid great slaughter on 16 June; the next day, they stormed and razed the town. It was left derelict for three years before rebuilding began in 1429.

Ústí nad Labem was again burned down in 1583 and was sacked by the Swedes in 1639 amid the Thirty Years' War. It also suffered grievously during the Seven Years' War and was near the 1813 Battle of Kulm between France and the alliance of Austria, Prussia, and Russia during the Napoleonic Wars. As late as 1830, its population was only 1400.

As part of the Kingdom of Bohemia, it was eventually incorporated into Austria and heavily industrialized over the 19th century. After the Compromise of 1867, it headed the Aussig District, one of Austrian Bohemia's 94 District Commissions (Bezirkshauptmannschaften). In the 1870s, with only 11,000 people, it was a major producer of woolen goods, linen, paper, ships, and chemicals and carried on a large trade in grain, fruit, mineral water, lumber, and coal. By 1900, large-scale immigration had boosted the population to nearly 40,000, mostly German, and added glassworking and stone to its trades. The local river port became the busiest in the entire Austro-Hungarian Empire, surpassing even the seaport in Trieste.

The factories of Aussig—as it was then known—were an early center of the National Socialism ("Nazi") movement. The German Workers' Party in Austria (Deutsche Arbeiterpartei in Österreich) was founded on 15 November 1903 and later gave rise to the Sudeten German Party and Austrian National Socialism. Their books continued to be printed in Ústí nad Labem even after the formation of Czechoslovakia in 1918. During the 1930 census, Ústí nad Labem was home to 43,793 residents: 32,878 considered German, 8,735 Czech or Slovak, 222 Jews, 16 Russians, and 11 Hungarians. Ústí nad Labem was ceded to Nazi Germany with the rest of the Sudetenland in October 1938 under the terms of the Munich Agreement and placed under the administration of the Regierungsbezirk Aussig of Reichsgau Sudetenland. On New Year's Eve of that year, the Nazis burnt down the local synagogue; a meat factory was later raised in its place. The Jewish community in Ústí nad Labem was mostly exterminated over the course of World War II amid the Holocaust. In April 1945, the city was severely bombed by the Allies.

Shortly after the war ended, on 31 July 1945, an explosion of the local ammunition depot triggered a pogrom of the German population, known as the Ústí massacre, mostly at the hands of out-of-town paramilitary groups. Whilst the official investigation blamed the explosion on German saboteurs, more recent historical work points towards it being a communist provocation, intended to effect the subsequent expulsion of Germans. Between 80 and a thousand people died in the event, with estimates varying widely, but being generally much higher than the official body count.

Under the terms of the Potsdam Conference and the Beneš decrees, the city was restored to Czechoslovakia and almost the totality of its previous population expelled as being German. In May 1948, the Communist government passed a new constitution declaring a people's republic. Communism continued until the 1989 fall of the Berlin Wall set off a series of events which are now known as the Velvet Revolution. Today, Ústí nad Labem is a major industrial city of the Czech Republic with substantial chemical, metallurgical, textile, food, and machine tool industries.

Matiční Street Wall
The city gained notoriety in the late 1990s when a 150 m wall was constructed along part of the Matiční Street separating houses on one side from the tenement blocks on the other. Since the latter were homes mainly to Romani, it turned into an international scandal. Mayor Ladislav Hruška promised local homeowners' representatives that the wall would be finished by the end of September, 1998. Foreign journalists travelled to Ústí nad Labem to investigate, and were told by councillors that the wall was not meant to segregate by race, but to keep respectable citizens safe from noise and rubbish coming from the opposite side of the street.

In September, city representatives announced that plans would be changed from a four-metre soundproof wall to a 1.8-metre wall of ceramic bricks, and a children's playground would also be constructed in front of the tenement blocks. Despite these changes, the Roma Civic Initiative and Deputy Prime Minister Vladimír Špidla vocally opposed the construction. The wall was criticised by U.S. Congressman Christopher Smith, and a delegation from the Council of Europe described it as a "racist" and drastic solution.

The new plans slated construction to begin 30 August 1999, but a decision by the district office delayed the move because a wall that large would require a permit, and threatened to damage the root systems of trees along Matični Street. On 5 October however, construction began regardless of the opposition by foreign observers and members of the Czech government. The following day, 50 Roma physically blockaded construction of the wall and dismantled parts that had already been set up. Nonetheless, the wall was completed on 13 October. Domestic and international pressure eventually persuaded the city to dismantle the wall, and it was demolished six weeks after it had been erected. The local zoo uses parts of this ceramic fence as a wall around its main entrance to this day. The original wall was only 1.8 metres high and a few more rows of ceramic parts were needed to make it higher. Matiční Street is now uninhabited and its buildings are scheduled for demolition.

Economy
Ústí nad Labem is the economic centre of the Ústí nad Labem Region and the seat of many industrial companies. The largest industrial employers with its headquarters in Ústí nad Labem and at least 500 employees are Spolek pro chemickou a hutní výrobu (chemical industry) and Strabag Rail (construction industry). Other large industrial companies in the city include Kone Industrial (manufacture of elevators), Severotisk (printing industry) and Pierburg (manufacture of car parts).

Road transport
The D8 motorway (part of the European route E55) from Prague to Dresden intersects the western border of Ústí nad Labem. The European route E442 from Liberec to Karlovy Vary, formed by first class road, also passes through the city.

Mariánský Bridge is a road bridge over the Elbe which was built over a period of five years and opened in 1998. It bleongs to the main landmarks of the city. International Association for Bridge and Structural Engineering ranked Mariánský Bridge between the 10 best structures of the world in the decade.

City mass transport
The city has a network of mass transport that includes bus and trolley bus lines. The city has its own transport company, Dopravní podnik města Ústí nad Labem.

Railway transport
Ústí nad Labem is an important railway node with four railway stations. The largest of these is Ústí nad Labem main railway station which is served by international EuroCity trains. Ústí nad Labem lies on the line from Prague to Děčín, which is part of several international lines, and thus the city has direct connections with Berlin, Budapest, Graz and Zürich. Lines of national importance are Prague–Cheb and Ústí nad Labem – Kolín.

River transport
The Elbe River Line is a junction with the West-European river lines opening access to Germany, Benelux countries, northern France and to important sea ports. Freight transportation and pleasure cruises are run on the water line section Pardubice – Chvaletice – Ústí nad Labem – Hřensko – Hamburg.

Air transport
An airport for small sports planes (ICAO code LKUL) is located northwest of the city. The nearest airports for airliners are in Prague (64 km) and Dresden, Germany (56 km).

Education
The city is home of the Jan Evangelista Purkyně University in Ústí nad Labem. This public university has about 8,500 students and with about 900 employees, it is one of the most important employers in the region.

Sport
The local ice hockey club HC Slovan Ústečtí Lvi. The football club is FK Viagem Ústí nad Labem, which plays at the Municipal Stadium.

The city hosts the Ústí nad Labem Half Marathon, one of the World Athletics Label Road Races.

During the 1950s and 1960s, the motorcycle speedway team AMK Ústí nad Labem raced in the city. It raced initially at a stadium in the Bukov area until it was converted into an athletics facility in 1960. Then the team raced at a track in the Klíše area from 1967 to 1968 and finally in Polepy. The team raced in the inaugural 1956 Czechoslovak Championship and won two silver medals and three bronze medals from 1961 to 1971.

Sights
The Střekov Castle is one of the main sights of Ústí nad Labem, and one of the most visited tourist destinations in the whole region. It is located in a southern suburb of the city. The castle was built in 1316–1319. With a break in 1945–1992, it has been owned by the Lobkowicz family since 1563.

The Church of the Assumption of the Virgin Mary was built in 1318 and is located in the city centre. It is well known for its leaning tower. The tower is 65 m high and its deviation, caused by bombing at the end of World War II, is 201 cm. It is the most leaning tower north of the Alps.

A significant landmark is the hill Větruše with an observation tower and the Větruše Castle, which was built in 1847 as a hotel and restaurant serving cultural and social purposes.

