explain xkcd - User contributions [en]

2117: Differentiation and Integration

2024-08-07T01:43:22Z

108.162.245.100: /* Explanation */ it was an old mathml parser, not html

{{comic
| number = 2117
| date = February 27, 2019
| title = Differentiation and Integration
| image = differentiation_and_integration.png
| titletext = "Symbolic integration" is when you theatrically go through the motions of finding integrals, but the actual result you get doesn't matter because it's purely symbolic.
}}

==Explanation==
This comic illustrates the old saying [https://mathoverflow.net/q/66377 "Differentiation is mechanics, integration is art."] It does so by providing a {{w|flowchart}} purporting to show the process of differentiation, and another for integration.

{{w|Derivative|Differentiation}} and {{w|Antiderivative|Integration}} are two major components of {{w|calculus}}. As many Calculus 2 students are painfully aware, integration is much more complicated than the differentiation it undoes.

However, Randall dramatically overstates this point here. After the first step of integration, Randall assumes that any integration can not be solved so simply, and then dives into a step named "????", suggesting that it is unknowable how to proceed. The rest of the flowchart is (we can assume deliberately) even harder to follow, and does not reach a conclusion. This is in contrast to the simple, straightforward flowchart for differentiation. The fact that the arrows in the bottom of the integration part leads to nowhere indicates that "Phone calls to mathematicians", "Oh no" and "Burn the evidence" are not final steps in the difficult journey. The flowchart could be extended by Randall to God-knows-where extents.

It should be noted that Randall slightly undermines his point by providing four different methods, and an "etc", and a "No"-branch for attempting differentiation with no guidelines for selecting between them.

===Differentiation===
'''{{w|Chain rule}}'''

For any <math> \frac{d}{dx}f(x)=f'(x)</math> and <math> \frac{d}{dx}g(x)=g'(x) </math>, it follows that <math> \frac{d}{dx}(f(g(x)))=f'(g(x))\cdot g'(x)</math>.

'''{{w|Power Rule}}'''

For any 𝑓(𝑥) = 𝑘 𝑥𝑎, it follows that 𝑓'(𝑥) = 𝑘 𝑎 𝑥𝑎−1.

'''{{w|Quotient rule}}'''

For any <math> \frac{d}{dx}f(x)=f'(x)</math> and <math> \frac{d}{dx}g(x)=g'(x) </math>, it follows that <math> \frac{d}{dx} \frac{f(x)}{g(x)}=\frac{f'(x)\cdot g(x)-f(x)\cdot g'(x)}{(g(x))^2}</math> if <math>g(x)\ne 0</math>.

'''{{w|Product rule}}'''

For any <math> \frac{d}{dx}f(x)=f'(x)</math> and <math> \frac{d}{dx}g(x)=g'(x) </math>, it follows that <math> \frac{d}{dx}(f(x)\cdot g(x))=f'(x)\cdot g(x)+f(x)\cdot g'(x)</math>.

===Integration===
'''{{w|Integration by parts}}'''

The "product rule" run backwards. Since <math>(uv)' = uv' + u'v</math>, it follows that by integrating both sides you get <math> uv = \int u dv + \int v du</math>, which is more commonly written as <math>\int u dv = uv - \int v du</math>. By finding appropriate values for functions <math>u, v</math> such that your problem is in the form <math>\int u dv</math>, your problem ''may'' be simplified. The catch is, there exists no algorithm for determining what functions they might possibly be, so this approach quickly devolves into a guessing game - this has been the topic of an earlier comic, [[1201: Integration by Parts]].

'''{{w|Integration by substitution|Substitution}}'''

The "chain rule" run backwards. Since d(f(u)) = f'(u) ⋅ du/dx ⋅ dx, it follows that f(u) = ∫ f'(u) ⋅ du/dx ⋅ dx. By finding appropriate values for functions f, u such that your problem is in the form ∫ f'(u) ⋅ du/dx ⋅ dx your problem ''may'' be simplified.

'''{{w|Cauchy's integral formula|Cauchy's Formula}}'''

Cauchy's Integral formula is a result in complex analysis that relates the value of a contour integral in the complex plane to properties of the singularities in the interior of the contour. <math> \frac{d^n}{da^n}f(a) = \frac{n!}{2\pi i} \oint_\gamma \frac{f(z)}{\left(z-a\right)^{n+1}}\, dz.</math> It is often used to compute integrals on the real line by extending the path of the integral from the real line into the complex plane to apply the formula, then proving that the integral from the parts of the contour not on the real line has value zero.

'''{{w|Partial_fraction_decomposition#Application_to_symbolic_integration|Partial Fractions}}'''

Partial fractions is a technique for breaking up a function that comprises one polynomial divided by another into a sum of functions comprising constants over the factors of the original denominator, which can easily be integrated into logarithms.

'''Install {{w|Mathematica}}'''

Mathematica is a modern technical computing system spanning most areas. One of its features is to compute mathematical functions. This step in the flowchart is to install and use Mathematica to do the integration for you. Here is a description about the [https://web.archive.org/web/20180727184709/http://reference.wolfram.com/language/tutorial/IntegralsThatCanAndCannotBeDone.html intricacies of integration and how Mathematica handles those]. (It would be quicker to try [https://www.wolframalpha.com Wolfram Alpha] instead of installing Mathematica, which uses the same backend for mathematical calculations.)

