text:
Précis of epistemology/Reason, a manual

Learning through reasoning what good principles teach
Science (or knowledge) is to give good proofs.

A good proof is a good observation or a good reasoning.

A good reasoning starts from good premises to arrive at a good conclusion by a clear and logical path.

Good premises are good principles or good observations, or conclusions already drawn from good reasoning.

Good conclusions are the statements or thoughts that help us to think well and to live well, the fruits of reason. A thought is a fruit of reason when it helps us to think well and to live well. What is well thought helps us to think well. What does not help us to think well is not well thought.

Reasoning reveals explicitly what principles implicitly determine, it develops what is enveloped, it unfolds what is folded, it unveils what is present but hidden until we have reasoned.

To reap the fruits of reason, we must learn through reasoning what good principles teach.

Ethics is fundamental to all sciences
To do a science we must always do it well. Think well, observe well, imagine well, speak well, will well, act well ...

Knowledge that is not good knowledge is not knowledge at all.

Epistemology is the knowledge about knowledge. It is fundamental for all knowledge, because we cannot develop knowledge without knowing how to recognize good knowledge.

Ethics is the knowledge about the good of the spirit. Since knowledge is a good of the spirit, epistemology is part of ethics.

All science is based on good principles. We recognize good principles by their fruits, their consequences which help us to think well and live well. To recognize good principles we must recognize their fruits. Since ethics is the knowledge about thinking well and living well, it is the knowledge which recognizes the fruits of reason. Such a knowledge is necessary to recognize all the principles of science. Ethics is therefore fundamental to all sciences.

Ethics is a science as soon as it gives good proofs, based on good principles and good observations.

Good principles are always good for everyone. If one spirit can reap the fruits of good principles, then all spirits can reap the same fruits. When we look for good principles, we look for a good for all spirits, we put into practice a great principle of ethics: the good of a spirit is to live for the good of all spirits.

The conditions of possibility of truth
For the truth of a statement to be determined, its meaning must be determined. The same statement can be sometimes true, sometimes false, according to the various ways of interpreting it.

To tell the truth it must be determined. We cannot know anything until we have clearly determined the concepts we use.

A concept can be defined from other more fundamental concepts. For example, a circle is a set of all points, in a plane, equidistant from the same point, the center of the circle. But for this definition to be useful, the concepts of set, point, plane, distance and equality must first be determined.

The most fundamental concepts are concepts that are not defined from other concepts. They are determined either empirically or theoretically.

A concept is determined empirically by determining the set of detectors, measuring instruments or observation devices intended to signal the presence of the concept.

Concepts are determined theoretically by giving principles (axioms and definitions) which enable us to reason with them. The most fundamental concepts are determined theoretically with axioms, the others with definitions.

For example, all distance measuring instruments empirically determine the concept of distance. Axioms which found a geometry determine it in a theoretical way.

In empirical science, we want our concepts to be determined both empirically and theoretically, because we want theories that explain our observations.

The principles of a theory determine a set of truths: theorems, logical consequences of axioms and definitions. These theoretical truths are truths by definition. They result from definitions of concepts. Axioms can be considered as definitions of fundamental concepts.

Any theory, any system of principles, however foolish, determines a set of truths by definition, provided it is non-contradictory. But such truths by definition are not enough to make true knowledge. We want good theories, based on good principles, which reveal their value when they bear fruit.

A remark on the precision of concepts: a concept is precisely determined when the truth of its application is determined in all cases. Such precision is rarely achieved and is not necessarily desirable. Conceptual vagueness, partial indeterminacy, can make the use of concepts more flexible and better adapted to reality. In order for a concept to be used to develop knowledge, it is sufficient that the truth of its application be determined in a certain domain, it does not have to be always determined.

What do we know about good principles?
Logic: we know all the logical principles (the rules of correct reasoning) or almost, because classical logic is enough to make all the sciences. Non-classical logics are not without interest, but they do not seem to be of fundamental importance.

Mathematics: we know principles (those of the theory of sets of Cantor, Zermelo ...) which are sufficient to found mathematical knowledge. We can prove (Gödel, Tarski ...) that our mathematical principles will never be sufficient to prove all the mathematical truths, that we can always find new principles which prove more, but in practice the principles that we already have are largely sufficient for almost all of our theoretical needs.

Fundamental physics: we do not know all the fundamental laws because we do not know how to unify quantum physics and the theory of gravity (general relativity) in a satisfactory way and because we do not know much about dark matter and dark energy. But our present theories can explain most physical phenomena. We have to question the universe as a whole, the Big Bang, the formation of galaxies, black holes, experiments in large particle accelerators ... to meet their limits.

Materialist sciences: we generally justify our explanations not with fundamental physical laws but with intermediate laws (fluid mechanics, population dynamics ...) that we sometimes manage to justify on the basis of more fundamental physical laws. We can always invent new ways of explaining with new intermediate principles. We cannot know in advance everything we will discover in this way.

Psychology: Fundamental physics does not explain the presence of spirit (the life of a consciousness): there are atoms and void - and spirits? It turns out that the electrical activity of the brains can make the spirit exist, but we do not know why. In practice we try to marry a psychological approach (what we know about the spirit because we are spirits) with a materialist approach (how the brain works).

A great principle of psychology: ee are conscious of ourselves in order to be able to act voluntarily on ourselves in an appropriate way.

Philosophy: we find good principles for thinking well and living well, but we never know everything. We can always invent new principles.

A great principle of epistemology: to know, we have to learn by reasoning what good principles teach.

A great principle of ethics: the good of a spirit is to live for the good of all the spirits.

A great principle of metaphysics: the whole being of a being is its being in a whole, or its being a whole, or both.

Next chapter: How the spirit works >>>

title: Précis of epistemology/Reason, a manual


text: User talk:80.4.213.141

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

title: User talk:80.4.213.141


text:
Wikibooks:Strategy guides

This is the policy for the inclusion of strategy guides and walkthroughs relating to video games. These should be placed on Shelf:Electronic games. See Reading room/Proposals for the ongoing discussion.

Historical background
Video game strategy guides and walk-throughs were banned from Wikibooks in 2007, although scholarly analysis of video games, or guides on the design of video games, have always been allowed. Following from this decision, most guides were migrated to StrategyWiki, an existing wiki with a compatible license.

Following the strong consensus at a community discussion in early 2021, this was reversed, as it was felt that it was no longer logical to exclude books on video games that described strategy. But there was also strong consensus that a policy would be needed to implement this change.

Overview of policy
The main consideration is that Wikibooks has a number of policies and guidelines relating to content, and nothing in this policy supersedes these. Instead it provides additional guidance for strategy guides, which should conform with the purpose of Wikibooks as an educational resource that allows textbooks, annotated texts, instructional guides, and manuals.

Copyright considerations

Wikibooks has strict rules regarding copyrights: see Copyrights for the full policy. In particular:

 * If you contribute text directly to Wikibooks, you irrevocably agree to license it for reuse under the CC BY-SA 3.0 and GFDL licenses.
 * Avoid plagiarism in your guide; you should never represent the work of another author as one's own.
 * If in any doubt over copyright, please ask somewhere like the assistance reading room.

The importing of text from external sources may be feasible under certain circumstances, which can be requested. But please note:

 * Compatible licenses of copyrighted sources for importing include CC BY-SA (versions 1.0, 2.0, 2.5, 3.0, but not version 4.0), and CC BY (all versions and ports, up to and including 4.0). Additionally, public domain and CC0 sources can be imported.
 * Be sure that any license requirements are complied with. For example, attribution of sources is often required by licenses.
 * Many sources will be copyrighted with licenses incompatible to that of Wikibooks. While providing citations to these are permitted, the copying of content is normally prohibited, unless under the doctrine of fair use. Fair use should only be applied as described by policy.
 * If importing from a source with a compatible license, make sure that the source itself does not have a different interpretation of copyright. For instance, StrategyWiki allows the use of video game handbook content verbatim in articles; but this is not allowed on Wikibooks (except in small portions), unless the video game handbook itself was released under a compatible license, which is usually not the case.

Screenshots and other media usage
The usage of media, such as images, videos and sounds, is strictly limited by policy: see Media for the full policy.

Media in Wikibooks usually fall into two broad categories:

Screenshots taken from copyrighted games are usually copyrighted non-free media; exceptions to this are rare, but the EULA (end-user license agreement) can be consulted to check this. Non-free media rules would normally apply to screenshots of popular games and series like Minecraft, Grand Theft Auto, and Pokémon.
 * Media in the public domain, or under a Commons-compatible free license. New media in this category should be uploaded to Wikimedia Commons; you can also search Commons for suitable media that are already there. Media at Commons may still be copyrighted, but their free licenses allow for their liberal reuse in books.
 * Non-free media, which must be uploaded to Wikibooks, and only used under the strict terms of the non-free media usage policy. Non-free media are copyrighted, and the policy for their reuse relates to the doctrine of fair use in US law. Trusted users needing to upload non-free media can apply for local uploader rights, and other users can ask for non-free media to be uploaded at requests for import.

Screenshots of open-source games may be in the public domain, or compatible with an appropriate free license; but carefully examine the EULA of open-source games as there can be exceptions. If a developer consents to donating screenshots, you may consider going through the Volunteer Response Team with them to make otherwise non-free media available under a free license.

Copyright issues relating to media can be complex, but a good place to ask would be Commons:Village pump/Copyright; but first read Commons:FAQ, Commons:Licensing, and Commons:Copyright rules by subject matter. To get around the restrictions on non-free media, graphical images and other media can be created to demonstrate gameplay, that are not screenshots nor restrictively copyrighted; examples of these can be found at Category:Video game gameplay at Commons.

Only include significant games
Please do not include strategy guides for video games that are likely not significant to the wider gaming community. Wikibooks is not a place to merely advertise your or your friend's game. A useful rule of thumb is to look at review aggregator websites such as Metacritic or OpenCritic; if a video game listed there has garnered reviews, there is every indication that the game is widely significant. Inclusion of games in other prominent review sites may also indicate significance. Games that pass the notability requirements of Wikipedia should also be acceptable.

If there is doubt whether a game is significant, this should be discussed at venues such as the projects reading room.

Employ an academic tone
In general, an academic tone should be employed in strategy guides. Overall, the style should be that of the academic writing that you might find in a textbook, research paper, or encyclopedia. In particular, advertising or self-promotion, or other types of endorsements, should be avoided; and a neutral point of view, which is a fair and neutral description of the facts, should always be presented.

It can be helpful to give a wider context to video game strategy, such as a historical or technical discussion of the game.

Avoid primary research
Wikibooks does not allow primary research, such as proposing new theories and solutions, and presenting original ideas. However, you can use video games as a primary source; and it is reasonable to describe elements of strategy that are likely to be encountered during a typical play-through of a game. But opinions about the relative benefits of strategies should be avoided, unless they are obvious, or else provided by citing reliable secondary or tertiary sources. It is preferable to provide references for citations of any secondary or tertiary sources.