In Krásné Březno part is located the Ústí nad Labem Zoo, founded in 1908. A notable building is the Krásné Březno Castle. It is formed by the Old Castle, built before 1568, and by the New Castle, built in the early 17th century and baroque modified in the first half of the 18th century. The castle is surrounded by an English park. Nowadays the castle serves as the seat of the branch of National Heritage Institute of the Czech Republic.

Notable people

 * Anton Raphael Mengs (1728–1779), German painter
 * Mimi Wagensonner (1897–1970), Austrian composer
 * Felix Weinberg (1928–2012), Czech-British physicist
 * Günther Herbig (born 1931), German conductor
 * Alfred Lipka (1931–2010), German violist
 * Vladimír Páral (born 1932), writer
 * Heinz Edelmann (1934–2009), Czech-German illustrator and designer
 * Milan Hejduk (born 1976), ice hockey player
 * Jiří Jarošík (born 1977), footballer
 * Jan Mertl (born 1982), tennis coach and player
 * Tomáš Černý (born 1985), footballer
 * Michal Neuvirth (born 1988), ice hockey player

Twin towns – sister cities
Ústí nad Labem is twinned with:
 * 🇩🇪 Chemnitz, Germany
 * 🏴 Halton, England, United Kingdom

Ústí nad Labem also cooperates with Dresden, Germany.

title: Ústí nad Labem


text:
Talk:Product rule

Proof
Hi,

the proof given for the product rule is a little hard to follow. What about rewriteing it, using the x -> x0 notation? And explaining the steps in more depth (I've had a little difficulty proving the tricky part to myself, when g(x)h(x) - g(x0)h(x0) becomes g(x) (h(x) - h(x0)) + h(x) ( g(x) - g(x0)).) PAStheLoD 18:49, 9 June 2006 (UTC)

---

Thanks, Anome. But if we have separate "formal" proofs that:

a) As δx -> 0, δy/δx -> dy/dx (what I have called "differentiation from first principles") b) The limit of a product is the product of the limits, and vice versa (what I have called the "product rule for limits")

then does that make the proof count as a formal one? Kidburla2002.

The justification is informal, because it uses infinitesimals in an informal way: it works under most reasonable assumptions, but what about unreasonable ones? To do it formally, you can either use limits formally (with δ - ε type stuff), or a formalised system of infinitesimals. The Anome

I changed the formal proof; I agree that it was hard to follow. I think it's much easier to prove this as δx -> 0 rather than as x -> x0. It also fits better with the initial "intuitive" proof, so it makes it clearer how make an intuitive proof rigorous. I tried to be fairly clear (without being pedantic) about definitions, hypotheses, conclusions, and where we use properties of limits. I also was careful to write it in a way that kept f and g in the same order, so that the proof will "work" in the non-commutative case. — Preceding unsigned comment added by Natkuhn (talk • contribs) 00:36, 26 December 2012 (UTC)

Disambiguation
Why was this moved? What is going at Product rule? -- Tarquin

I am not aware of any other product rule, so a disambiguation like product rule (calculus) is unnecessary and I will move back to product rule. AxelBoldt 19:54 18 Jun 2003 (UTC)

The Product Rule is a proper name Pizza Puzzle

There's plenty of other identities which can be called product rules. There must be 15 of them involving curls and divs and grads and what not. But I'm an opponent of pre-emptive disambiguation, so I say leave it unless there's a real need to move it. Certainly you can call this one the product rule, and get away with it. -- Tim Starling 14:24 19 Jun 2003 (UTC)

There is another "product rule" that is often used in combinatorics. (see Rule of product) which (i think) would be better listed as "product rule" than "rule of product". --Hughitt1 23:09, 3 April 2006 (UTC)

"Informal justification" section
I took away the "informal justification" section, for the following reasons: Revolver 03:21, 20 Feb 2004 (UTC)
 * 1) It is virtually identical in essence to the argument using differentials at the beginning of the article. The steps are almost identical, just put "delta-u" instead of "du", or vice versa.
 * 2) There is already a rigorous proof given. In other words, there is already a "formal" derivation (using differentials), and a rigorous proof (using difference quotients, so unless this is really a new idea (and it isn't, see #1), why is it necessary?

???
''This can also be seen as a barber shop analogy. For example, in the above example, u stands at one side while v takes the haircut.''
 * Is this just silly vandalism, or am I really missing something??? Revolver 16:11, 30 September 2005 (UTC)

 * I couldn't understand this either and have removed it. Eric119 08:34, 16 October 2005 (UTC)

Leibniz and intuition
The argument of Leibniz can be adapted to provide better intuition than more common proofs.

Let ρ be an infinitesimal, and write g(x+ρ) as g(x)+g′(x)ρ+O(ρ2); likewise for f(x+ρ). Then (f·g)(x+ρ) will be
 * $$\begin{align}

(f \cdot g)(x+\rho) &{}= f(x+\rho) \cdot g(x+\rho) \\ &{}= \left( f(x)+f'(x)\rho+O(\rho^2) \right) \cdot \left( g(x)+g'(x)\rho+O(\rho^2) \right) \\ &{}= f(x)g(x) + f'(x)g(x)\rho + f(x)g'(x)\rho + O(\rho^2) \end{align}$$ When we take the derivative we subtract off the f(x)g(x) term, divide by ρ, and discard any remaining terms containing ρ — which will be the O(ρ2) bit. In other words, the product rule is merely the distributive law followed by the discarding of higher order terms. This also quickly explains the result for a multiple product; any product term with two or more derivatives will be higher order, thus discarded.

The derivative here is, of course, a non-standard analysis version, of the form
 * $$ f'(x) = \mathrm{standard}\left( \frac{f(x+\rho) - f(x)}{\rho} \right), $$

where ρ is an infinitesimal.

Sometimes this idea is conveyed with a picture of areas.

░░░░░░░░░░░░█ ▒▒▒▒▒▒▒▒▒▒▒▒░ ▒▒▒▒▒▒▒▒▒▒▒▒░ ▒▒▒▒▒▒▒▒▒▒▒▒░ ▒▒▒▒▒▒▒▒▒▒▒▒░ ▒▒▒▒▒▒▒▒▒▒▒▒░ ▒▒▒▒▒▒▒▒▒▒▒▒░

Here the top right corner corresponds to the higher order terms in ρ. --KSmrqT 04:26, 28 January 2007 (UTC)

multiplication signs, please
Formulas like
 * $$(fg)'=f'g+fg' \,$$
 * $${d\over dx}(uv)=u{dv\over dx}+v{du\over dx}$$

are hard to decipher for the beginner.

Multiplication signs
 * $$(f\cdot g)'=f'\cdot g+f\cdot g' \,$$
 * $${d\over dx}(u\cdot v)=u\cdot {dv\over dx}+v\cdot {du\over dx}$$

make it easier.

Bo Jacoby (talk) 21:43, 11 June 2008 (UTC).

For vector functions
I'm new (sorry about sloppy/lack of code) and not a math-whiz (sorry if this is irrelevant), but in reading this article I thought it might be worth noting in the section "For vector functions" the difference between dot and cross products in this circumstance. Or perhaps just denoting the equation with the appropriate multiplication sign as noted above would suffice.

... that is :

$$(f\times g)'=f'\times g+f\times g' \,$$ and not
 * $$(f\times g)'=f'\times g+g'\times f \,$$

7yl4r (talk) 23:17, 24 February 2009 (UTC)

Associativity ?
I think it is very unfortunate that associativity is assumed throughout this article without ever mentioning it, then, under Generalisations suddenly the sequence of multiplicands is observed. IMHO the product rule should NEVER be stated

 * $$\dfrac{d}{dx}(u\cdot v)=u\cdot \dfrac{dv}{dx}+v\cdot \dfrac{du}{dx}$$

but only

 * $$\dfrac{d}{dx}(u\cdot v)=u\cdot \dfrac{dv}{dx}+\dfrac{du}{dx} \cdot v$$

or with brackets, where necessary

 * $$\dfrac{d}{dx}(u\cdot v)=u\cdot \dfrac{d}{dx} v + (\dfrac{d}{dx} u) \cdot v$$

This would be appreciated by everyone who works with operators (quantum mechanics), matrices, or just wants to apply the product rule to vector products (as 7yl4r found out already). In this respect, unfortunately, many of the math lines need reworking - anyone? 213.68.42.99 (talk) 13:15, 19 January 2010 (UTC)

 * Did you mean commutativity rather than associativity? Commutativity of multiplication says ab = ba.  Associativity is something else. Michael Hardy (talk) 14:21, 19 January 2010 (UTC)

 * The problem with stating it right is that students will go and forget that they aren't supposed to differentiate the v, very silly I know. Also it makes the two bits look different even though multiplication is commutative for what they're doing. Dmcq (talk) 16:34, 19 January 2010 (UTC)

source for barrow
See "The early mathematical manuscripts of Leibniz", Gottfried Wilhelm Leibniz, translated by J. M. Child; page 29, footnote 58. Tkuvho (talk) 15:44, 18 October 2010 (UTC)

 * that citation you gave said nothing of the sort, perhaps you could point to the version of the book as well or indicate the place better? I'm not doubting Barrow made a geometrical analogue but lots of people did geometrical bits of calculus before it was formalised in an algebraic way. Also I would have thought more should be put into Barrow's article about his geometric lectures before branching out as it were? Dmcq (talk) 16:11, 18 October 2010 (UTC)

 * The 2008 edition is online. I am not sure what you mean above.  You were not able to find footnote 58 on pages 28-29?  Tkuvho (talk) 17:23, 18 October 2010 (UTC)
 * I see what has happened. Okay I've read it and in it Leibniz is accused of plagiarism as far as I can see. Even though Leibniz quite evidently got it wrong in the first instance. That is so self-evidently silly I don't know why Child wrote it. However I can see the attribution of the product rule to Barrow. I still believe you would be much better off adding a bit about Barrow's geometrical lectures in his article for the reasons I gave above, Child wrote a book about them. Dmcq (talk) 18:22, 18 October 2010 (UTC)

 * I read Child somewhat differently. My guess is that Barrow had it first, Leibniz discovered it independently, and Newton I don't know.  You seem to know more about Barrow's geometrical lectures than I do; why don't you give it a try?  I can participate if you wish.  Tkuvho (talk) 20:08, 18 October 2010 (UTC)

Leibniz did not make this error
The claim that "Leibniz himself made this error initially" is absurd, though it has been recycled in some publications more interested in amusing the students than historical accuracy. Leibniz specifically says that he would like to determine whether the rule d(xy)=dxdy applies, makes a few calculations, and a few lines later rejects this. This was dealt in detail in Margaret Baron's book (last chapter). Tkuvho (talk) 15:54, 18 October 2010 (UTC)
 * Well stick the citation in saying it is disputed. What's there is from the same J M Child book you referenced above to support Barrow. Dmcq (talk) 16:17, 18 October 2010 (UTC)

 * I can't follow your grammar. The source for the product rule is Baron, not Child.  Tkuvho (talk) 17:32, 18 October 2010 (UTC)

 * I really wish you would put proper citations into the article. Just copy another citation to see how to do them. If you can give details on the talk page then you can do enough for that. You've put in an addition but left the original citation that contradicts what you said. That citation references Michelle Cirillo who used Child's book and another one by Antonella Cupillari
 * As to Baron I just had a look and all it seems to say about that is that he 'eventually' worked it out. Dmcq (talk) 18:00, 18 October 2010 (UTC)

 * I see. I might have misread Baron.  If so, my apologies.  Do you have the impression Baron would agree with the claim that Leibniz originally made a mistake?  Tkuvho (talk) 20:09, 18 October 2010 (UTC)

 * Note that Leibniz clearly states that he wishes to examine whether d(xy)=dxdy, or not. He proceeds to examine it, and comes to a different conclusion.  This much emerges clearly from Baron's documentation.  By no stretch of the imagination can this be described as an "error" on Leibniz's part.  Does Cupillari actually examine the evidence?  Cirillo certainly does not.  Tkuvho (talk) 20:59, 18 October 2010 (UTC)

 * Well I think apocryphal is too strong when he spent a couple of pages on the idea and only wrote the correct result 10 days later. See Church (2008) page 100 on about it. Especially the bit on page 101 where he says it appeared to be proved but was incorrect. Incidentally Child on page 102 seems to attribute the original idea to Fermat and then Barrow and Newton, I'm not sure what that's about. Dmcq (talk) 21:54, 18 October 2010 (UTC)

 * Let's get back to Barrow later. If Leibniz wrote that "it appeared to be proved but was incorrect", doesn't it show that he never actually claimed d(xy)=dxdy as true?  And if the text we are quoting is a running diary where he felt free to improvise over a 10 day period, isn't it ungenerous to claim that Leibniz made an "error" as if there was a mistake in his research paper?  Tkuvho (talk) 02:33, 19 October 2010 (UTC)
 * Well if you can phrase it better please do. However apocryphal refers to stories which didn't happen in real life. He though it might be true and later found out it wasn't. If he thought 7 times 8 might be 54 and later corrected it to 56 I know what I'd call it. Anyway we should be following the sources and not giving our own opinions, if there is dissent about the point then put the citation for the other point of view into the article as well please or else show how the citation that's there is unreliable or plain wrong. As far as I can see it is not plain wrong but a matter of interpretation. Dmcq (talk) 12:27, 19 October 2010 (UTC)
 * Can you suggest a wording based on your reading of the secondary sources? As far as Cirillo is concerned, I don't think we need to relate to her as she has no pretense of being a historian.  In fact we should delete the reference.  Tkuvho (talk) 12:53, 19 October 2010 (UTC)

Example
Isn't the law only valid for constant-mass systems?123Mike456Winston789 (talk) 22:05, 19 January 2011 (UTC)

 * For constant mass you would get F=m(dv/dt), i.e. F=ma. The formulation F=dp/dt is more general. Tkuvho (talk) 22:16, 19 January 2011 (UTC)

$$\text{st}$$ notation
I'm not versed in non-standard analysis. What does the $$\text{st}$$ notation in the "Using non-standard analysis" section mean? (I'd look it up in a non-standard analysis article but somehow I'm not comfortable editing a section about which I know very little.) 67.158.43.41 (talk) 05:57, 4 January 2011 (UTC)

 * st denotes the standard part function. I have added a few words and links to the section to provide some minimal context. It still is rather bare I'm afraid. — Tobias Bergemann (talk) 08:58, 4 January 2011 (UTC)

 * Thank you, that's all I wanted. The st page is itself bare, but ah well. Maybe one day I'll have to read a book on non-standard analysis. 67.158.43.41 (talk) 13:17, 9 January 2011 (UTC)

 * What is your background exactly? To get some motivation it may be helpful to start at Ghosts of departed quantities.  The reason standard part function is "bare" is because it relies on other pages to explain the background.  In an encyclopedia, one can't expect each article to expain every point.  Tkuvho (talk) 01:44, 10 January 2011 (UTC)

incorrect calculation in the section "using nonstandard analysis"
The calculation in the section "using nonstandard analysis" is incorrect. Tkuvho (talk) 15:20, 13 February 2011 (UTC)
 * Explain. At first glance, it looks acceptable, although verifying
 * $$\operatorname{st}\frac {\mathrm du \cdot \mathrm dv}{\mathrm dx} = 0$$
 * is not obvious. — Arthur Rubin  (talk) 17:18, 13 February 2011 (UTC)

 * That step is OK, but the calculation confuses the infinitesimal dx, dy and their role in the symbolic notation for the derivative dy/dx. Dropping the mention of "st" in the last line is not correct if dy/dx is literally a ratio, since it may not even be real.  Tkuvho (talk) 17:22, 13 February 2011 (UTC)

 * OK, I can see the problem, related to the confusion of the non-standard analysis expression:
 * $$\operatorname{st}\frac {\mathrm du} {\mathrm dx}$$
 * and the calculus expression
 * $$\frac {\mathrm du} {\mathrm dx}$$
 * But what to do about it? — Arthur Rubin  (talk) 17:25, 13 February 2011 (UTC)

 * What Keisler and others usually do is use $$\Delta x$$ for the initial infinitesimal. Once the derivative is defined as the standard part of $$\Delta y/\Delta x$$, one defines dy by setting dy = f'(x)dx, where $$dx=\Delta x$$.  Tkuvho (talk) 17:29, 13 February 2011 (UTC)