Citing secondary or tertiary sources may also help in maintaining a neutral point of view; any controversial statements should be shown within the context of other opinions, and reliable sources should help in providing these.

title: Wikibooks:Strategy guides


text:
Wikibooks:Collections/Unicode Character Reference

Unicode Character Reference

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text:
MediaWiki:Gadget-markAdmins-data.js

// Main gadget script located at MediaWiki:Gadget-markAdmins.js

mw.hook('userjs.script-loaded.markadmins').fire({	//local admins	"Atcovi": ["sysop",],	"Az1568": ["sysop"],	"JackPotte": ["sysop"],	"Jusjih": ["sysop"],	"Kittycataclysm": ["sysop"], /*expires 7 Apr 2025*/	"Leaderboard": ["sysop"],	"MarcGarver": ["sysop", "checkuser", "steward"],	"Mrjulesd": ["sysop"],	"SHB2000": ["sysop"],	"Xania": ["sysop", "checkuser"],	//stewards	"Ajraddatz": ["steward"],	"Albertoleoncio": ["steward"],	"AmandaNP": ["steward"],	"AntiCompositeNumber": ["steward"],	"Base": ["steward"],	"Bsadowski1": ["steward"],	"DerHexer": ["steward"],	"Elton": ["steward"],	"EPIC": ["steward"],	"HakanIST": ["steward"],	"Hasley": ["steward"],	"Hoo man": ["steward"],	"JJMC89": ["steward"],	"Johannnes89": ["steward"],	"Jon Kolbert": ["steward"],	"Martin Urbanec": ["steward"],	"Masti": ["steward"],	"Matanya": ["steward"],	"Melos": ["steward"],	"Mykola7": ["steward"],	"RadiX": ["steward"],	"Sakretsu": ["steward"], "Schniggendiller": ["steward"], "Sotiale": ["steward"], "Stryn": ["steward"], "Superpes15": ["steward"], "Tegel": ["steward"], "Teles": ["steward"], "Tks4Fish": ["steward"], "Vermont": ["steward"], "Vituzzu": ["steward"], "Wim b": ["steward"], "Yahya": ["steward"], "Xaosflux": ["steward"], "علاء": ["steward"], //global sysops "1234qwer1234qwer4": ["global_sysop"], "Ameisenigel": ["global_sysop"], "Amire80": ["global_sysop"], "Billinghurst": ["global_sysop"], "Biplab Anand": ["global_sysop"], "CptViraj": ["global_sysop"], "DARIO SEVERI": ["global_sysop"], "Daniuu": ["global_sysop", "ombudsman"], "DannyS712": ["global_sysop"], "Fehufanga": ["global_sysop"], "Iluvatar": ["global_sysop"], "Infinite0694": ["global_sysop"], "LR0725": ["global_sysop"], "MF-Warburg": ["global_sysop"], "MdsShakil": ["global_sysop", "ombudsman"], "Minorax": ["global_sysop", "ombudsman"], "Mtarch11": ["global_sysop"], "Praxidicae": ["global_sysop"], "Renvoy": ["global_sysop", "ombudsman"], "Samuele2002": ["global_sysop"], "Savh": ["global_sysop"], "Syunsyunminmin": ["global_sysop"], "Turkmen": ["global_sysop"], "Uncitoyen": ["global_sysop"], "WhitePhosphorus": ["global_sysop"], "WikiBayer": ["global_sysop"], //ombudsman "AGK": ["ombudsman"], "Ameisenigel": ["ombudsman"], "Doǵu": ["ombudsman"], "Emufarmers": ["ombudsman"], "Faendalimas": ["ombudsman"], "Nehaoua": ["ombudsman"], "RoySmith": ["ombudsman"], "だ＊ぜ": ["ombudsman"], });

title: MediaWiki:Gadget-markAdmins-data.js


text:
MediaWiki:Gadget-markAdmins.js

/** // /* eslint indent:["error","tab",{"outerIIFEBody":0}] */ /* global jQuery:false, mediaWiki:false*/
 * Flag administrators and special user group members with a letter
 * in parenthesis behind links that go into their user namespace.
 * @Documentation: https://commons.wikimedia.org/wiki/MediaWiki_talk:Gadget-markAdmins.js
 * E.g. Didym -> Didym (A)
 * @rev 3 (14:14, 20 August 2019 (UTC))
 * @author Euku - 2005, PDD, Littl, Guandalug
 * @author Didym - 2014
 * @author Rillke  - 2014
 * @contributor Perhelion - 2017
 * @contributor Perhelion - 2017

(function (mw, $) { 'use strict';

var markAdmins = mw.libs.markAdmins = { config: {}, defaults: { groups: { 'sysop': { label: 'A', legacyName: 'Admin', legacyLabelId: 'atxt', enabled: true },			'oversight': { label: 'OS', legacyName: 'Oversight', legacyLabelId: 'oversighttxt', enabled: true },			'checkuser': { label: 'CU', legacyName: 'Checkuser', legacyLabelId: 'checkusertxt', enabled: true },			'bureaucrat': { label: 'B', legacyName: 'Bureaucrat', legacyLabelId: 'bureautxt', enabled: true },			'steward': { label: 'S', legacyName: 'Steward', legacyLabelId: 'stewtxt', enabled: true },			'ombudsman': { label: 'Omb', legacyName: 'Ombudsman', legacyLabelId: 'ombtxt', enabled: true },			'global_sysop': { label: 'GS', legacyName: 'Global Sysop', legacyLabelId: 'gstxt', enabled: true }		},		runOn: ['Special', '', 'User', 'User_talk', 'Project', 'File', 'Help'], runOnHistory: true, runOnTalk: true, runOnDiff: true },

init: function (users) { markAdmins.users = users; // Wait for user configuration through their .js		// Not adding as a gadget dependency because user .js		// is sometimes invalid and fails loading and // gadget dependencies do not offer failed/error options $.when(mw.loader.using('user'), $.ready).then(markAdmins.mergeConfig, markAdmins.mergeConfig); },

mergeConfig: function { // Merge new configuration var optionsConfig = mw.user.options.get('markAdminCfg'), cfg = $.extend(				true,				markAdmins.config,				markAdmins.defaults,				window.markAdminCfg || {},				optionsConfig ? JSON.parse(optionsConfig) : {}			);

markAdmins.ns = mw.config.get('wgNamespaceNumber'); cfg.markSubpages = !!window.marksubpages; cfg.dontMarkMyself = window.dontmarkmyself ? mw.config.get('wgUserName') : false;

// Namespace run conditions if (!(cfg.runOn.indexOf(mw.config.get('wgCanonicalNamespace')) !== -1 || cfg.runOnHistory && mw.config.get('wgAction') === 'history' || cfg.runOnTalk && markAdmins.ns % 2 || cfg.runOnDiff && !!mw.util.getParamValue('diff'))) return;

// Hook-up content loading mw.hook('wikipage.content').add(function ($c) { markAdmins.addLabels($c); }); // bind },

destroy: function { markAdmins.nodes.forEach(function (n) {			$(n).remove;		}); },	nodes: [], // for reInit

reInit: function { markAdmins.fullPageProcessed = 0; markAdmins.destroy; markAdmins.mergeConfig; },

addLabels: function ($content) { // Right, the configuration evaluation is here // It might be possible to use Ajax for page // navigation in future. var cfg = this.config, noSubpages = !cfg.markSubpages || !!({ Prefixindex: 1, Allpages: 1 })[mw.config.get('wgCanonicalSpecialPageName')], isUserpage = [2, 3].indexOf(this.ns) !== -1, reUserpage = /^\/wiki\/User([ _]talk)?:(.+)/, enabledGroups = {}, marker = {}, previousUser, anchors, node = document.createElement('b');

node.className = 'adminMark';

// Filter enabled groups (Do it here and not later on each anchor) Object.keys(cfg.groups).forEach(function (g, grpCfg) {			grpCfg = cfg.groups[g];			if (grpCfg.enabled) enabledGroups[g] = grpCfg;		});

if (!this.fullPageProcessed) $content = mw.util.$content || $content; if (!$content[0]) return;

anchors = $content[0].getElementsByTagName('a'); // Add also the userpage link if (isUserpage && !this.fullPageProcessed &&			((isUserpage = document.getElementById('ca-nstab-user')) && (isUserpage = isUserpage.getElementsByTagName('a')))) { anchors = Array.from(anchors); anchors.push(isUserpage[0]); }		this.fullPageProcessed = true;

if (cfg.dontMarkMyself) marker[cfg.dontMarkMyself] = '';

for (var i = 0, len = anchors.length; i < len; ++i) { var a = anchors[i], m = a.getAttribute('href');

if (!m) continue; // Extract user page ( /wiki/User_talk:Foo/subpage -> Foo/subpage ) m = m.match(reUserpage); if (!m || !m[2]) continue; // Extract user var userM = m[2], user = userM.replace(/[/#].*/, ''), isMainUserpageLink = user === userM; user = decodeURIComponent(user); // Two consecutive links to the same user? Don't mark followups! previousUser = previousUser === user && !!m[1]; // isUsertalkLink if (previousUser) continue; // only once

userM = marker[user]; if (userM === undefined) { userM = ''; // User groups of selected user, polish user name m = this.users[user.replace(/_/g, ' ')]; if (!m) continue;

for (var g = 0; g < m.length; g++) { var grpCfg = enabledGroups[m[g]]; if (!grpCfg) continue; // String concatenation is oftentimes faster in modern browsers, // so using Arrays and joining them finally seems advantage. // But we would need an additional IF, so there is no gain. if (userM) userM += '/'; userM += grpCfg.label; }				marker[user] = userM ? [userM] : userM; }			// Are there markers at all? if (!userM) continue; // Don't mark certain pages, except link to user main page. // Does the link go to the main user page or, if linking subpages is enabled, // is it not a page that is just listing subpages? if (!isMainUserpageLink && noSubpages) continue; // Check finished, now append node marker[user][1] = this.markUser(marker[user], a, node); // Required for consecutive user link check previousUser = user; } // end loop },

markUser: function (mark, a, node) { if (mark[1]) { node = mark[1].cloneNode(1); } else { node = node.cloneNode(1); node.appendChild(document.createTextNode('\u00A0(' + mark[0] + ')')); }		a.appendChild(node); this.nodes.push(node); // for reInit return node; }

};

mw.hook('userjs.script-loaded.markadmins').add(markAdmins.init);

}(mediaWiki, jQuery));

title: MediaWiki:Gadget-markAdmins.js


title: User:Minorax/common.js


text:
User:Gifnk dlm 2020/sandbox

=1=

=2= =3= Mathematics of the Jewish Calendar/Printable version Practice Problems in LOLGraphics/Printable version

title: User:Gifnk dlm 2020/sandbox


text: Template:InactiveOnlineGame

title: Template:InactiveOnlineGame


text:
Template:InactiveOnlineGame/doc

Usage
This template should be added to the main pages of Wikibooks that discuss inactive online and/or multiplayer games to tell that it may not be able to play the game in question anymore but that it is still preserved as a historic reference.

title: Template:InactiveOnlineGame/doc


text:
Calculus/Change of variables

The Jacobian matrix and the change of variables are proven to be extremely useful in multivariable calculus when we want to change our variables. They are extremely useful because if we want to integrate a function such as $$\iint_Re^{\frac{x+y}{x-y}}\,dA$$, where $$R$$ is the trapezoidal region with vertices $$(1,0),(2,0),(0,-2),(0,-1)$$, it would be helpful if we can substitute $$x+y$$ as $$u$$ and $$x-y$$ as $$v$$ because $$e^{\frac{u}{v}}$$ is easier to be integrated. However, we need to be familiar with integration, transformation, and the Jacobian, which the latter two will be discussed in this chapter.

Transformation
Let us start with an introduction to the process of variable transformation. Assume that we have a function $$f(x,y)$$. We want to calculate the expression:"$\iint_R f(x,y)\,dA$" in which $$R$$ is a region in $$xy$$-plane. (Another notation for $$dA$$ is $$dxdy$$. ($$dA$$ here is not differential.) ) However, the area of $$R$$ is too complicated to be written out in terms of $$x,y$$. So, we want to change the variables so that the area of $$R$$ can be more easily expressed. Furthermore, the function itself is too hard to be integrated. It would be much easier if the variables can be changed to more convenient ones, Assume there are two more variables $$u,v$$ that have connections with variables $$x,y$$ that satisfy:"$x=x(u,v)\quad\text{and}\quad y=y(u,v).$" The original integral can be rewritten into:"\frac{\partial(x,y)}{\partial(u,v)}\right" in which $$S$$ is another region in $$uv$$-plane transformed from the region $$R$$ in $$xy$$-plane. The purpose of this section is to have us understand the process of this transformation, excluding the $$\left| \frac{\partial(x,y)}{\partial(u,v)}\right|$$ part. We will discuss the purpose and meaning of $$\left| \frac{\partial(x,y)}{\partial(u,v)}\right|$$ in the next section.

Introduction
In fact, we have already encountered two examples of variable transformation in $$\mathbb R^3$$. The first example is using polar coordinates in integration while the second one is using spherical coordinates in integration. Using polar coordinates in integration is a change in variable because we effectively change the variables $$x,y,z$$ into $$r,\theta, z$$ with relations:"$x=r\cos\theta\quad y=r\sin\theta\quad z=z$"As a result, the function being integrated $$f(x,y)$$ is transformed into $$f(x(r,\theta,z),y(r,\theta,z),z(r,\theta,z))$$, thus giving us:"$\iiint_R f(x,y,z)dV=\iiint_S f(r\cos\theta,r\sin\theta,z)dz\ r\ dr\ d\theta$, which is the formula for polar coordinates integration. (It will be proved later)"

The second example, integration in spherical coordinates, offers a similar explanation. The original variables $$x,y,z$$ and the transformed variables $$\rho,\theta,\phi $$ have the relations:"$\begin{cases} x=\rho\sin\phi\cos\theta \\ y=\rho\sin\phi\sin\theta \\ z=\rho\cos\phi \end{cases}$"These relations can give us that"$\iiint_R f(x,y,z)dV=\iiint_S f(\rho\sin\phi\cos\theta,\rho\sin\phi\sin\theta,\rho\cos\phi)\rho^2\sin\phi \ d\rho\ d\theta\ d\phi$, which is the formula for spherical coordinates integration. (It will be proved later)"

Generalization
We understand the transformation from Cartesian coordinates to both polar and spherical coordinates. However, those two are specific examples of variable transformation. We should expand our scope into all kinds of transformation. Instead of specific changes, such as $$x=r\cos\theta,y=r\sin\theta\text{ and }z=z$$, we will talk about general changes. Let's start from two variables.