"Common Error" - really?
The article says "It is a common error, when studying calculus, to suppose that the derivative of (uv) equals (u ′)(v ′)". I dispute that. I don't see why it's intuitive (and the article doesn't say), and in my experience most students simply follow whatever recipe they're given. But leaving aside my personal opinion, it's unreferenced. It's been sitting there since the very first version of this article (Nov 2002) when WP was in its infancy, and it's time it was removed. The reference for Leibniz' error can simply be moved to the history section. Adpete (talk) 11:52, 25 February 2013 (UTC)
 * OK I'm removing it. Even in the unlikely event that it's true, it's not encyclopedic and (to me) is out of place in an otherwise excellent article. Adpete (talk) 06:34, 26 February 2013 (UTC)

Formatting error.
I'm getting the error "Failed to parse(unknown function '\begin'): {\begin{aligned}d(u\cdot v)&{}=(u+du)\cdot (v+dv)-u\cdot v\\&{}=u\cdot dv+v\cdot du+du\cdot dv.\end{aligned}}" in the first line of the discovery section. I don't know enough about wiki formatting math equations to fix this or if this might just be an error with my browser. Someone with better experience of formatting maths expressions should probably have a look at fixing it. — Preceding unsigned comment added by 110.20.69.222 (talk) 07:56, 10 February 2014 (UTC)

Rigorous to Direct?
Two proofs of the Product Rule are provided. As far as I can see they are both rigorous and they are both nicely presented. However, the title of the second proof, `Rigorous Proof', suggests, to me at least, that the proof by factoring is not rigorous. The second proof proceeds directly from the definition of the derivative. I suggest changing the title to `Direct Proof'. Michealefr (talk) 08:24, 13 September 2015 (UTC)

Wikipedia_talk:WikiProject_Mathematics
Please take a look at Wikipedia_talk:WikiProject_Mathematics. Dennui (talk) 06:19, 23 April 2019 (UTC)

https://en.wikipedia.org/wiki/Wikipedia_talk:WikiProject_Mathematics/Archive/2019/Apr#Article_product_rule Dennui (talk) 13:53, 5 July 2020 (UTC)

Is this limit superfluous?
In the final line of the proof it seems to me that the limit applied to f(x) [i.e., "\lim_{\Delta x\to 0} f(x)"] shouldn't be there. After all, f(x)=f(x) by definition. This is a minor thing but if it's superfluous it makes the proof more complicated than it needs to be. Joe in Australia (talk) 21:39, 27 October 2022 (UTC)
 * The author of the proof was mechanically distributing the limits over sums and products. Then the limit in question simplifies in the next step. Whether to simplify it a step earlier, during the distribution step, is a matter of taste. For what it's worth, I prefer the current version, over the version that you seem to be proposing. Regards, Mgnbar (talk) 23:21, 27 October 2022 (UTC)

comparison to it's product rule: cross term
the current articles doesn't discuss dudv and why this is not used In differentiable deterministic calculus but is required in stochastic calculus. Ito's product rule is not cited or referenced in article which is therefore missing discussion of important concepts. Rdsk2014 (talk) 08:22, 5 November 2023 (UTC)

 * It seems that what you call dudv is a differential form of order 2. This has nothing to do with the product rule, and this is described in details in the linked article. Product rule is about differentiation of a product of functions, not about the exterior product of differential forms. D.Lazard (talk) 12:06, 5 November 2023 (UTC)

plausible search not supported
If I google for "product rule for derivatives", this article should be among the first hits presented. As it is, it doesn't appear at all. So, this expression ought to be incorporated into the article in some way, perhaps as part of a list of keywords. Big-picture takeaway: I guess we can't just blithely assume that a garrulous / comprehensive article on a topic will automatically contain the most plausible / relevant search strings for it. Maybe a deliberate program of inclusion of keywords for every article needs to be undertaken. Kontribuanto (talk) 14:58, 5 March 2024 (UTC)

title: Talk:Product rule


text:
Online encyclopedia

An online encyclopedia, also called an Internet encyclopedia, is a digital encyclopedia accessible through the Internet. Examples include Encarta from 2000 to 2009, Wikipedia since 2001, the Encyclopædia Britannica since 2016, and Encyclopedia.com since 1998.

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Other digitization projects have made progress in other titles; one example is Easton's Bible Dictionary (1897) digitized by the Christian Classics Ethereal Library.

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title: Online encyclopedia


text:
PPM

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title: PPM


text:
Talk:Quotient rule

Regarding Limitations
$$f(x) = \frac{|x|}{|x|}$$ is not at all differentiable at $$x = 0$$; it's not even defined there. 138.89.101.39 (talk) 00:18, 10 April 2009 (UTC)

Informal proof
Out of curiosity, what exactly is there about the informal proof that makes it informal? Cburnett 23:22, 3 Feb 2005 (UTC)
 * I guess it's considered an informal proof because it's based off the product rule, while the "formal" proof is based off the difference quotient, which is more direct. I changed the headings to be more specific. - Evil saltine 05:11, 4 Feb 2005 (UTC)

This was proofed by Sir Al-Hinai Quotient, who was regarded as the leader of classical mathematics- at his time. —Preceding unsigned comment added by 128.243.220.21 (talk) 02:11, 18 May 2009 (UTC)
 * The proof is informal because it contains a hidden assumption, that the derivative of f exists. David Radcliffe 02:44, 26 May 2007 (UTC)

Total differential proof
The total differential proof uses the fact that the derivative of 1/x is −1/x2. But without the quotient rule, one doesn't know the derivative of 1/x, without doing it directly, and once you add that to the proof, it doesn't seem as "elegant" anymore, but without it, it seems circular. Revolver 16:03, 30 September 2005 (UTC)
 * I would vote to remove that proof altogether. Too many proofs in this article, and they are lengthy and using lots of calculations. Oleg Alexandrov 17:44, 2 October 2005 (UTC)
 * I guess I'll go ahead and get rid of it. I'll save the section incase anyone wants it back. (Frazz 18:04, 7 February 2007 (UTC))

 * On second thought, it's fine really. (Frazz 16:30, 8 February 2007 (UTC))

 * This total differential proof uses knowledge of what the derivative of 1/x is. If you know that, you can prove the quotient rule in two lines using the product and chain rules, not having to go through a huge mumbo-jumbo of differentials.  I really don't know why such a proof is not on this page and numerous complicated ones are. 67.149.103.91 (talk) 04:24, 17 June 2010 (UTC)

Fix needed in a proof
A proof of the quotient rule is not complete. Why $$h(x)$$≠ 0 does implies $$h(x+\Delta x)\ne 0$$? I have seen a proof of the quotient rule based on a characterization theorem (don't know its proper English name). --Matikkapoika 01:25, 15 November 2006 (UTC)

 * I know I'm no expert and am probably spouting rubbish, but would it be that $$h(x)\neq 0\Rightarrow h(x+\Delta x)\neq 0$$ as $$\Delta x\to 0$$, i.e. $$h(x+\Delta x)\to h(x)\neq 0$$? 118.90.51.62 (talk) 02:13, 22 June 2008 (UTC)

From the product rule
The text:

The rest is simple algebra to make f'(x) the only term on the left hand side of the equation and to remove f(x) from the right side of the equation.

on my PC, I can't see the dash after the f

SethMould 07:53, 16 July 2007 (UTC)

Mnemonic Section
A simple mnemonic can be used to recall the quotient rule when solving derivatives by hand: "lo Dhi, minus hi dlo, cross the line and square below"

Extra '('
There is an extra open brackets in the third line of the first proof. —Preceding unsigned comment added by 201.95.65.32 (talk) 17:47, 15 December 2009 (UTC)

From the Product Rule
As noted above, the simplest proof is from the product rule. A proof from the product rule used to be part of the article, and since it helps in memorizing the rule, I don't see why this short proof is not in the article.

The following text works:

The shortest proof comes from the product rule. It is not hard to prove by various means that $$\frac{d}{dx} \left ( \frac{1}{x} \right) = \left ( \frac{-1}{x^2}\right) $$. For a short proof of this fact, note that $$x^{-1} = e^{-\ln (x)} $$ and apply the chain rule. Another direct application of the chain rule tells us that $$\frac{d}{dx} \left( \frac{1}{g} \right ) = \frac{-g'}{g^2}$$. The quotient rule then follows:
 * $$\frac{d}{dx} \left ( \frac{f}{g}\right) = \frac{d}{dx} \left ( f \cdot\frac{1}{g}\right)

= f'\cdot \frac{1}{g} - f \cdot\frac{g'}{g^2} = \frac{f'g - fg'}{g^2} $$

Mistake
The very first definition of quotient rule is wrong. There is mistake of sign. — Preceding unsigned comment added by 184.17.73.223 (talk) 23:16, 15 January 2012 (UTC)

Mistake
The second derivative was incorrect. Denominator should be to the power of 3 to make numerator consistent with maths and h-prime term in the numerators 3rd term should have been squared. From formating these look like purely clerical errors. Double checked with maple13. — Preceding unsigned comment added by 94.195.250.134 (talk) 19:58, 25 March 2012 (UTC)
 * I agree. I added a fact tag for now. Rlendog (talk) 00:18, 21 December 2018 (UTC)

Jumps between g(x) an j(x) or equivalents in successive rendered(TeX) examples.
In the introducing article, jumps occur between g and j functions every other rendered example (once j then g then j ...); they appear to be the same. I don't have the expertise to affirm it is wrong in the current form.

In the Proof>Algebraic_Proof section, the jump occurs following the introduction of f'(x) as the question that is to be answered. j(x) and h(x) are used as the functions comprising f(x) when they're divided. Taking 1/delta x as a factor and combining the two remaining fractions shouldn't (algebraically) affect j(x) in any way contrary to what's demonstrated. In this case, g(x) is substituted for j(x) for no apparent reason.

My observations may not be accurate so I need either a confirmation or a clarification. In case of the earlier response, I'll rewrite the TeX code. — Preceding unsigned comment added by Enamex (talk • contribs) 17:28, 24 September 2012 (UTC)

Derivation (The Uncomplicated One)
I couldn't help noticing that the Quotient Rule article is a stub. Here's a possible proof that doesn't fall into the same pithole that the previous proof fell through.

 * Let some function $$f(x)=\frac{g(x)}{h(x)}$$.
 * We wish to find $$\frac{d}{dx}f(x)$$, and this is equivalent to $$\frac{d}{dx}\frac{g(x)}{h(x)}$$.
 * We also know that $$g(x)=f(x)*h(x)$$.
 * By the product rule, we can say that $$\frac{d}{dx}g(x)=h(x)*\frac{df(x)}{dx}+\frac{dh(x)}{dx}*f(x)$$.
 * From the equation above, $$\frac{d}{dx}f(x)=\frac{\frac{dg(x)}{dx}-\frac{dh(x)}{dx}*f(x)}{h(x)}$$
 * Because $$f(x)=\frac{g(x)}{h(x)}$$, the right side simplifies to $$\frac{\frac{dg(x)}{dx}-\frac{dh(x)}{dx}*\frac{g(x)}{h(x)}}{h(x)}$$, which simplifies to $$\frac{d}{dx}f(x)=\frac{\frac{dg(x)}{dx}*h(x)-\frac{dh(x)}{dx}*g(x)}{h^2(x)}$$, completing the derivation.

There's another proof stemming from the original definition of the derivative that is also quite simple to understand (and rigorous). The article would benefit by its addition. Gummygoodness (talk) 04:23, 11 March 2013 (UTC)

 * At a glance it looks good. My only criticism is that "*" is not normally used for multiplication in maths, in my experience. I think just remove the "*" and have the multiplication implied. Adpete (talk) 11:13, 11 March 2013 (UTC)

Rollback
This article should be rolled back to just before the first edit of July 11, 2016 when someone deleted a big chunk without explaining why and also put in an incorrect formula (the formula was correct before this edit). Jrheller1 (talk) 05:13, 19 August 2016 (UTC)

 * I added implicit differentation and higher order formulas section (which was the useful information contained in the version before July 11, 2016). So I don't think rolling back would be useful anymore. Jrheller1 (talk) 22:07, 19 August 2016 (UTC)

What happened to this article?
Looking back several years in the history, this article used to be much better. It had a lot more info and was better written. But somewhere along the line, it seems to have been gutted and remade poorer. I'm working on fixing it up some, and I'll add a todo list in a moment with a couple things I think it needs. --Deacon Vorbis (talk) 16:02, 23 April 2017 (UTC)

one little comma
One of my edits has now been reverted twice. I've put it back once more and started this section to help clear up the issue. The sentence in question (with the comma I added -- the one immediately preceding "and solving for...") is:

According to any grammar guide I can find (such as point #1 in this guide), a comma is necessary here because we have two independent clauses joined by a coordinating conjunction. Even if one could argue it's not strictly necessary, it would still appear to be preferred with the comma. --Deacon Vorbis (talk) 23:56, 23 April 2017 (UTC)

 * The following is User:Jrheller1's response, which I've copied here from my talk page in order to keep the discussion in one place:
 * (signing just for completeness) --Deacon Vorbis (talk) 02:28, 24 April 2017 (UTC)
 * (signing just for completeness) --Deacon Vorbis (talk) 02:28, 24 April 2017 (UTC)

 * Rule 1 applies because there are two independent clauses in this sentence -- there are two subjects, each with their own verbs:
 * subject 1 is "two implicit differentiations"
 * verb 1 is "yields"
 * subject 2 is "Solving for $$f''$$
 * verb 2 is "yields"
 * Rule 13 doesn't apply because this is not a compound predicate. An equivalent everyday example would be something like "Alice ran to the bakery, and Bob ran to the cafe".  The example you gave isn't equivalent because it only has one subject ("We").

 * Also, again, please undo your latest revert, as it's in violation of WP:3RR. --Deacon Vorbis (talk) 02:40, 24 April 2017 (UTC)

 * I made the wording less awkward in that sentence. Hopefully that will resolve this issue.  Jrheller1 (talk) 03:04, 24 April 2017 (UTC)

Which changes are not helpful?
@Deacon Vorbis: On the examples and proofs:
 * 1) I've been trying to keep a long equation separate from long wording, so I have been breaking them up in an effort to improve readability.
 * 2) I usually see a fraction to the left of a long equation (such as for a limit, derivative, or integral, as examples) that it is factored out of, so I have been trying to do that.
 * 3) (I understand that the last equation in a section, etc, should have a period at the end of it, so I have left periods there after the first time they got reverted.)

Could you please tell me which changes you are not finding helpful, and why? Thank you - I appreciate the cooperation. -- Bwefler (talk) 15:58, 1 February 2018 (UTC)

 * Formulas can certainly be set apart from text, but they should usually be by themselves, not with other running text, and on Wikipedia, the standard is to do so by indenting with a  (although there's some talk about this changing, but that doesn't really matter for now).  Throwing in a bunch of scattered   s is certainly not the way to do it.  However, a simple statement of the product rule isn't an especially long or complex formula, and setting too many apart tends to break up the flow of the text.  As for the fraction, it was originally separated on the right hand side, so it's probably good to leave it there.  It could have just as easily been taken out on the left also, but multiplication is commutative, and it's fine either way – there's no particular good reason to change it.  –Deacon Vorbis (carbon • videos) 16:21, 1 February 2018 (UTC)

 * Acknowledged. -- Bwefler (talk) 16:33, 1 February 2018 (UTC)

Higher Order Formulas
The higher order formula seems incorrect. At a minimum, the denominator should be h^3, and even with the g/h terms in the numerator that would only get to h^2. I am not quite getting the same numerator either, but I may have made an error or there may be another algebraic step needed. For now I added a citation needed tag, but if this can't be supported by a reliable source or by showing more intermediate steps it needs to be removed. Rlendog (talk) 00:21, 21 December 2018 (UTC)

 * What's there is correct. If you expand out the $$f' = (g/h)'$$ term in the rightmost expression, it will be as you're expecting.  I'm inclined to remove the  tag per WP:CALC.  –Deacon Vorbis (carbon • videos) 00:49, 21 December 2018 (UTC)