We consider a change of variables that is given by a transformation $$T$$ from the $$uv$$-plane to the $$xy$$-plane. In other words,"$T(u,v)=(x,y)$, where $(x,y)$ is the original or old variables and $(u,v)$ is the new ones."In this transformation, $$x,y$$ are related to $$u,v$$ by the equations"$x=g(u,v)\quad y=h(u,v)$"We usually just assume that $$T$$ is a $$C^1$$ transformation, which means that $$g,h$$ have continuous 1st-order partial derivatives. Now, time for some terminologies.

 * If $$T(u_1,v_1)=(x_1,y_1)$$, the point $$(x_1,y_1)$$ is called the image of the point $$(u_1,v_1)$$.
 * If no two points have the same image, like functions, $$T$$, the transformation, is called one-to-one (or injective).
 * $$T$$ transforms region $$S$$ into region $$R$$. $$R$$ is called the image of $$S$$. The transformation can be described as:

$$T(S)=R$$

$$T^{-1}(R)=S$$
 * If $$T$$ is one-to-one, then, like functions, it has an inverse transformation $$T^{-1}$$ from the $$xy$$-plane to the $$uv$$-plane, with relation

Regions
Recall that we have established the transformation $$T(S)=R$$, where $$S$$ is the region in the $$uv$$-plane while $$R$$ is the region in the $$xy$$-plane. If we are given the region $$S$$ and transformation $$T$$, we are expected to calculate the region $$R$$. For example, a transformation is defined by the equations"$x=u^2-v^2\quad y=2uv$"Find the image of $$S$$, which is defined as $$S=\{ (u,v):0 \le u \le 1,\ 0 \le v \le 1 \} $$. In this case, we need to know the boundaries of the region $$S$$, which is confined by the lines:

$$u=0\quad u=1\quad v=0\quad v=1$$

If we can redefine the boundaries using $$x,y$$ instead of $$u,v$$, we effectively will find the image of $$S$$.

$$\begin{align} \text{When }\quad & u=0 \quad (0\le v\le 1) \\ & \begin{cases} x=-v^2 \\ y=0 \\ \end{cases} \quad \text{(substitution)}\\ \text{Thus, }\quad & y=0 \quad (-1\le x\le 0) \\ \end{align} $$$$\begin{align} \text{When }\quad & u=1 \quad (0\le v\le 1) \\ & \begin{cases} x=1-v^2 \\ y=2v \\ \end{cases} \\ \text{Thus, }\quad & x=1-\frac{y^2}{4} \quad (0\le x\le 1) \\ \end{align} $$$$\begin{align} \text{When }\quad & v=0 \quad (0\le 0\le 1) \\ & \begin{cases} x=u^2 \\ y=0 \\ \end{cases} \\ \text{Thus, }\quad & y=0 \quad (0\le x\le 1) \\ \end{align} $$$$\begin{align} \text{When }\quad & v=1 \quad (0\le u\le 1) \\ & \begin{cases} x=u^2-1 \\ y=2u \\ \end{cases} \\ \text{Thus, }\quad & x=\frac{y^2}{4}-1 \quad (-1\le x\le 0) \\ \end{align} $$

As a result, the image of $$S $$ is $$R=\{(x,y):0\le y\le 2,\ \frac{y^2}{4}-1\le x\le 1-\frac{y^2}{4}\} $$ We can use the same method to calculate $$S $$ from $$R $$.

The Jacobian
The Jacobian matrix is one of the most important concept in this chapter. It "compromises" the change in area when we change the variables so that after changing the variables, the result of the integral does not change. Recall that at the very beginning of the last section, we reserved the explanation of $$\left| \frac{\partial(x,y)}{\partial(u,v)}\right|$$ from $$\iint_S f(x(u,v),y(u,v))\left| \frac{\partial(x,y)}{\partial(u,v)}\right| \ du\ dv$$ here. To actually start explaining that, we should review some basic concepts.

Review "u-substitution"
Recall that when we are discussing $$u $$-substitution (a simple way to describe "integration by substitution for single-variable functions"), we use the following method to solve integrals."$\int_a^bf(x)\ dx=\int_c^df(x(u))\ \frac{dx}{du}\ du \quad\text{where }c=x(a),d=x(b) $"For example,"$\int \frac{\sin(\ln (x))}{x}\ dx $" $$\begin{align} \text{Let } \quad & u=\ln(x) \\ \text{Thus, } \quad & \frac{du}{dx}=\frac{1}{x} \\ \Rightarrow\ & du=\frac{1}{x}dx \\ \end{align} $$ $$\begin{align} \int \frac{\sin(\ln (x))}{x}\ dx & = \int\sin(\ln (x))\ \Big(\frac{1}{x}\ dx\Big) & \quad\text{rearrangement}\\ & = \int\sin(u)\ du & \quad \text{let }u=\ln(x)\implies du=\frac{1}{x}\ dx \\ & = -\cos(u)+C & \quad\text{integration} \\ & = -\cos(\ln(x))+C & \quad\text{resubstitution} \\ \end{align} $$ If we add endpoints into the integral, the result will be: $$\begin{align} \int_e^{e^2} \frac{\sin(\ln (x))}{x}\ dx & = \int_e^{e^2}\sin(\ln (x))\ \Big(\frac{1}{x}\ dx\Big) & \quad\text{rearrangement}\\ & = \int_1^2\sin(u)\ du & \quad \text{remember }u=\ln(x)\text{ and }du=\frac{1}{x}\ dx \\ & = \Big[-\cos(u)\Big]_1^2 & \quad\text{integration} \\ & = \cos(1)-\cos(2) \\ \end{align} $$ If we look carefully at the "rearrangement" and "remember" part in the solution, we find that we effectively changed our variable from $$x $$ to $$u $$ through this method:"$\int_a^bf(x)dx=\int_{x=a}^{x=b}f(x(u))\ d(x(u))=\int_{u=x(a)}^{u=x(b)}f(x(u))\ \frac{dx}{du}\ du $, which is what we have mentioned above." The appearance of the term $$\frac{dx}{du} $$ not only is a mathematical product of deduction, but also serves a intuitive purpose. When we change our function from $$f(x) $$ to $$f(x(u)) $$, we also change the region we are integrating, which can be seen by looking at the endpoints. This change of region is either "stretched" or "condensed" by a factor of $$\frac{du}{dx} $$. To counter this change, $$\frac{dx}{du} $$ is deduced to compromise (recall that $$\frac{dx}{du}\left(\frac{du}{dx}\right)=1$$). We can simply think this term as a compromise factor that counters the change of region due to a change of variables.

Now, let us put our focus back to two variables. If we change our variables from $$x,y$$ to $$u,v$$, we also change the region we are integrating, as demonstrated in the previous section. So, continuing our flow of thought, there should also be a term deduced to counter the change of region. In other words: $$\iint_Rf(x,y)\,dA_1=\iint_Sf(x(u,v),y(u,v)) \underbrace{\frac{dA_1}{dA_2}}_{\text{informal term}}\,dA_2 $$

Note that the symbols used here are for intuitive purpose and not for official use. Official terms will be introduced later in the chapter, but for now, we use these terms for better understanding. In this case, when we change the function from $$f(x,y) $$ to $$f(x(u),y(u)) $$, we "stretched" or "condensed" the area of our region, by a factor of $$\frac{dA_2}{dA_1} $$; therefore, we need to counter the change with a factor of $$\frac{dA_1}{dA_2} $$. The Jacobian matrix for two variables is basically an expression for calculating $$\frac{dA_1}{dA_2} $$ in terms of $$u,v$$, so that we are able to integrate the new integral after transformation, since the function involved in the new integral can only in terms of $$u,v$$, but not $$x,y$$ (we need to express $$x$$ and $$y$$ in terms of $$u,v$$).

Double integrals
Now, it is time for us to deduce the Jacobian matrix. In the review above, we already established informally that the Jacobian matrix for two variables is basically $$\frac{dA_1}{dA_2} $$, with $$dA_1$$ being the infinitesimally small area in the region $$R$$ in the $$xy$$-plane and $$dA_2$$ being the infinitesimally small area in the region $$S$$ in the $$uv$$-plane. Since we are changing our variables from $$x,y$$ to $$u,v$$, we should describe $$dA_1$$ and $$dA_2$$ in terms of $$u,v$$ over a region in $$uv$$-plane.

Let us start with $$dA_2$$ first because it is easier to calculate. We start with a small rectangle $$S_0$$, which is a part of $$S$$, in the $$uv$$-plane whose lower left corner is the point $$(u_0,v_0)$$ and whose dimensions are $$\Delta u,\Delta v$$. Thus, the area of $$S_0$$ is"$\Delta A_2=\Delta u\Delta v$"The image of $$S_0$$, in this case let's name it $$R_0$$, is in the $$xy$$-plane according to the transformation $$T(S_0)=R_0$$. One of its boundary points is $$(x_0,y_0)=T(u_0,v_0) $$. We can use a vector $$\mathbf{r}$$ to describe the position vector of $$R_0$$ of the point $$(u,v)$$. In other words, $$\mathbf{r}$$ can describe the region $$R_0$$ given that"$\mathbf{r}(u,v)=x(u,v)\mathbf{i}+y(u,v)\mathbf{j}\quad\text{where }u_0\le u\le u_0+\Delta u,\ v_0\le v\le v_0+\Delta v$"The region $$R_0$$ now can be described in terms of $$u,v$$. The next step is to utilize the position vector $$\mathbf{r}(u,v)$$ to calculate its area $$dA_1$$.

The shape of the region $$R_0$$ after transformation $$R_0=T(S_0)$$ can be approximated, which is a parallelogram. As we learnt in algebra, the area of a parallelogram is defined to be the product of its base and height. However, this definition cannot help us with our calculations. Instead, we will use the cross product to determine its area. Recall that the area of a parallelogram formed by vectors $$\mathbf{a}$$ and $$\mathbf{b}$$ can be calculated by taking the magnitude of the cross product of the two vectors."\mathbf{a}\times \mathbf{b}"In this parallelogram, the two vectors $$\mathbf{a}$$ and $$\mathbf{b}$$ are, in terms of $$u,v$$: $$\mathbf{a}=\mathbf{r}(u_0+\Delta u,v_0)-\mathbf{r}(u_0,v_0) \quad \text{ and }\quad \mathbf{b}=\mathbf{r}(u_0,v_0+\Delta v)-\mathbf{r}(u_0,v_0)$$ It seems very similar to the definition of partial derivatives:

$$\mathbf{r}_u=\lim_{\Delta u\rightarrow 0}\frac{\mathbf{r}(u_0+\Delta u,v_0)-\mathbf{r}(u_0,v_0)}{\Delta u}\quad\text{ and }\quad \mathbf{r}_v=\lim_{\Delta v\rightarrow 0}\frac{\mathbf{r}(u_0,v_0+\Delta v)-\mathbf{r}(u_0,v_0)}{\Delta v}$$ As a result, we can approximate that:

$$\begin{align} & \mathbf{a}=\mathbf{r}(u_0+\Delta u,v_0)-\mathbf{r}(u_0,v_0)\approx \Delta u \mathbf{r}_u \quad \text{ and }\quad \\ & \mathbf{b}=\mathbf{r}(u_0,v_0+\Delta v)-\mathbf{r}(u_0,v_0)\approx \Delta v \mathbf{r}_v \\ \end{align}$$ Now, we calculate $$\mathbf{r}_u,\mathbf{r}_v$$, given that $$\mathbf{r}(u,v)=x(u,v)\mathbf{i}+y(u,v)\mathbf{j}$$:

$$\begin{align} & \mathbf{r}_u=x_u(u,v)\ \mathbf{i}+y_u(u,v)\ \mathbf{j}=\frac{\partial x}{\partial u}\ \mathbf{i}+\frac{\partial y}{\partial u}\ \mathbf{j} \quad\text{ and } \\ & \mathbf{r}_v=x_v(u,v)\ \mathbf{i}+y_v(u,v)\ \mathbf{j}=\frac{\partial x}{\partial v}\ \mathbf{i}+\frac{\partial y}{\partial v}\ \mathbf{j} \\ \end{align}$$ We can calculate $$\Delta A_1=||\mathbf{a}\times\mathbf{b}||$$ (we take absolute value to prevent negative area). You can review the cross product in Chapter. Note that the inner bar of || is for calculating the magnitude (or norm) while the outer bar of || is for taking the absolute value. $$\begin{align} \Delta A_1 & = ||\mathbf{a}\times\mathbf{b}|| \\ & = ||(\Delta u\ \mathbf{r}_u)\times(\Delta v\ \mathbf{r}_v)|| & \quad \text{approximation} \\ & = ||\mathbf{r}_u\times\mathbf{r}_v||\Delta u\Delta v \\ & = \begin{vmatrix} \begin{vmatrix} \det\begin{pmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ \frac{\partial x}{\partial u} & \frac{\partial y}{\partial u} & 0 \\ \frac{\partial x}{\partial v} & \frac{\partial y}{\partial v} & 0 \\ \end{pmatrix} \\ \end{vmatrix} \end{vmatrix}\Delta u\Delta v & \quad\text{cross product} \\ & = \begin{vmatrix} \begin{vmatrix} \det\begin{pmatrix} \frac{\partial x}{\partial u} & \frac{\partial x}{\partial v} \\ \frac{\partial y}{\partial u} & \frac{\partial y}{\partial v} \\ \end{pmatrix} \mathbf{k} \end{vmatrix} \end{vmatrix} \Delta u\Delta v & \quad\text{evaluation} \\ & = \begin{vmatrix} \det\begin{pmatrix} \frac{\partial x}{\partial u} & \frac{\partial x}{\partial v} \\ \frac{\partial y}{\partial u} & \frac{\partial y}{\partial v} \\ \end{pmatrix}\underbrace{|\mathbf k|}_{1} \end{vmatrix} \Delta u\Delta v \\ \end{align}$$ Then, we can substitute our newly deduced terms. $$\frac{\Delta A_1}{\Delta A_2}=\frac{\begin{vmatrix} \det\begin{pmatrix} \frac{\partial x}{\partial u} & \frac{\partial x}{\partial v} \\ \frac{\partial y}{\partial u} & \frac{\partial y}{\partial v} \\ \end{pmatrix} \end{vmatrix} \Delta u\Delta v}{\Delta u\Delta v}=\begin{vmatrix} \det\begin{pmatrix} \frac{\partial x}{\partial u} & \frac{\partial x}{\partial v} \\ \frac{\partial y}{\partial u} & \frac{\partial y}{\partial v} \\ \end{pmatrix} \end{vmatrix} = \left| \frac{\partial x}{\partial u}\frac{\partial y}{\partial v}-\frac{\partial x}{\partial v}\frac{\partial y}{\partial u} \right|$$ Finally, we derived the absolute value of Jacobian. The definition of Jacobian is as follows:

We will then use the Jacobian in the change of variables in integrals. The absolute value is added to prevent a negative area. $$\begin{align} \iint_Rf(x,y)\,dA & \approx \sum_{i=1}^m \sum_{j=1}^n f(x_i,y_j)\Delta A \\ & \approx \sum_{i=1}^m \sum_{j=1}^n f(x(u_i,v_j),y(u_i,v_j))\Delta A_2 \\ \text{Since } & \Delta A_2 \approx \left| \frac{\partial (x,y)}{\partial (u,v)} \right|\Delta u\Delta v \\ \sum_{i=1}^m \sum_{j=1}^n f(x(u_i,v_j),y(u_i,v_j))\Delta A_2 & \approx \sum_{i=1}^m \sum_{j=1}^n f(x(u_i,v_j),y(u_i,v_j))\ \left| \frac{\partial (x,y)}{\partial (u,v)} \right|\Delta u\Delta v \\ & \approx\iint_Sf(x(u,v),y(u,v))\ \left| \frac{\partial (x,y)}{\partial (u,v)} \right|\ du\ dv \\ \end{align}$$ Here is the theorem for the change of variables in a double integral and we have explained intuitively why and how it works, but the above explanations are not proof of this theorem. In particular, we make some approximation, while the statement in the following theorem is equality, and not approximation. The actual proof is quite complicated and advanced, and thus not included here.

{ Example. }

{Choose correct expression(s) for the Jacobian $$\frac{\partial (u,v)}{\partial (x,y)}$$ in which $$x=2u$$ and $$y=5v$$.} + $$\frac{1}{10}$$ - $$\begin{vmatrix}0&\frac{1}{2}\\ \frac{1}{5}&0\end{vmatrix}$$ - $$\begin{vmatrix}2&0\\ 0&5\end{vmatrix}$$ - $$\begin{vmatrix}0&2\\ 5&0\end{vmatrix}$$ - $$10$$
 * Since $$x=2u\implies u=\frac{x}{2}$$ and $$y=5v\implies v=\frac{y}{5}$$,
 * $$\frac{\partial (u,v)}{\partial (x,y)}$$=\begin{vmatrix}\frac{1}{2}&0\\0&\frac{1}{5}\end{vmatrix}=\frac{1}{10}.
 * This is the expression for $$\frac{\partial (u,v)}{\partial (y,x)}$$.
 * This is the expression for $$\frac{\partial (x,y)}{\partial (u,v)}$$.
 * This is the expression for $$\frac{\partial (y,x)}{\partial (u,v)}$$.
 * This is the expression for $$\frac{\partial (x,y)}{\partial (u,v)}$$.

{Choose correct expression(s) for the integral $$\iint_D (x+y)(5x+2y)$$ in which $$D$$ is a region bounded by $$y=1-x,x=2-y,x=\frac{-5y}{3}$$ and $$y=\frac{1-5x}{2}$$. - $$3\int_1^2\int_0^1 uv\,du\,dv$$ - $$-3\int_0^1\int_1^2 uv\,du\,dv$$ - $$-\frac{1}{3}\int_0^1\int_1^2 uv\,du\,dv$$ + $$\frac{1}{3}\int_1^2\int_0^1 uv\,du\,dv$$ + $$\frac{1}{3}\int_0^1\int_1^2 uv\,du\,dv$$
 * type="[]"}
 * Let $$v=x+y$$ and $$u=5x+2y$$, and the bounds in a transformed region via this change of variables are $$1\le v\le 2$$ and $$0\le u\le 1$$.
 * Then, the Jacobian is $$\frac{\partial (u,v)}{\partial (x,y)}=5-2=3.$$
 * Thus, $$\iint_D (x+y)(5x+2y)=\frac{1}{3}\iint_D (x+y)(5x+2y)\left|\frac{\partial (u,v)}{\partial (x,y)}\right|=\frac{1}{3}\int_1^2\int_0^1 uv\,du\,dv$$
 * Let $$u=x+y$$ and $$v=5x+2y$$, and the bounds in a transformed region via this change of variables are $$1\le u\le 2$$ and $$0\le v\le 1$$.
 * Then, the Jacobian is $$\frac{\partial (u,v)}{\partial (x,y)}=2-5=-3.$$
 * Thus, $$\iint_D (x+y)(5x+2y)=\frac{1}{|-3|}\iint_D (x+y)(5x+2y)\left|\frac{\partial (u,v)}{\partial (x,y)}\right|=\frac{1}{3}\int_0^1\int_1^2 uv\,du\,dv$$
 * Let $$u=x+y$$ and $$v=5x+2y$$, and the bounds in a transformed region via this change of variables are $$1\le u\le 2$$ and $$0\le v\le 1$$.
 * Then, the Jacobian is $$\frac{\partial (u,v)}{\partial (x,y)}=2-5=-3.$$
 * Thus, $$\iint_D (x+y)(5x+2y)=\frac{1}{|-3|}\iint_D (x+y)(5x+2y)\left|\frac{\partial (u,v)}{\partial (x,y)}\right|=\frac{1}{3}\int_0^1\int_1^2 uv\,du\,dv$$
 * Let $$v=x+y$$ and $$u=5x+2y$$, and the bounds in a transformed region via this change of variables are $$1\le v\le 2$$ and $$0\le u\le 1$$.
 * Then, the Jacobian is $$\frac{\partial (u,v)}{\partial (x,y)}=5-2=3.$$
 * Thus, $$\iint_D (x+y)(5x+2y)=\frac{1}{3}\iint_D (x+y)(5x+2y)\left|\frac{\partial (u,v)}{\partial (x,y)}\right|=\frac{1}{3}\int_1^2\int_0^1 uv\,du\,dv$$
 * Let $$u=x+y$$ and $$v=5x+2y$$, and the bounds in a transformed region via this change of variables are $$1\le u\le 2$$ and $$0\le v\le 1$$.
 * Then, the Jacobian is $$\frac{\partial (u,v)}{\partial (x,y)}=2-5=-3.$$
 * Thus, $$\iint_D (x+y)(5x+2y)=\frac{1}{|-3|}\iint_D (x+y)(5x+2y)\left|\frac{\partial (u,v)}{\partial (x,y)}\right|=\frac{1}{3}\int_0^1\int_1^2 uv\,du\,dv$$

{Choose correct statement(s) from the following statements.} + If both $$u$$ and $$v$$ are independent from both $$x$$ and $$y$$, the Jacobian $$\frac{\partial (u,v)}{\partial (x,y)}=0$$. + If both $$u$$ and $$v$$ are independent from both $$x$$ and $$y$$, the Jacobian $$\frac{\partial (x,y)}{\partial (u,v)}=0$$. + If $$u$$ is independent from both $$x$$ and $$y$$, while $$v$$ is dependent from both $$x$$ and $$y$$, the Jacobian $$\frac{\partial (u,v)}{\partial (x,y)}=0.$$ + If $$u$$ is independent from both $$x$$ and $$y$$, while $$v$$ is dependent from both $$x$$ and $$y$$, the Jacobian $$\frac{\partial (x,y)}{\partial (u,v)}=0.$$ - If $$v=ku$$ in which $$k$$ is a real number, then the Jacobian $$\frac{\partial (u,v)}{\partial (x,y)}=k.$$
 * This condition implies $$\frac{\partial u}{\partial x}=\frac{\partial u}{\partial y}=\frac{\partial v}{\partial x}=\frac{\partial v}{\partial y}=0$$.
 * Thus, the Jacobian equals zero.
 * Since both $$u$$ and $$v$$ are independent from both $$x$$ and $$y$$ implies that both $$x$$ and $$y$$ are independent from both $$u$$ and $$v$$. (They have the same meaning)
 * This condition implies $$\frac{\partial x}{\partial u}=\frac{\partial x}{\partial v}=\frac{\partial y}{\partial u}=\frac{\partial y}{\partial v}=0$$.
 * Thus, the Jacobian equals zero.
 * The condition $$u$$ is independent from both $$x$$ and $$y$$ implies that $$\frac{\partial u}{\partial x}=\frac{\partial u}{\partial y}=0$$.
 * Thus, the first row of the Jacobian a zero row, and it can be seen that the Jacobian equals zero.
 * The condition $$u$$ is independent from both $$x$$ and $$y$$ implies that $$\frac{\partial u}{\partial x}=\frac{\partial u}{\partial y}=0$$.
 * Thus, the second row of the Jacobian a zero row, and it can be seen that the Jacobian equals zero.
 * Since $$\frac{\partial v}{\partial x}=k\cdot\frac{\partial u}{\partial x}$$ and $$\frac{\partial v}{\partial y}=k\cdot\frac{\partial u}{\partial y}$$ in this case,
 * the Jacobian equals $$\frac{\partial u}{\partial x}\cdot k\cdot\frac{\partial u}{\partial y}-\frac{\partial u}{\partial y}\cdot k\cdot\frac{\partial u}{\partial x}=k\left(\frac{\partial u}{\partial x}\frac{\partial u}{\partial y}-\frac{\partial u}{\partial x}\frac{\partial u}{\partial y}\right)=0$$
 * So, this statement is incorrect unless $$k=0$$, i.e., it is not always correct and thus not a correct statement.

Triple integrals
If we continue our flow of thoughts, we can also find the Jacobian for three variables. Suppose there is a function $$f(x,y,z)$$. $$x,y,z$$ has relations with $$u,v,w$$, which are"$x=x(u,v,w),\quad y=y(u,v,w),\quad \text{and}\quad z=z(u,v,w)$"$$R$$ is a region in the $$xyz$$-space, and $$S$$ is a region in the $$uvw$$-space, with transformation $$T(S)=R$$.