 * I don't think that's obvious from the formulas shown. I am ok with removing the tag without an explicit source, but then the intermediate formulas need to be shown. Rlendog (talk) 17:36, 21 December 2018 (UTC)

 * Well, I added in one extra step (I think it was there at some point in the past but got removed somewhere along the way). I'm not sure if that's what you're referring to though.  Earlier I also removed the plain $$f = g/h$$ substitution at the end as not really necessary.  As it stands now, it gives an expression for $$f''$$ partially in terms of its lower derivatives, as advertised.  Someone could always sub in more, but I'm not sure how illuminating that would be.  –Deacon Vorbis (carbon • videos) 19:03, 21 December 2018 (UTC)

Avoiding comma abuse in a statement of two sequential steps
The statement "Differentiating fh=g twice and then solving for the second derivative of f" has exactly the same structure as "We laid out our music and snacks and then began to study". The result of the differentiation should be given as a parenthetical statement. Jrheller1 (talk) 17:28, 8 March 2019 (UTC)

A sentence describing the two steps of a single mathematical procedure should not be composed of two independent clauses. If it appears that the two steps are independent clauses, then the sentence should be rewritten so that it is clear they are not independent clauses. This is exactly what I did with my edit. I will be reverting back to my last edit if there are no objections on the talk page. Jrheller1 (talk) 17:06, 9 March 2019 (UTC)

 * There is absolutely no reason why you can't break something like this up into two independent clauses. Moreover, what constitutes a "single mathematical procedure" is somewhat arbitrary and subjective.  There are two conceptually separate steps here: 1) differentiating, and 2) solving for one term.  Looking back over the history, someone else had complained about the terseness, and so I added a step back in as a compromise (not even remembering why it was gone in the first place).  Awkwardly forcing it to be a single predicate by placing part inside parentheses isn't appropriate.  –Deacon Vorbis (carbon • videos) 17:41, 9 March 2019 (UTC)
 * I forgot to mention also that your analysis of the sentence structure is completely incorrect. See the older discussion further up the talk page.  –Deacon Vorbis (carbon • videos) 17:42, 9 March 2019 (UTC)
 * I forgot to mention also that your analysis of the sentence structure is completely incorrect. See the older discussion further up the talk page.  –Deacon Vorbis (carbon • videos) 17:42, 9 March 2019 (UTC)

 * I should have used rule 14: "Don't put a comma between the two nouns, noun phrases, or noun clauses in a compound subject or a compound object". "Differentiating fh=g twice and then solving for the second derivative of f" is a compound subject, so "Differentiating fh=g twice, and then solving for the second derivative of f" would be wrong.

 * My preferred form of this sentence is a compound subject ("Differentiating fh=g twice [with result here] and then solving for the second derivative of f"), a single verb (yields), and a single object (the final equation for the second derivative of f).

 * Your preferred form is two clauses ("Differentiating fh=g twice yields x" and "solving for the second derivative of f yields y"). Why do you think this form is better?  Jrheller1 (talk) 19:39, 9 March 2019 (UTC)

 * I also don't understand why you insist on placing a comma between the two clauses in your preferred form. For a two-step procedure, "Step 1 yields intermediate result A and then step 2 yields final result B" seems fine to me.  Even though there is no subordinating conjunction between the first clause and the second clause, the second clause is obviously dependent on the first clause; and grammar references state that no comma should be used between an independent clause followed by a dependent clause.  Of course, commas must be used in a three-step procedure: "Step 1 yields intermediate result A, then step 2 yields intermediate result B, and step 3 yields final result C". Jrheller1 (talk) 21:11, 9 March 2019 (UTC)

 * This really should settle the issue. The comma doesn't belong in either of the two possible sentence structures.  In the two clause structure, the first clause is independent because it can be a sentence by itself.  The second clause is dependent because it cannot be a sentence by itself ("solving for the second derivative of f yields y" is meaningless without the first clause).  But the compound subject, verb, object structure is better because it doesn't use the verb "yields" twice.  So I will be reverting back to my last edit unless someone objects on the talk page. Jrheller1 (talk) 03:11, 10 March 2019 (UTC)

 * No, this doesn't settle it. It's not currently a compound subject.  You  turn it into a compound subject as you indicated, in which case a comma wouldn't be needed, but to do that, you either have to remove the intermediate result (which I put back in ages ago because someone (reasonably) complained about the terseness, as I indicated), or you have to add it in as some sort of parenthetical, which would be awkward.  For crissakes, this is getting absolutely ridiculous.  Please just drop this already.  –Deacon Vorbis (carbon • videos) 03:23, 10 March 2019 (UTC)

 * My last edit (which you reverted) has the compound subject, verb, object form.  In the current two clause form, the comma still shouldn't be there since it is an independent clause followed by a dependent clause. Adding the intermediate result as a parenthetical is not awkward at all. Jrheller1 (talk) 03:30, 10 March 2019 (UTC)

 * It's two independent clauses joined by a coordinating conjunction (and). Seriously, drop this; it's way past ridiculous at this point.  Both our times would be better served doing other things.  –Deacon Vorbis (carbon • videos) 03:46, 10 March 2019 (UTC)

As I stated above, the current form is an independent clause followed by a dependent clause, so a comma between them is wrong. You have been attempting to edit war an incorrect comma into text that I originally wrote from April 2017 up to the present. I have no trouble admitting when I am wrong, but in this case, you are definitely wrong, so you are the one who needs to drop this. Jrheller1 (talk) 04:25, 10 March 2019 (UTC)

 * And you're still wrong; it's two independent clauses separated by a coordinating conjunction. There's nothing more to say about this.  –Deacon Vorbis (carbon • videos) 16:34, 10 March 2019 (UTC)

 * As I have stated multiple times above, the second clause ("solving for the second derivative of f yields y") is meaningless without the first clause, so it is dependent on the first clause. You are clearly wrong and clearly have a problem admitting this. Jrheller1 (talk) 16:39, 10 March 2019 (UTC)

 * It doesn't matter whether or not it's conceptually meaningful or meaningless. Syntactically, it's independent.  Just fucking stop already.  –Deacon Vorbis (carbon • videos) 17:04, 10 March 2019 (UTC)

 * The second clause is obviously dependent on the first. But even if you assume that the second clause is independent, there still should not be a comma between the clauses because this is a list of two items.  The sentence "Alice is 8 and Bob is 9" is composed of two independent clauses.  But since it is a list of two items (the ages of two individuals), there should be no comma between the clauses. Jrheller1 (talk) 18:34, 10 March 2019 (UTC)

Your behavior here is very disruptive. The reason I removed the intermediate result back in 2017 was to try to get you to stop edit warring the incorrect comma into the text I wrote. My latest edit is an improvement over my text from 2017 since it eliminates the double use of the verb "yields". Jrheller1 (talk) 18:34, 10 March 2019 (UTC)

 * Before you go throwing stones, the version that was in place had been reached by compromise between myself and another editor. You came in here way later and decided to change it.  You were reverted.  You attempted to discuss, which is great, but you were unwilling to accept the fact that you didn't convince (even remotely) the only other editor in the discussion (myself) that the change was a good one.  The only disruption here was initiated by you.  Since you've been unwilling to accept the version that was in place, I've listed this for input at WP:3O.  Please refrain from making any further changes until there's been feedback from there.  –Deacon Vorbis (carbon • videos) 19:02, 10 March 2019 (UTC)

 * You're being very dishonest about your behavior here. The "version that was in place" or "compromise between myself and another editor" was simply my text with your incorrect comma added.  I wrote two versions of the sentence.  Version 1 is the two clauses without the unnecessary comma.  Version 2 is the improvement from March 8 with a compound subject, single "yields", and single object.  The only thing I'm unwilling to accept is the unnecessary comma that you keep trying to edit war into my text.   You have made at least 7 edits forcing this unnecessary comma into my text.  If that's not disruptive, I don't know what is. Jrheller1 (talk) 19:33, 10 March 2019 (UTC)

 * It's not WP:OWN. Other people can come in later and modify it.  The change I made most recently was in response to another editor's concern about omitted steps.  I've backed up my claim about the correctness of a comma with evidence from grammar guides, while you've continued to shift the goalposts about why it's needed.  Your latest reasoning about a list of two items doesn't apply because this isn't a list.  If that reasoning were valid, you could apply it to  sentence with two independent clauses joined by a coordinating conjunction.  Moreover, the example you used  require a comma because it has two independent clauses joined by a coordinating conjunction.  You  get away with omitting the comma there because each clause is so short, but that's irrelevant to the case at hand.  –Deacon Vorbis (carbon • videos) 20:32, 10 March 2019 (UTC)

 * You're being intentionally obtuse. I know very well that nobody owns anything on Wikipedia.  What I clearly meant was that I wrote the text of the sentence.  The only edit you ever made to that sentence was repeatedly adding the incorrect comma.  Back when you were edit warring the unnecessary comma into the sentence in April 2017, I removed the "yields $$fh+2f'h'+fh=g$$" in an attempt to stop you from edit warring the unnecessary comma into the sentence.  Sometime between then and December 21, 2018, some other editor corrupted the formula.  Then on December 21, 2018, you put back the text I wrote (including the "yields $$fh+2f'h'+fh=g$$") plus your unnecessary comma.