To calculate the Jacobian for three variables, we go through a similar process. The process of transformation will be: a rectangular prism with dimensions $$\Delta u,\Delta v,\Delta w$$ in the $$uvw$$-space to a parallelepiped in the $$xyz$$-space and a volume of $$\Delta V_2=\Delta u\Delta v\Delta w$$. The parallelepiped can be described with the position vector:"$\mathbf{r}(u,v,w)=x(u,v,w)\ \mathbf{i}+y(u,v,w)\ \mathbf{j}+z(u,v,w)\ \mathbf{k}$"The three sides of the parallelepiped can be described by the position vector as: $$\begin{align} & \mathbf{a}=\mathbf{r}(u+\Delta u,v,w)-\mathbf{r}(u,v,w), \\ & \mathbf{b}=\mathbf{r}(u,v+\Delta v,w)-\mathbf{r}(u,v,w),\quad\text{and} \\ & \mathbf{c}=\mathbf{r}(u,v,w+\Delta w)-\mathbf{r}(u,v,w). \\ \end{align}$$ Since the derivatives of $$\mathbf{r}$$ are defined as: $$\begin{align} & \mathbf{r}_u = \lim_{\Delta u\rightarrow 0}\frac{\mathbf{r}(u+\Delta u,v,w)-\mathbf{r}(u,v,w)}{\Delta u}, \\ & \mathbf{r}_v = \lim_{\Delta v\rightarrow 0}\frac{\mathbf{r}(u,v+\Delta v,w)-\mathbf{r}(u,v,w)}{\Delta v},\quad\text{and} \\ & \mathbf{r}_w = \lim_{\Delta w\rightarrow 0}\frac{\mathbf{r}(u,v,w+\Delta w)-\mathbf{r}(u,v,w)}{\Delta w}. \\ \end{align}$$ The three vectors $$\mathbf{a},\mathbf{b},\mathbf{c}$$ can be similarly approximated into: "$\mathbf{a}\approx\Delta u\ \mathbf{r}_u,\quad\mathbf{b}\approx\Delta v\ \mathbf{r}_v,\quad\text{and}\quad\mathbf{c}\approx\Delta w\ \mathbf{r}_w$"Since the position vector $$\mathbf{r}$$ is $$\mathbf{r}(u,v,w)=x(u,v,w)\ \mathbf{i}+y(u,v,w)\ \mathbf{j}+z(u,v,w)\ \mathbf{k}$$, the partial derivatives for $$\mathbf{r}$$ are: $$\begin{align} & \mathbf{r}_u=\frac{\partial x}{\partial u}\ \mathbf{i}+\frac{\partial y}{\partial u}\ \mathbf{j}+\frac{\partial z}{\partial u}\ \mathbf{k}, \\ & \mathbf{r}_v=\frac{\partial x}{\partial v}\ \mathbf{i}+\frac{\partial y}{\partial v}\ \mathbf{j}+\frac{\partial z}{\partial v}\ \mathbf{k},\quad\text{and} \\ & \mathbf{r}_v=\frac{\partial x}{\partial w}\ \mathbf{i}+\frac{\partial y}{\partial w}\ \mathbf{j}+\frac{\partial z}{\partial w}\ \mathbf{k} \\ \end{align}$$ Recall that the volume of a parallelepiped determined by the vectors $$\mathbf{a},\mathbf{b},\mathbf{c}$$ is the magnitude of their scalar triple product:"(\mathbf{a}\times\mathbf{b})\ \cdot\ \mathbf{c}"We just need to substitute the vectors with what we have yielded. $$\begin{align} \Delta V_1=|(\mathbf{a}\times\mathbf{b})\ \cdot\ \mathbf{c}| & = |(\Delta u\ \mathbf{r}_u)\times(\Delta v\ \mathbf{r}_v)\ \cdot\ \Delta w\ \mathbf{r}_w| \\ & = |\mathbf{r}_u\times\mathbf{r}_v\ \cdot\ \mathbf{r}_w|\Delta u\Delta v\Delta w \\ & = \begin{vmatrix} \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ \frac{\partial x}{\partial u} & \frac{\partial y}{\partial u} & \frac{\partial z}{\partial u} \\ \frac{\partial x}{\partial v} & \frac{\partial y}{\partial v} & \frac{\partial z}{\partial v} \\ \end{vmatrix} \ \cdot\ \begin{pmatrix} \frac{\partial x}{\partial w} \\ \frac{\partial y}{\partial w} \\ \frac{\partial z}{\partial w} \\ \end{pmatrix}\end{vmatrix} \Delta u\Delta v\Delta w & \quad\text{cross product}\\ & = \begin{vmatrix} \begin{pmatrix} \frac{\partial y}{\partial u}\frac{\partial z}{\partial v}-\frac{\partial z}{\partial u}\frac{\partial y}{\partial v} \\ \frac{\partial z}{\partial u}\frac{\partial x}{\partial v}-\frac{\partial x}{\partial u}\frac{\partial z}{\partial v} \\ \frac{\partial x}{\partial u}\frac{\partial y}{\partial v}-\frac{\partial y}{\partial u}\frac{\partial x}{\partial v} \\ \end{pmatrix} \ \cdot\ \begin{pmatrix} \frac{\partial x}{\partial w} \\ \frac{\partial y}{\partial w} \\ \frac{\partial z}{\partial w} \\ \end{pmatrix} \end{vmatrix}\Delta u\Delta v\Delta w \\ & = \left| \frac{\partial x}{\partial w}\frac{\partial y}{\partial u}\frac{\partial z}{\partial v}-\frac{\partial x}{\partial w}\frac{\partial y}{\partial v}\frac{\partial z}{\partial u}+ \frac{\partial x}{\partial v}\frac{\partial y}{\partial w}\frac{\partial z}{\partial u}-\frac{\partial x}{\partial u}\frac{\partial y}{\partial w}\frac{\partial z}{\partial v}+ \frac{\partial x}{\partial u}\frac{\partial y}{\partial v}\frac{\partial z}{\partial w}-\frac{\partial x}{\partial v}\frac{\partial y}{\partial u}\frac{\partial z}{\partial w} \right| \Delta u\Delta v\Delta w & \quad\text{dot product}\\ & = \left| \frac{\partial x}{\partial u}\bigg( \frac{\partial y}{\partial v}\frac{\partial z}{\partial w}-\frac{\partial y}{\partial w}\frac{\partial z}{\partial v} \bigg) + \frac{\partial x}{\partial v}\bigg( \frac{\partial y}{\partial w}\frac{\partial z}{\partial u}-\frac{\partial y}{\partial u}\frac{\partial z}{\partial w} \bigg) + \frac{\partial x}{\partial w}\bigg( \frac{\partial y}{\partial u}\frac{\partial z}{\partial v}-\frac{\partial y}{\partial v}\frac{\partial z}{\partial u} \bigg) \right| \Delta u\Delta v\Delta w & \quad\text{rearrangement}\\ & = \begin{vmatrix} \begin{vmatrix} \frac{\partial x}{\partial u} & \frac{\partial x}{\partial v} & \frac{\partial x}{\partial w} \\ \frac{\partial y}{\partial u} & \frac{\partial y}{\partial v} & \frac{\partial y}{\partial w} \\ \frac{\partial z}{\partial u} & \frac{\partial z}{\partial v} & \frac{\partial z}{\partial w} \\ \end{vmatrix} \end{vmatrix} \Delta u\Delta v\Delta w & \quad\text{cross product}\\ \end{align}$$ Thus, $$\frac{\Delta V_1}{\Delta V_2}=\frac{\begin{vmatrix} \begin{vmatrix} \frac{\partial x}{\partial u} & \frac{\partial x}{\partial v} & \frac{\partial x}{\partial w} \\ \frac{\partial y}{\partial u} & \frac{\partial y}{\partial v} & \frac{\partial y}{\partial w} \\ \frac{\partial z}{\partial u} & \frac{\partial z}{\partial v} & \frac{\partial z}{\partial w} \\ \end{vmatrix} \end{vmatrix} \Delta u\Delta v\Delta w}{\Delta u\Delta v\Delta w}= \begin{vmatrix} \begin{vmatrix} \frac{\partial x}{\partial u} & \frac{\partial x}{\partial v} & \frac{\partial x}{\partial w} \\ \frac{\partial y}{\partial u} & \frac{\partial y}{\partial v} & \frac{\partial y}{\partial w} \\ \frac{\partial z}{\partial u} & \frac{\partial z}{\partial v} & \frac{\partial z}{\partial w} \\ \end{vmatrix} \end{vmatrix}$$.

The absolute value is added to prevent a negative volume. $$\begin{align} \iiint_Rf(x,y,z)dV & \approx \sum_{i=1}^m \sum_{j=1}^n \sum_{k=1}^p f(x_i,y_j,z_k)\Delta V \\ & \approx \sum_{i=1}^m \sum_{j=1}^n \sum_{k=1}^p f(x(u_i,v_j,w_k),y(u_i,v_j,w_k),z(u_i,v_j,w_k))\Delta V_2 \\ \text{Since } & \Delta V_2 \approx \left| \frac{\partial (x,y,z)}{\partial (u,v,w)} \right|\Delta u\Delta v\Delta w \\ \sum_{i=1}^m \sum_{j=1}^n \sum_{k=1}^p f(x(u_i,v_j,w_k),y(u_i,v_j,w_k),z(u_i,v_j,w_k))\Delta V_2 & \approx \sum_{i=1}^m \sum_{j=1}^n \sum_{k=1}^p f(x(u_i,v_j,w_k),y(u_i,v_j,w_k),z(u_i,v_j,w_k))\ \left| \frac{\partial (x,y,z)}{\partial (u,v,w)} \right|\Delta u\Delta v\Delta w \\ & \approx\iiint_S f(x(u,v,w),y(u,v,w),z(u,v,w))\ \left| \frac{\partial (x,y,z)}{\partial (u,v,w)} \right|\ du\ dv\ dw \\ \end{align}$$ Then, we have the following theorem which is analogous to the theorem for double integrals. Again, we should aware that the above explanations are not proof of this theorem.

{ Example. }

{Choose correct statement(s) from the following statements. - $$\frac{\partial (x,y,u)}{\partial (x,y,v)}=\frac{\partial u}{\partial v}.$$ - $$\iiint_D xyz\left|\frac{\partial (u,v,w)}{\partial (x,y,z)}\right|\,dx\,dy\,dz$$ gives the 4-dimensional volume under the graph of $$f(x,y,z)=xyz$$ over the region $$D$$ in $$xyz$$-space. + $$\iiint_D (uvw)\,du\,dv\,dw$$ gives the 4-dimensional volume under the graph of $$f(u,v,w)=uvw$$ over the region $$D$$ in $$uvw$$-space. + $$\iiint_D \left|\frac{\partial (x,y,z)}{\partial (u,v,w)}\right|\,du\,dv\,dw$$ gives the volume of the region $$D'$$ in $$xyz$$-space, that is mapped from the region $$D$$ in $$uvw$$-space by a transformation satisfying the conditions mentioned in the theorem about change of variables in a triple integral. Now, we understand the purpose and the derivation of the Jacobian. It is time to apply this new knowledge to some examples. The first two examples consist of the change of coordinates from the Cartesian coordinate system into the polar coordinate system and the change of Cartesian to spherical coordinates.
 * type="[]"}
 * $$\frac{\partial (x,y,u)}{\partial (x,y,v)}=\begin{vmatrix}1&\frac{\partial x}{\partial y}&\frac{\partial x}{\partial v}\\\frac{\partial y}{\partial x}&1&\frac{\partial y}{\partial v}\\\frac{\partial u}{\partial x}&\frac{\partial u}{\partial y}&\frac{\partial u}{\partial v}\end{vmatrix}$$
 * This is not equal to $$\frac{\partial u}{\partial v}$$ in general.
 * The absolute value of the Jacobian is an extra term. Without it, this is true.
 * By the proposition about 4-dimensional volume given by triple integration, this is true.
 * This integral equals $$\iiint_{D'} 1\,dx\,dy\,dz$$ which is the volume of the region $$D'$$ in $$xyz$$ space,
 * since the function involved in the above integral is $$f(x,y,z)\equiv 1$$ (constant function), the value of function still equals one after changing variables $$x,y$$ and $$z$$ to $$u,v$$ and $$w$$ respectively by setting $$x=x(u,v,w),y=y(u,v,w)$$ and $$z=z(u,v,w)$$,
 * and we need to insert an extra term to the integral, namely absolute value of the Jacobian.