 * I'm not "shifting the goalposts". I have always known that the comma was unnecessary.  I don't think about grammar rules when I'm writing; it's something that becomes intuitive after a while.  You are forcing me to waste time looking for the specific grammar rules showing that the comma is unnecessary.  Now I have found them.  First rule: no comma between independent clause followed by dependent clause.  Second rule: no comma between items in a two-item list.  The sentence definitely is a list (more specifically, an ordered list) of two steps in a mathematical procedure.  The reason "Alice is 8 and Bob is 9" shouldn't have a comma is precisely because it is a two-item list, not because the clauses are short.  "Alice is wealthy, but Bob is poor" definitely requires a comma.

 * This is such a minor issue that this debate over grammar is really a waste of time. But your behavior is totally responsible for this waste of time.  If you (or any other editor) had placed an unnecessary comma somewhere in a sentence you wrote, I probably would not have even bothered to remove it.  I definitely would have immediately dropped the issue if my edit was reverted. Jrheller1 (talk) 22:12, 10 March 2019 (UTC)

 * Pictogram voting comment.png 3O Response: Commas do often come up in punctuation disputes. I also tend to look at this as an inline list which is introduced with For example,.  A list with two items does not need a serial comma before the and as might be used with a longer list.  However, instead of simple terms, the list items here are independent clauses.  (Despite the second clause representing a secondary step in the process being described, it is not grammatically subordinate to the previous clause.)  While a comma isn't required as a list separator, I believe that a separator is required for the clauses.  From the lede of independent clause is this sentence:  Independent clauses can be joined by using a semicolon or by using a comma followed by a coordinating conjunction (and, but, for, or, nor, so, yet, etc.).  Thus this could be ; then or, and then.  You can find some common examples of an and then construction at then. Please note that the punctuation introducing the list should be raised from a comma to a colon; there needs to be greater separation between For example and the rest of the sentence than the separation between the clauses.  (This would also be consistent with an example near the top of the article.)  It might be worthwhile to have an wikicomment following the comma to explain to editors why it is there.  This is a non-binding third opinion but I hope that it helps!  If you want further clarification, I would recommend asking at Reference desk/Language or Wikipedia talk:WikiProject Guild of Copy Editors. – Reidgreg (talk) 12:16, 12 March 2019 (UTC)

 * This doesn't really address the issue. The text I want is: "differentiating $$fh=g$$ twice (resulting in $$fh + 2f'h' + fh = g$$) and then solving for $$f$$ yields final result".  The text the other editor wants is: "differentiating $$fh=g$$ twice yields $$fh + 2f'h' + fh = g,$$ and then solving for $$f$$ yields final result".  It is obviously better to avoid two uses of the word "yields" in the same sentence. Jrheller1 (talk) 01:01, 21 March 2019 (UTC)

 * Also: the text I prefer is closer to the stable version (from April 23, 2017 to December 21, 2018) than the text DV prefers. If someone other than DV does not explain why DV's preferred version is better, I will be reverting back to my preferred version. Jrheller1 (talk) 16:21, 21 March 2019 (UTC)

Cause of edit warring over a comma
The thing that bugs me about this is that Deacon Vorbis has been consistently trying to blame me for the edit warring over a comma in this article. See the comment he left on my talk page back in April 2017 and also his comments in the discussion above.

I wrote a sentence. He added an unnecessary comma to the sentence I wrote. Then I reverted his addition of an unnecessary comma. At this point, WP:BRD indicates that the reverted editor should start a discussion if they wish to challenge the revert. But since the issue is so minor, any reasonable editor would have just dropped the issue at this point.

Then he started the edit war by reverting my deletion of his comma without any discussion. The fact is, he is totally responsible for the edit warring over a comma in this article. And he is still insisting that his edit-warred-in version with his unnecessary comma prevail, even though my latest version is definitely an improvement (it gets rid of the double use of the word "yields" in a single sentence).

If and when I revert his latest revert, the purpose will be to get rid of the double use of the word "yields" and also to remove any possible ambiguity over the need for a comma. He has no good reason to object other than wanting to get his way. Jrheller1 (talk) 18:24, 11 March 2019 (UTC)

I don't want to waste the time of any other editors with this silly dispute. What I do want is for Deacon Vorbis to end the edit war by not reverting me when I restore my improvement later this week. Jrheller1 (talk) 00:28, 12 March 2019 (UTC)

 * Your edits aren't improvements; your use of a comma is incorrect. I've explained this many, many times already. I've pointed you to a grammar guide with very clear reasoning (any other guide will tell you the same thing, so I'm not even cherry picking here).  You've just continued to deny reality about this while offering no evidence to support your position.  I don't see what else I can say to convince you, but I  ask for a WP:3O, so please stop threatening to continue this stupid edit war and just wait to see what comes out of that.  –Deacon Vorbis (carbon • videos) 00:38, 12 March 2019 (UTC)

 * As I just explained above, you started the edit war, so it is your responsibility to end it by not reverting me. My edit from March 8 that you reverted has a compound subject with two nouns.  There is a grammar rule clearly stating that there should not be a comma between the two nouns.  Even the current form (comma between the two clauses) has always "looked" wrong to me, and now I have found the grammar rule explaining why: no comma between the items in a two-item list. Jrheller1 (talk) 00:59, 12 March 2019 (UTC)

 * You agree this is a silly dispute. So why do you want to waste the time of other editors with it?  Do you think they will appreciate having their time wasted? Jrheller1 (talk) 00:59, 12 March 2019 (UTC)

 * It's not a fucking list. A sentence cannot just be a list.  It's two independent fucking clauses joined by a coordinating fucking conjunction.  As I already pointed out, if your reasoning were valid, then any fucking sentence with two independent fucking clauses joined by a coordinating fucking conjunction would be a list of two items and have the comma omitted.  But then there would be no need for the pretty basic rule about having a comma there in the first place.  It doesn't matter if it "looks" wrong to you.  It matters what people who know what they're talking about advise, and they advise that there should be a comma there.  You yourself said you don't pay attention to rules and just go by what feels intuitively right, but your intuition can fail, and it has failed here.  Your edit was completely unnecessary; its only purpose was because you have some weird fucking obsession with the fact that it got rephrased back in a way that required the comma, because another editor didn't like the omitted step.  You were nowhere to be found when that was going on, and it had been in place for months before you dropped back in to stir this same shit up again.  I've said all there is to say – multiple times.  I'm done discussing this until someone can step in and offer another view.  –Deacon Vorbis (carbon • videos) 01:15, 12 March 2019 (UTC)
 * I'm not getting involved in the specifics on this, but offer my sympathies and a little advice. I am also a strong proponent of WP:BRD, but note that this is an editing practise and has not gained community consensus as a guideline or policy.  Editors are under no requirement to follow BRD.  I would also note that in a dispute between two editors, if one is edit warring then by definition they both are.  I know that can suck if all you're trying to do is keep the article stable, but sometimes you have to live with what you feel is an "inferior" version of an article until you can reach some form of consensus.  Edit warring is disruptive, period, no matter how correct you believe you are, or how much good work you may have done on the article.  If you're unable to resolve the dispute on the article talk page and have to go through dispute resolution, it's more work for other editors to go through the diffs and edit warring looks bad for all parties involved.  I was claiming BRD while in a dispute with an editor on my first GA, and ended up edit warring which led to the article being put under protection and almost jeopardized its appearance on the main page.  Even if I was right, I was going about it the wrong way.  Sometimes you have to live with what you may feel is an inferior version of an article while patiently going through discussions or, if it comes to it, months of dispute resolution.
 * Ideally, disputes can be resolved on the article talk page, which requires less time and effort. Assume good faith, listen and be open-minded, and don't respond adversarially.  I feel this is especially important with the little things, like a comma, as few volunteers will spend their time to help resolve deadlocks over such.  Try not to take things personally, and if you do get upset then find something else to work on for a while; there's plenty to do on Wikipedia.  If you have time, there are a ton of essays at  with advice. – Reidgreg (talk) 12:20, 12 March 2019 (UTC)