Change of coordinate system
Sometimes, we may change the region we are integrating over to another region in other coordinate system. This can simplify the computation of integrals, especially when the region in Cartesian coordinate system is related to circle, e.g. sphere, cone, circle, etc.

Let us start with the change of coordinates from the Cartesian coordinate system into the polar coordinate system.

title: Calculus/Change of variables


text:
Wikibooks:Reading room/Archives/2021/January

Quickbooks Online Book?
Can I create a book on how to use Quickbooks Online-a cloud-based accounting software? Would this topic be alllowed? Thank you. Starlord1990 (discuss • contribs) 12:39, 24 December 2020 (UTC)
 * I think it's fine. Leaderboard (discuss • contribs) 08:51, 2 January 2021 (UTC)
 * You will need to be cautious to avoid copyright material. For example, you won't be able to use screenshots of the software. QuiteUnusual (discuss • contribs) 13:48, 5 January 2021 (UTC)

Centering in a cell in a table
Hi, tables can be used to create an illustration as in this section. Observe that in the middle row, first column, "Oberon MediaWiki.Markup command ... downarrow" is left justified in the cell. In my trial, "text-align: center" failed to center the content in the cell. Is this a limitation of MediaWiki translation of markup to HTML? Can anyone suggest how to center in this instance? Thanks ... PeterEasthope (discuss • contribs) 17:57, 16 January 2021 (UTC)
 * I've tried something; not sure if it's the effect you wanted. --Pi zero (discuss • contribs) 18:50, 16 January 2021 (UTC)
 * Good! I should have thought of "align" rather than "text-align". Thanks! ... PeterEasthope (discuss • contribs) 19:06, 16 January 2021 (UTC)

Hide title on main page
Hi! I’m working at Norwegian WB trying to revive the project. I have used a copy of your layout with some adjustments on the main page. Is the extention that enables the magic word NOTITLE enabled by default? I haven’t been able to make it go away. What can I do to hide the page title? — Abuluntu  (talk) 10:42, 22 January 2021 (UTC)

 * I think this is done through CSS. At MediaWiki:Common.css it has the following: . I think this switches off the title. You need to do the equivalent for yours. -- Jules  (Mrjulesd) 17:49, 22 January 2021 (UTC)
 * Great! Thanks! I haven’t been granted permissions yet by meta, but when they do I’ll try it. — Abuluntu  (talk) 07:05, 23 January 2021 (UTC)

Renaming Book Pages
How do I rename a Book Page? When I change the name of the Book Page, it seems that the link to the old page is broken and I have copy the old content to the new page that was created as a result of renaming. TIA!

--Rjbfigueroa (discuss • contribs)

 * Can you give an example of a page you wanted to move, and where you wanted to move it to, so we can see what happened? Fwiw, here's how to move a page, if you're using Vector skin (in the desktop interface, rather than the mobile one): View the page you want to move (not its talk page). There should be a control bar at the top of the browser, and way over on the far right of it there should be a dropdown menu.  The only option on that menu should be either "rename" or "move"; use that.  — Preceding unsigned comment added by Pi zero (talk • contribs) 17:26, 19 February 2018 (UTC)

I would guess the OP is changing the name of the page in the link expecting it to change the actual name of the linked page. The result is a red link (of course). Moving the page, as described above, is the way to resolve it. QuiteUnusual (discuss • contribs) 18:01, 29 January 2021 (UTC)

'Reading Room' is not a good label for this area. 'Forum' is better
I realize that the label 'Reading room' plays on the metaphor of library and books, but it does not accurately describe this part of the Wiki.

I would label this area as a 'Forum' -- a public place for discussion and (in some cases) deliberation.

This is just my opinion and suggestion and I leave it to others on what to do with it.

(I was going to put this comment in the 'Discussion' tab for 'Reading Room' but I do not have edit privileges for that page.)

--Russell Cameron Thomas (discuss • contribs) 20:30, 20 January 2021 (UTC)

 * Here are names for a common discussion area on different wikimedia sisters. Some projects have a single name for all the central stuff, while others may slpit off a few or many functions into pages named differently; it's a tough call which other names are worth noting.
 * {| class="wikitable"

! sister ! discussion ! other stuff
 * Wikibooks
 * Reading room (12) previously: Staff lounge
 * Wikimedia Commons
 * Village pump (3)
 * Help desk ; Administrators' noticeboard (4) ; Community portal (+1)
 * Wikidata
 * Project chat
 * Wikinews
 * Water cooler (5)
 * AAA (Admin Action Alerts)
 * Wikipedia
 * Village pump (6)
 * Help desk, Reference desk, Questions, etc.
 * Wikiquote
 * Village pump
 * Reference desk (+1)
 * Wikisource
 * Scriptorium
 * Wiktionary
 * Beer parlour
 * Tea room, Grease pit ; Information desk, Etymology scriptorium
 * Wikiversity
 * Colloquium
 * Wikivoyage
 * Travellers' pub
 * Tourist office
 * }
 * My own opinion of some of these: "Beer parlour" is a tad too eccentric; "Colloquium" is a bit academically snobbish (and I say this as someone with a PhD); "Village pump" is, meh; "Water cooler" has the merit of distinguishing itself from "Village pump" but, otherwise, meh; and "Scriptorium" is pretty cool. For whatever perspective any of this offers on what name would work best here.  I'm not, btw, personally filled with boundless enthusiasm for "forum". Brand recognition (i.e., sticking with a name that's been in use for a long time) has some advantage, though it oughtn't prevent all change. --Pi zero (discuss • contribs) 23:54, 20 January 2021 (UTC) [Later added Wikivoyage, as suggested below. --Pi zero (discuss • contribs) 14:12, 21 January 2021 (UTC)]
 * Would like to add the Wikivoyage term "Travellers' pub", since it invokes the feeling of an informal gathering. Perhaps "Book club" or "Reader's circle" or something similar would be appropriate? I also agree that Scriptorium would be cool. --Mbrickn (discuss • contribs) 10:01, 21 January 2021 (UTC)
 * "Forum" seems to be too generic, I'd prefer a name that suits our project well - which is why I would agree to "Scriptorium". The term fits our project and is also easier to reference the "Scriptorium" than the "Reading Room" (shorter word, basically). —Atcovi (Talk - Contribs) 14:04, 21 January 2021 (UTC)
 * I've added Wikivoyage (oops); indeed, "Travellers' pub" is pretty good. I suppose Wikivoyage's "Destinations" page is somewhat analogous to our Wikibooks Stacks. One of the difficulties with "Reading room" might be that some libraries use the term "reading room" to describe a somewhat different facility than others do.  Imho Scriptorium doesn't fit, because properly a scriptorium is a room where scribes go to write — kind of the opposite problem from "Reading room" which, in many libraries, is a place readers go to read. At opportunity, perhaps we should hunt down the discussion that accompanied changing "Staff lounge" to "Reading room". --Pi zero (discuss • contribs) 14:12, 21 January 2021 (UTC)
 * There's "Common room"; possible drawbacks that come to mind are maybe-somewhat-imperfect fit to the function, and vague similarity to the name of a different sister (Commons). Still, having thought of it I'm finding it hard to dismiss as it has a certain something. --Pi zero (discuss • contribs) 14:21, 21 January 2021 (UTC)
 * Break Out Area; Coffee Room (my local library calls it "Book Rest Café"); Creche?!; Editors' Meet... Maybe we should run a "competition?" QuiteUnusual (discuss • contribs) 14:59, 21 January 2021 (UTC)
 * You're looking for this. Leaderboard (discuss • contribs) 19:09, 21 January 2021 (UTC)
 * Oh! Thanks, Leaderboard.  Skimming, the one point I see that doesn't seem to have been much remarked this time around is that the name "Reading room" helps to keep everyone's minds focused clearly on books. --Pi zero (discuss • contribs) 19:35, 21 January 2021 (UTC)
 * Tourist office
 * }
 * My own opinion of some of these: "Beer parlour" is a tad too eccentric; "Colloquium" is a bit academically snobbish (and I say this as someone with a PhD); "Village pump" is, meh; "Water cooler" has the merit of distinguishing itself from "Village pump" but, otherwise, meh; and "Scriptorium" is pretty cool. For whatever perspective any of this offers on what name would work best here.  I'm not, btw, personally filled with boundless enthusiasm for "forum". Brand recognition (i.e., sticking with a name that's been in use for a long time) has some advantage, though it oughtn't prevent all change. --Pi zero (discuss • contribs) 23:54, 20 January 2021 (UTC) [Later added Wikivoyage, as suggested below. --Pi zero (discuss • contribs) 14:12, 21 January 2021 (UTC)]
 * Would like to add the Wikivoyage term "Travellers' pub", since it invokes the feeling of an informal gathering. Perhaps "Book club" or "Reader's circle" or something similar would be appropriate? I also agree that Scriptorium would be cool. --Mbrickn (discuss • contribs) 10:01, 21 January 2021 (UTC)
 * "Forum" seems to be too generic, I'd prefer a name that suits our project well - which is why I would agree to "Scriptorium". The term fits our project and is also easier to reference the "Scriptorium" than the "Reading Room" (shorter word, basically). —Atcovi (Talk - Contribs) 14:04, 21 January 2021 (UTC)
 * I've added Wikivoyage (oops); indeed, "Travellers' pub" is pretty good. I suppose Wikivoyage's "Destinations" page is somewhat analogous to our Wikibooks Stacks. One of the difficulties with "Reading room" might be that some libraries use the term "reading room" to describe a somewhat different facility than others do.  Imho Scriptorium doesn't fit, because properly a scriptorium is a room where scribes go to write — kind of the opposite problem from "Reading room" which, in many libraries, is a place readers go to read. At opportunity, perhaps we should hunt down the discussion that accompanied changing "Staff lounge" to "Reading room". --Pi zero (discuss • contribs) 14:12, 21 January 2021 (UTC)
 * There's "Common room"; possible drawbacks that come to mind are maybe-somewhat-imperfect fit to the function, and vague similarity to the name of a different sister (Commons). Still, having thought of it I'm finding it hard to dismiss as it has a certain something. --Pi zero (discuss • contribs) 14:21, 21 January 2021 (UTC)
 * Break Out Area; Coffee Room (my local library calls it "Book Rest Café"); Creche?!; Editors' Meet... Maybe we should run a "competition?" QuiteUnusual (discuss • contribs) 14:59, 21 January 2021 (UTC)
 * You're looking for this. Leaderboard (discuss • contribs) 19:09, 21 January 2021 (UTC)
 * Oh! Thanks, Leaderboard.  Skimming, the one point I see that doesn't seem to have been much remarked this time around is that the name "Reading room" helps to keep everyone's minds focused clearly on books. --Pi zero (discuss • contribs) 19:35, 21 January 2021 (UTC)

Moving Wikimania 2021 to a Virtual Event

''. Apologies if you are not reading this message in your native language. . ''

Wikimania will be a virtual event this year, and hosted by a wide group of community members. Whenever the next in-person large gathering is possible again, the ESEAP Core Organizing Team will be in charge of it. Stay tuned for more information about how you can get involved in the planning process and other aspects of the event. Please read the longer version of this announcement on wikimedia-l.

ESEAP Core Organizing Team, Wikimania Steering Committee, Wikimedia Foundation Events Team, 15:15, 27 January 2021 (UTC)

Project Grant Open Call
This is the announcement for the Project Grants program open call that started on January 11, with the submission deadline of February 10, 2021. This first open call will be focussed on Community Organizing proposals. A second open call focused on research and software proposals is scheduled from February 15 with a submission deadline of March 16, 2021.

For the Round 1 open call, we invite you to propose grant applications that fall under community development and organizing (offline and online) categories. Project Grant funds are available to support individuals, groups, and organizations to implement new experiments and proven ideas, from organizing a better process on your wiki, coordinating a campaign or editathon series to providing other support for community building. We offer the following resources to help you plan your project and complete a grant proposal:

 * Weekly proposals clinics via Zoom during the Open Call. Join us for #Upcoming_Proposal_Clinics|real-time discussions with Program Officers and select thematic experts and get live feedback about your Project Grants proposal. We’ll answer questions and help you make your proposal better. We also offer these support pages to help you build your proposal:
 * Video tutorials for writing a strong application
 * General planning page for Project Grants
 * Program guidelines and criteria

Program officers are also available to offer individualized proposal support upon request. Contact us if you would like feedback or more information.

We are excited to see your grant ideas that will support our community and make an impact on the future of Wikimedia projects. Put your idea into motion, and submit your proposal by February 10, 2021!

Please feel free to get in touch with questions about getting started with your grant application, or about serving on the Project Grants Committee. Contact us at projectgrantswikimedia.org. Please help us translate this message to your local language. MediaWiki message delivery (discuss • contribs) 08:00, 28 January 2021 (UTC)


text:
Wikijunior talk:Kings and Queens of England/The Anglo-Saxons

When one writes for children, one should to write simply. I see, from the first few sentences, that editors here have not grasped the idea as fully as they might. It is possible to test some issues by preparing two versions. That supposes one has a classroom or two at hand. I see many issues of communication fail where testing would have shown the problem immediately. As a text to help people become better writers, I suggest:
 * Do not use contractions in prose. What does it cost to write "is not" instead of "isn't" or "it is" instead of "it's"?
 * Do not use circumlocutions (ready-made phrases). For "At this point in time" why not "then" or "now" (as appropriate).
 * Short sentences with verbs in straightforward tenses. Not long sentences with verbs in exotic tenses. I counted one sentence as 49 words. If you think that's OK for young readers, then you are wrong.
 * Vagueness.
 * Detail appropriate to younger ages. It is quite obvious that the level of detail in this article is far greater than needed by any junior. This is important because, for less expert readers, the time needed to process long complex texts goes up exponentially with increased length. I liked the sentence "Defeated, the Vikings retreated"!
 * Personally, I do not like the mixture of historically verified facts with "stories".
 * Nash, Walter 1986. English usage: a guide to first principles. Routledge. ISBN 0-7102-1200-3

Macdonald-ross (discuss • contribs) 10:54, 7 March 2021 (UTC)


text:
User talk:RacoonyRE

{| cellpadding="10" cellspacing="5" style="width: 95%; background-color: inherit; margin-left: auto; margin-right: auto"  '''This discussion page is not actively monitored. You can write to me best on Meta or in the German Wikipedia.'''
 * style="background-color: #FDFDFD; border: 1px solid #777777; -moz-border-radius-topleft: 8px; -moz-border-radius-bottomleft: 8px; -moz-border-radius-topright: 8px; -moz-border-radius-bottomright: 8px;" colspan="2" |

'''Dear visitors, I attach importance to a friendly tone (Wikiquette). '''  The best thing to do is to watch my discussion page. (Click here)
 * You can create a new post here. Please always sign with -- ~ (two hyphens, four tildes) or by clicking on the icon shown below '''
 * If you do not want others to read your message, you can also write me an Email.
 * I basically reply where the discussion started so that it doesn't get torn apart.

title: User talk:RacoonyRE


text:
Mirad Thesaurus

Welcome to Mirad Thesaurus, a thematically-ordered vocabulary of Mirad, a serious constructed language. This book is an accompaniment to Mirad Grammar.

title: Mirad Thesaurus


text:
Mirad Thesaurus/Universe

=The Universe (Ha Mor)=

Universe

 * mor=universe, cosmos
 * mora=universal, cosmic
 * moray=universally
 * moreb=master of the universe, god
 * morweb=male god
 * moreyb=godess
 * moreba=divine
 * moreban=divinity
 * morebam=heaven
 * morebama=celestial
 * morted=progenitor of the universe, god
 * morsaxut=creator of the universe, god
 * mortun=cosmology
 * mortuna=cosmological
 * mortut=cosmologist
 * moruin=cosmogony
 * moruina=cosmogonic
 * moruit=cosmogonist
 * morij=big bang, beginning of the universe
 * morij tuin=big bang theory
 * byi morij=since forever, since the beginning of time
 * moruj=end of the universe
 * buu moruj=for eternity, until the end of time, forever
 * gwamor=macrocosm
 * gwamor teaxar=telescope
 * gwomor=microcosm
 * gwomor teaxer=microscope
 * geizona mor=parallel universe
 * morxer=universalize
 * morxen=universalization
 * morxwa=universalized
 * hyumor=alternate universe

Star

 * mar=star
 * mara=astral, stellar
 * maraya=starry
 * marika=star-studded
 * maruka=starless
 * marar=astrolabe
 * martyez=astromancy
 * marnagtyen=astrometry
 * mardraf=star chart
 * marmaf=galaxy
 * marmafa=galactic
 * ebmarmafa=intergalactic
 * marman=starlight
 * marmek=stardust
 * marmeg=comet
 * marmeyg=meteorite
 * marsiyn=asterisk
 * martun=astronomy
 * martuna=astonomical
 * martut...astronomist
 * martuin=astrology
 * martuina=astrological
 * martuit=astrologist
 * maruka=starless
 * maryan=constellation
 * maryena=starlike
 * marpaput=astronaut, cosmonaut
 * marpur=star ship
 * marpureb=star ship captain
 * marnaap=star system
 * marman=star light
 * marmanxwa=starlit
 * marnaud=star ray, star beam
 * mar bi David=star of David

Sun

 * maar=sun
 * maara=solar
 * maara azon=solar power
 * maaryafon=solar power
 * maaryafon zyevmays=solar power panel
 * maaryafonxwa=solar-powered
 * maaraya=sunny
 * maarika=sunny
 * maarfyifren=heliolatry, sun worshiping
 * maarman=sunlight
 * maarmanx=sunshine
 * marmanxwa=sunlit
 * maarmotozaxer=to suntan
 * maarmoynar=sunshade
 * maarnagar=heliometer
 * maarnaud=sun ray, sunbeam
 * maarovar=parasol
 * maarovyel=sunblock, sunscreen (lotion)
 * maartun=heliology
 * maarubuzpus=heliotrope
 * maaryap=sunrise
 * maaryop=sunset
 * maaryupen=heliotaxis
 * maaryuzem=heliosphere
 * maarzena=heliocentric
 * maar yibnod=solstice

Planet

 * mer=planet
 * amera=northern
 * amernod=north pole
 * yibamer=far north, arctic
 * omera=southern
 * omernod=south pole
 * yibomer=far south, antartic
 * imera=eastern
 * Imerem=Orient
 * umera=western
 * Umerem=Occident
 * aomera=north-south
 * aomernad=longitude
 * uimera=east-west
 * uimernad=latitude
 * emernad=meridian, equator
 * emernada=meridional, equatorial
 * zyamera=global
 * zyamerxen=globalization
 * merizon=compass direction
 * merizondar=compass
 * mernod=pole
 * mernoda=polar
 * mernodan=polarity
 * mernodxer=to polarize
 * mernodxen=polarization
 * mernodxwa=polarized
 * merovnod=antipode
 * mersindraf=atlas
 * mersindrafxen=cartography
 * mertun=geology
 * mertuna=geological
 * mertut=geologist
 * merzen=planet core
 * meram=planetarium
 * merzyun=globe, sphere
 * eymerzyun=hemisphere
 * obemera=tropical
 * obemerem=tropics
 * Obemernad=Tropic of Cancer
 * obemera mapil=tropical storm
 * oyebmer=exoplanet
 * yebmer=endoplanet

Earth

 * meir=earth
 * meira=terrestrial
 * meirat=earthling
 * yizmeirat=extraterrestrial
 * meir zen=earth core
 * meir absayeb=earth crust
 * meir ebsayeb=earth mantle

World

 * mir=world, environment, sphere, realm
 * mira=world (adj.)
 * Ha gwa nyaza twob be ha mir.=The richest man in the world.
 * Nas zyube ha mir.=Money makes the world go round.
 * Mirad=World language, Mirad
 * Mirader=to say in Mirad
 * Miradu has!=Say it in Mirad!
 * Miradaler=to speak in Mirad
 * Duven et Miradale?=Do you speak in Mirad?
 * hyamira=worldwide, global
 * hyumir=other world
 * hyumira=other-worldly, exotic
 * fyamir=heaven
 * fyamira=heavenly, celestial
 * fyamirat=celestial creature, heavenly host
 * mira dropek=world war
 * aa mira dropek=first world war
 * ea mira dropek=second world war
 * jomir=afterworld, metasphere, hereafter
 * ebjomir=limbo
 * eyjomir=purgatory
 * obmir=underworld
 * obmirat=underworld creature, demon
 * fyomir=hell
 * fyomira=infernal
 * fyomirat=demon
 * Pu fyomir!=Go to hell!
 * tobmir=world of men, earth
 * tujdin mir=dream-world
 * tutmir=academia, scholastic world
 * tutmira=academic, scholarly, scholastic
 * vyamir=real world
 * vyamira=real-world
 * vyomir=fantasy-land
 * vyomira=fantastic
 * yizmir=world beyond, metasphere, metaworld
 * yizmira=metaspheric, other-worldly, supernatural
 * yizmirat=supernatural being, ghost, spirit
 * totmir=world of the gods, heaven, realm of the gods
 * totmira=celestial, heavenly
 * megmir=lithosphere
 * yeybmegmira=intratelluric
 * dommir=urbane world
 * milmir=hydrosphere
 * mirteasyen=world view, Weltanschau
 * zyamira=world-wide
 * zyamira neaf=world-wide web, WWW, internet
 * oyeybmira=extramundane
 * miruj=end of the world
 * miruja=apocalyptic
 * Ha Mira Nasam=The World Bank
 * Ha Mira Bak Naab=The World Health Organization
 * yuzmir=environment
 * yuzmira=environmental

Moon

 * mur=satellite, moon
 * muar=moon, Earth's moon
 * muara=lunar
 * muara pur=lunar vehicle
 * muarman=moonlight
 * muarmanx=moonshine
 * muarmanxwa=moonlit
 * muarpuer=land on the moon
 * muarpuen=moon landing
 *  mur noog=moon phase
 *  eymur=crescent, half moon
 *  eymuravul=croissant

title: Mirad Thesaurus/Universe


text:
Associative Composition Algebra/Split-quaternions

There are at least three portals leading to split quaternions: the dihedral group of a square, matrix products in M(2,R), and the modified Cayley-Dickson construction. The work of Max Zorn on split octonions showed the necessity of including split real AC algebras in the aufbau of the category.

The development through the dihedral group was started with a lemma in the Introduction, and is completed with exercises below.

Or one can start with a basis {1, i, j, k} taken from M(2,R), where the identity matrix is one, $$\begin{pmatrix}0 & 1 \\ 1 & 0 \end{pmatrix}$$ is j, $$\begin{pmatrix}0 & 1 \\- 1 & 0 \end{pmatrix}$$ is i, and $$\begin{pmatrix}1 & 0 \\ 0 & -1 \end{pmatrix}$$ is k. Some practice with matrix multiplication shows they are anticommutative like division quaternions, but some products differ:
 * j2 = +1 = k2,  j k = − i.

Then the real AC algebra of split-quaternions uses coefficients w, x, y, z ∈ R to express an element, its conjugate, and the quadratic form N:
 * $$q = w + xi + yj + zk,\quad q^* = w - xi - yj - zk, \quad N(q) = w^2 + x^2 - y^2 - z^2 .$$

Exercises
1. What are the involutions on a square ?

2. As reflections, what is the angle of incidence of the axes of reflection ?

3. The composition of these reflections has what angle of rotation ?

Computations
Insight into the structure and dynamics of split-quaternions is available through elementary computational exercises. These exercises use j2 = +1 = k2 and jk = −i, contrary to the quaternion group, which is expressed with the same letters i, j, k, but which here refer to the dihedral group of a square instead.
 * 1) For r = j cos θ + k sin θ, show that r2 = +1 = −r r*.
 * 2) Compute ir.
 * 3) Recall  that  = (q t* + t q*)/2. Show  = real part of q t*
 * 4) Definition: q and t are orthogonal when  = 0.
 * 5) Show that for any theta, r and ir are orthogonal.
 * 6) Let p = i sinh a + r cosh a. Show that p2 = +1 for any a and r.
 * 7) Let v = i cosh a + r sinh a. Show that v2 = −1.
 * 8) For a given a and r, show that p and v are orthogonal.
 * 9) Let m = p exp(bp) = sinh b + p cosh b. Show that m m* = −1.
 * 10) Let w = exp(bp) = cosh b + p sinh b. Show that m is orthogonal to w.
 * 11) Show that m is orthogonal to v.
 * 12) For any θ, a, and b defining r, p, w, v, and m, the set {m, w, v, ir} is an orthonormal basis.
 * 13) If u is a unit,  = uu* .


text: Category:Book:Human Body

title: Category:Book:Human Body


text:
User:BFDIFan707

Hi

title: User:BFDIFan707


text:
User talk:JJPMaster

Welcome!
Welcome to Wikibooks, JJPMaster!

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You do not need technical skills to contribute. You can easily change most books. Please introduce yourself, and let us know what interests you.