This is nuts
Please stop. I thought this was settled, and yet you've reignited the edit war. There was even a third opinion generously offered. And I realize it's not binding, but it's the best we've got. The version you've reverted to was not the "stable" one. Please, please, please just let this one go. –Deacon Vorbis (carbon • videos) 19:27, 24 March 2019 (UTC)

Changes in notation that got reverted
I kept on changing $$f(x)=g(x)/h(x)$$ to $$h(x)=f(x)/g(x)$$ even though it’s just notation and not an improvement to the article because the latter is commonly used in textbooks and online resources, but it turns out the former has been long established on this page. Sorry for the unnecessary changes. Also, I replaced k with $$\Delta x$$ in the limits since normally $$\Delta x$$ or h but not k is used to represent the change that approaches zero in the derivative, but since h was taken for a function, the remaining common choice was $$\Delta x$$. Please take note though that my other edits also got reverted when only the changes in notation were unnecessary. 49.147.83.13 (talk) 16:49, 21 May 2020 (UTC)

 * Your other changes were mostly not improvements either. There were already a couple reasonably representative examples in place, we don't need more (similar) ones.  Having too many runs afoul of WP:NOTTEXTBOOK.  Likewise, additional proofs are pretty unneeded at this point, and probably more importantly, your "proof" using logarithmic differentiation is only even valid when both functions are positive.  Also, the section headers don't have to state every bit that goes into them; they just have to serve as a means to help tell themselves apart.  On a side note, I find the "reciprocal rule" a silly extra rule that serves no purpose.  Most Calculus texts that I've seen don't even bother to name or prove it separately.  –Deacon Vorbis (carbon • videos) 17:22, 21 May 2020 (UTC)

 * Thank you for explaining that you disagree with the logarithmic differentiation proof and find the reciprocal rule quite trivial. 49.147.83.13 (talk) 17:27, 21 May 2020 (UTC)

title: Talk:Quotient rule


text:
Talk:Multiverse

Prove of multi-verse Or is that the edge of the universe isn’t the end?
if you stir a cup of water, do you get a vortex? No if you put that same cup of water on a plate and spin it do you get a centrifuge with all of the heavy elements, expanding to the side with the big bang and eternal inflation it is safe to say the universe is spinning inside of a larger entity Tony Alan Ratliff 2600:1014:B1E9:6AB7:BD29:2E58:D33C:B971 (talk) 14:43, 2 January 2024 (UTC)

 * This is a page for discussion of the article, not its subject. If you have a modification you’d like to make, you may. OverzealousAutocorrect (talk) 18:28, 15 February 2024 (UTC)

The obvious solution
it's not that hard to understand. The nature of the multiverse or whatever you want to label it is obvious. Initially there are a nearly infinite number of potential universes, a timeline for every configuration of quantum events. We can refer to these as "potentials" as time progresses and quantum states are determined they become fixed. Or "realized". At that instant any potential that is not consistent with the quantum states that have been realized become "unfulfilled" and are effectively erased because they are not consistent with really happened. Which we can call " reality" so initially there was a potential for anything that was within the confines of the natural laws ( which we don't have a complete understanding of yet. Nor have we even discovered all of its mechanisms of influence) so as time goes by and quantum states are realized massive amounts of potentials are changed to unfulfilleds, like the opposite of a ripple. Eventually there will come a time when the last free willed entity/observer will exert its last influence upon the universe and at that point all that follows will be strictly cause and effect. Retroactively validating itself as the predetermined destiny of the universe and everything in it. Since it was the only potential that perfectly and exactly was composed of the configuration of quantum states that have been realized..

Heres an example. As of today all the stuff that happened in the past really happened. There were other things that could have happened but they didn't, so they arent reality. When i make my choice of what to eat for breakfast there are trillions of alternative dimensions desperately depending on what choice i make. As i exercise my free will and choose cornflakes every universe in which i didn't is instantly erased. The me that picked eggs and the me that picked waffles is snuffed out of existence. They aren't chilling somewhere else living out their alternative configuration somewhere, their entire reality is retroactively erased,they never happened. Likewise every universe in which i picked cornflakes BUT you did too is deletes, as well as any other that isn't consistent with the choices i made AS WELL AS those that everyone Else made. We all have an equal share in authoring what will ultimately be the final story when all is said and done. Because even the things that an aborted child would have maybe done are as impactful as the 100 year life of a great leader. It's like an exponentially reducing pool of possibilities that Eventually reach the very last one. And that one "wins" or whatever. Anyway if you don't understand how it works after reading this then all the potentials in which you were smart Have become unfulfilleds. 2601:201:8B00:4F00:44D3:A986:76BD:FE9E (talk) 23:52, 21 January 2024 (UTC)

 * What do you mean by "nearly infinite"? 2001:9E8:460F:6200:34BF:6372:F526:466C (talk) 10:58, 14 February 2024 (UTC)
 * Again, this is a page for discussing the article, not its contents. If you have a change you’d like to make to the article, you may suggest it here. OverzealousAutocorrect (talk) 18:30, 15 February 2024 (UTC)
 * you have inspired me to exercise my free will and eat a chicken for dinner. thank you! 🍗 38.15.195.81 (talk) 21:02, 25 March 2024 (UTC)

Not hypothetical
The framing of the opening sentence does not match my understanding of the use of the word "multiverse". It is a metaphysical claim that there really is a multiplicity of universes. --Russell E (talk) 12:28, 14 March 2024 (UTC)

Popular but erroneous definition in the page
The correct description according to ontoaxiomatics = physical axiomatics:

 * multiverse: mathematically related universes = family of universes

Infinitely many multiverses exist.

 * omniverse: the impossible set of all multiverses: many are mutually exclusive = the omniverse has no common axiomatic engagement

The axiomatic system of all axiomatic systems doesn't exist.

The set (non-engaged collection) of all sets doesn't exist.

The omniverse cannot be described neither by a human = non-fundamental mathematical, because it includes nontrivially (without easy informational patterns) infinite parts and mathematical descriptions.

Also the Everything and the unitary (single) logical foundations don't exist as unitary logical wholes for the same reasons.

The core/main definition is wrong. It's simply popular and biasedly soothing = in medieval times the Earth and nowadays the multiverse is the center of the supposedly functionally unitary Everything = which is wrong.

___________

Encyclopedias must inform and not merely promote popular biases.

title: Talk:Multiverse


text:
Paars

Paars is a commune in the Aisne department in Hauts-de-France in northern France.

title: Paars


text:
User:Pazzer

Pazzer aka Steve Parrett. From Symonds Yat West, Herefordshire, England. Interests include biogeography, anthropology and local history. Currently studying Geography and Environmental Management at The University of The West of England Bristol.

My contributions include:

 * Chepstow
 * England
 * Forest of Dean
 * Goodrich
 * Hereford
 * Herefordshire
 * Manic Street Preachers
 * Monmouth
 * Monmouthshire
 * River Wye
 * Symonds Yat
 * Tintern Abbey
 * Welsh Bicknor
 * Whitchurch, Herefordshire

title: User:Pazzer