If you already contribute at other Wikimedia projects, our Wikimedia Orientation should quickly get you started. (Would you like to provide feedback on this message?)

  Getting help   Made a mistake?   Goodies, tips and tricks
 * Read the Using Wikibooks book for a friendly introduction to the project, or our help pages for more information.
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-- Jules (Mrjulesd) 12:43, 8 March 2021 (UTC)

title: User talk:JJPMaster


text:
User talk:Iifar

Welcome!
Welcome to Wikibooks, Iifar!

 First steps tutorial Wikibooks is for collaborative development of free textbooks.

You do not need technical skills to contribute. You can easily change most books. Please introduce yourself, and let us know what interests you.

If you already contribute at other Wikimedia projects, our Wikimedia Orientation should quickly get you started. (Would you like to provide feedback on this message?)

  Getting help   Made a mistake?   Goodies, tips and tricks
 * Read the Using Wikibooks book for a friendly introduction to the project, or our help pages for more information.
 * You can get friendly help from the community in the user assistance room or our IRC channel.
 * Upload freely licensed files to Wikimedia Commons. You may request permission to upload fair use files locally. Please include author and source information and a non-free use rationale for non-free files.
 * You can restore a previously saved version.
 * Pages should follow the  naming convention.
 * Need to rename a page? Use the move tab (is available once your account is 4 days old - until then, ask for help).
 * To request a page be deleted, add to the top of the page.
 * Was a page you made deleted? Please read the deletion policy, and check the deletion log to find out why. Also check the RFD archives if applicable. You can request undeletion at WB:RFU, or ask the administrator who deleted the page.
 * Please fill in the edit summary and preview your edits before saving.
 * Sign your name on discussion pages by typing ~~~~
 * User scripts can make many tasks easier. Look at the Gadgets tab of my preferences; check off the boxes for the scripts you want, and hit save!

-- Jules (Mrjulesd) 22:15, 8 March 2021 (UTC)

title: User talk:Iifar


text:
Wikibooks:Requests for deletion/Exercise as it relates to Disease

Exercise as it relates to Disease/Feeling hot for health
I was going to delete it, but has seemingly approved it, so I'm bringing it here instead. The content in this article isn't quite suitable to me (it clearly looks like a work in progress, and the talk page of the parent book would be better?) Leaderboard (discuss • contribs) 18:03, 21 September 2019 (UTC)
 * 1)  It now looks like a complete advertising added in an existing book. JackPotte (discuss • contribs) 21:04, 21 September 2019 (UTC)
 * 2)  per JackPotte, just inserted to advertise Bikram yoga program, no real connection to book. -- Jules  (Mrjulesd) 21:35, 14 October 2019 (UTC)
 * 3)  per above --DannyS712 (discuss • contribs) 09:32, 30 March 2020 (UTC)


text:
Wikibooks:Requests for deletion/Euclid's Elements

Euclid's Elements
Flagged as a new book more than 5 years ago but no development and no pages.--ЗAНИA talk 03:04, 2 January 2020 (UTC)
 * , this is pretty uncontroversial I feel. No pages beyond a title page. -- Jules (Mrjulesd) 16:57, 3 January 2020 (UTC)


text:
Salute, Jonathan!/Capitul 1 - with notes

Contenete - Capitul 1 - Capitul 2 - Capitul 3

Unesim may (1 may)
Un mann sta in un cité. Li mann scri un jurnale. Li mann vide un cité.

Esque li mann sta in un cité? Yes, il sta in un cité.

Esque li mann sta in...un mann? No, il ne sta in un mann. Il sta in un cité.

Esque li mann sta in un tren? No, il ne sta in un tren. Il sta in un cité. Li mann sta in un cité.

Esque li mann scri un jurnale? Yes, il scri un jurnale.

Esque li jurnale scri li mann? No, li jurnale ne scri li mann. Un jurnale ne scri. Un mann scri. Li mann scri li jurnale.

Esque li cité vide li mann? No, li cité ne vide li mann. Un cité ne vide. Un mann vide. Esque li mann vide li cité? Yes, li mann vide li cité.

U sta li mann? Il sta in un cité. Quo li mann scri? Il scri un jurnale. Quo li mann vide? Il vide un cité. Il sta in un cité, e il scri un jurnale, e il vide un cité.

Esque li mann sta in un jurnale? No, il scri in un jurnale. Esque li mann vide un mann? No, il ne vide un mann; il vide un cité.

Li mann es grand. Li mann es bon, e li mann es inteligent. Il pensa. Il pensa pri li cité. Il pensa: “Quo es li cité? Esque li cité es bon? Esque li cité es grand?”

Esque li mann scri un cité? No, il ne scri un cité; un cité es grand. Il scri un jurnale; un jurnale ne es grand.

Esque li jurnale pensa pri li mann? No, un jurnale ne pensa. Un mann pensa. Il pensa pri li jurnale, e pensa pri li cité.

E il scri in li jurnale. Il scri pri li cité. Il scri: “Li cité es bon, e li cité es grand.”

Il pensa: “Li cité es bon”; il pensa que li cité es bon. Il pensa: “Li cité es grand”; il pensa que li cité es grand. Il pensa pri li cité, e il pensa pri li jurnale.

Quo es li jurnale? Li jurnale es u li mann scri; il scri in li jurnale. In li jurnale, li mann scri pri li cité. In li jurnale, li mann ne scri pri li tren; li mann sta in li cité, ne in li tren. Il ne pensa pri li tren; il pensa pri li cité, li cité Munich.

Quo es li cité? It es Munich. U es Munich? It es u li man sta. U es li mann? Il es in Munich.

Yes, Munich es un grand cité, e un bon cité. Li mann pensa que Munich es un bon cité, e il pensa que it es un grand cité. Li mann es inteligent. Il di: “Salute, Munich!” Il es un bon mann!

Li mann sta e pensa: “U es li tren?” Il vide...il vide li tren! Il pensa: "Li tren!" Nu il ne pensa pri li jurnale e ne pensa pri li cité; il pensa pri li tren!

Tri (3) verbes + verbe "es"

 * Verbes in a: sta, pensa
 * Verbes in i: scri
 * Verbes in e: vide
 * Verbe "es": es

esque

 * esque = ?
 * Un mann sta in un cité.
 * Esque un mann sta in un cité?

·esim

 * un = 1
 * un·esim
 * du = 2
 * du·esim
 * tri = 3
 * tri·esim

 * May 2019 es 2019.05.
 * Unesim may es 2019.05.01.
 * Duesim may es 2019.05.02.
 * Triesim may es 2019.05.03.

li, un

 * li = 1
 * Un mann sta in un cité. Li mann scri in un jurnale. Li mann vide un tren. = 1
 * un = +1
 * Un mann sta in un cité. Un mann scri in un jurnale. Un mann vide un tren. = 3

pri

 * Li mann pensa pri Munich.
 * Li mann scri pri Munich.
 * Li mann scri: "Yo vide un tren!" Il scri pri un tren.

Vocabularium

 * bon: 👍
 * cité
 * es
 * esque
 * grand
 * in
 * it
 * inteligent
 * jurnale
 * li
 * mann
 * no ↔ yes
 * pensa: 💭
 * pri
 * que
 * quo
 * salute
 * scri
 * sta
 * tren
 * u
 * un = 1
 * vide
 * yes ↔ no

Di:

 * c = k
 * capitul: kapitul
 * scri: skri
 * ci, ce: tsi, tse
 * cité: tsité


text:
Wikibooks:Requests for deletion/Fire Simulation for Engineers

Fire Simulation for Engineers
I propose to delete this book, because it is now outdated and orphaned. The tools that are described in the text have new incompatible releases, and we do not have anymore the resources needed to update the book.

 * Although it is outdated, it is good reference point for anyone who is still using the old versions.
 * Nominated by Encik Tekateki (discuss • contribs) 08:51, 29 November 2020 (UTC)
 * The book should be updated instead. Leaderboard (discuss • contribs) 21:43, 3 December 2020 (UTC)


text:
Wikibooks:Requests for deletion/Monopoly

Monopoly rules
Rules stated do not match official rules. — Preceding unsigned comment added by 119.18.0.7 (talk • contribs)
 * 119.18.0.7: Please could you give examples? -- Jules (Mrjulesd) 10:44, 3 June 2020 (UTC)


text:
Wikibooks:Requests for deletion/Centrelink help

Centrelink help
This page had delete for a few days, but as the author is still working hard on it, the least we can do is give them a chance.
 * 1)  It could complement , but according to What is Wikibooks? it looks like an inadmissible advertising booklet for a private company. JackPotte (discuss • contribs) 21:08, 7 February 2020 (UTC)
 * 2)  while this could be viewed as a breaking the "advertising or self-promotion" rule I think this is pretty borderline in this instance. -- Jules  (Mrjulesd) 20:59, 8 February 2020 (UTC)
 * 3)  Afaics, it's a how-to about dealing with a government agency — not a private company.  Other concerns, such as overall organization, or current lack of completion of some sections, are not RFD-worthy. --Pi zero (discuss • contribs) 00:01, 9 February 2020 (UTC)


text:
Wikibooks:Requests for deletion/Intellectual Property and the Internet

Intellectual Property and the Internet/Arab Spring
In the course of trying to form a mental image of everything that already exists in this very raw, semi-coherent book that I seem to have adopted (or that has adopted me, we break-up and get back together several times per month still), I just can't see any path forward for this particular page. While I can see how the Arab Spring could be used to illustrate real-world implications of public policy regarding content ownership, there's no clay to mold here. I'm exceedingly new here and will take what's said on the matter as a lesson in the Wikibooks mindset, more than anything. Thanks in advance for any thoughts you choose to share. —  ⚞ 🐈ℛogueScholar  ₨Talk🗩 ⚟    $My recent mischief$   08:26, 26 October 2019 (UTC)
 * The treatment is too dry and encyclopedic.
 * Much of what's there is visually-driven in a way that I interpret as very "un-Wikibooks-like."
 * With no pre-existing content tying the events to IP, it would take more effort to bring the prose into focus under that lens than it would to write the content from scratch. If/when I do that, I'd want that work to happen on a page with a broader geographic and temporal scope.
 * per nom, just an imported article from en.wikipedia in 2012 with some minor cleanup. It has little to do with "Intellectual Property and the Internet" as it is written, and so it is irrelevant to the book topic in this form. -- Jules (Mrjulesd) 11:17, 26 October 2019 (UTC)


text:
Wikibooks:Requests for deletion/Dividedness

Dividedness
Feels out of scope and unsuitable for Wikibooks. Other examples by same author: Strictly better. Leaderboard (discuss • contribs) 07:51, 31 January 2021 (UTC)
 * This book, I believed not fulfilling Wikibooks criteria Encik Tekateki (discuss • contribs) 11:34, 31 January 2021 (UTC)
 * Original research (both of them, to be clear). --Pi zero (discuss • contribs) 13:30, 31 January 2021 (UTC)
 * both of them, I cant believe they're anything more than someone fooling around. -- Jules (Mrjulesd) 15:12, 31 January 2021 (UTC)


text:
Wikibooks:Requests for deletion/Sunvox Reference

Sunvox Reference
This book is a stub for 6 years and no further additions are seen to improve the book content beside the general intro. Alternatively, Sunvox Tutorial at least provide slightly more details.
 * Nominated by Encik Tekateki (discuss • contribs) 04:24, 3 January 2021 (UTC)
 * Inclined to delete this, has nothing useful in it. Leaderboard (discuss • contribs) 20:38, 3 January 2021 (UTC)


text:
Wikibooks:Requests for deletion/GCSE Physics

GCSE Physics
Nominate to delete this as the book is remained a stubs over 8 years. I already moved to partly developed contents to 9-1 Physics/Conservation of energy Also GCSE has moved to 9-1 curriculum in 2020.

Proposed deletion by Encik Tekateki (discuss • contribs) 02:37, 17 January 2021 (UTC)

 * Looks as if the small amount of content produced for this book was moved to another book. --Pi zero (discuss • contribs) 03:30, 17 January 2021 (UTC)
 * Not a useful book and GCSE programmes change every year and vary by exam board. --ЗAНИA [[Image:Flag_of_the_Isle_of_Mann.svg|15px]]talk 09:53, 17 January 2021 (UTC)
 * Once moved "GCSE Physics/Energy for the Home/Heating Houses" to 9-1 Physics/Conservation of energy, there is no content other than empty sections.--Jusjih (discuss • contribs) 06:17, 19 January 2021 (UTC)