'''{{w|Riemann integral|Riemann Integration}}'''

The Riemann integral is a definition of definite integration. Elementary textbooks on calculus sometimes present finding a definite integral as a process of approximating an area by strips of equal width, as in <math>\sum_{i=0}^{n-1} f(t_i) \left(x_{i+1}-x_i\right).</math>, and then taking the limit as the strips become narrower. Riemann integration removes the requirement that the strips have equal width, and so is a more flexible definition. However there are still many functions for which the Riemann integral doesn't converge, and consideration of these functions leads to the {{w|Lebesgue integration|Lebesgue integral}}. Riemann integration is not a method of calculus appropriate for finding the anti-derivative of an elementary function.

'''{{w|Stokes' Theorem}}'''

Stokes' theorem is a statement about the integration of differential forms on manifolds. <math>\int_{\partial \Omega}\omega=\int_\Omega d\omega\,.</math> It is invoked in science and engineering during control volume analysis (that is, to track the rate of change of a quantity within a control volume, it suffices to track the fluxes in and out of the control volume boundary), but is rarely used directly (and even when it is used directly, the functions that are most frequently used in science and engineering are well-behaved, like sinusoids and polynomials).

'''{{w|Risch Algorithm}}'''

The Risch algorithm is a notoriously complex procedure that, given a certain class of symbolic integrand, either finds a symbolic integral or proves that no elementary integral exists. (Technically it is only a semi-algorithm, and cannot produce an answer unless it can determine if a certain symbolic expression is {{w|Constant problem|equal to 0}} or not.) Many computer algebra systems have chosen to implement only the simpler Risch-Norman algorithm, which does not come with the same guarantee. A series of extensions to the Risch algorithm extend the class of allowable functions to include (at least) the error function and the logarithmic integral. A human would have to be pretty desperate to attempt this (presumably) by hand.

'''{{w|Bessel function}}'''

Bessel functions are the solution to the differential equation x2 d2y/dx2 + x dy/dx + (x2 - n2) y = 0, where n is the order of the Bessel function. Though they do show up in some engineering, physics, and abstract mathematics, in lower levels of calculus they are often a sign that the integration was not set up properly before someone put them into a symbolic algebra solver.

'''Phone calls to mathematicians'''

This step would indicate that the flowchart user, desperate from failed attempts to solve the problem, contacts some more skilled mathematicians by phone, and presumably asks them for help. The connected steps of "Oh no", "What the heck is a Bessel function?" and "Burn the evidence" may suggest the possibility that this interaction might not play out very well and could even get the caller in trouble.
Specialists and renowned experts being bothered - not to their amusement - by strangers, often at highly inconvenient times or locations, is a common comedic trope, also previously utilized by xkcd (for example in [[163: Donald Knuth]]).

'''Burn the evidence'''

This phrase parodies a common trope in detective fiction, where characters burn notes, receipts, passports, etc. to maintain secrecy. This may refer to the burning of one's work to avoid the shame of being associated with such a badly failed attempt to solve the given integration problem. Moreover, such a poor attempt at integration could be viewed as a 'crime against mathematics', with the working of the problem being criminal 'evidence' that must be destroyed to avoid exposure as the culprit. Alternatively, it could be an ironic hint to the fact that in order to find the integral, it may even be necessary to break the law or upset higher powers, so the negative consequences of a prosecution can only be avoided by destroying the evidence.

'''{{w|Symbolic integration}}'''

Symbolic integration is the basic process of finding an antiderivative function (defined with symbols), as opposed to numerically integrating a function. The title text is a pun that defines the term not as integration that works with symbols, but rather as integration as a symbolic act, as if it were a component of a ritual. A symbolic act in a ritual is an act meant to evoke something else, such as burning a wooden figurine of a person to represent one’s hatred of that person. Alternatively, the reference could be seen as a joke that integration might as well be a symbol, like in a novel, because Randall can't get any meaningful results from his analysis.

==Transcript==
:[Two flow charts are shown.]

:[The first flow chart has four steps in simple order, one with multiple recommendations.]
:DIFFERENTIATION
:Start
:Try applying
::Chain Rule
::Power Rule
::Quotient Rule
::Product Rule
::Etc.
:Done?
::No [Arrow returns to "Try applying" step.]
::Yes
:Done!

[The second flow chart begins like the first, then descends into chaos.]
:INTEGRATION
:Start
:Try applying
::Integration by Parts
::Substitution
:Done?
:Haha, Nope!

:[Chaos, Roughly from left to right, top to bottom, direction arrows not included.]
::Cauchy's Formula
::????
::???!?
::???
::???
::?
::Partial Fractions
::??
::?
::Install Mathematica
::?
::Riemann Integration
::Stokes' Theorem
::???
::?
::Risch Algorithm
::???
::[Sad face.]
::?????
::???
::What the heck is a Bessel Function??
::Phone calls to mathematicians
::Oh No
::Burn the Evidence
::[More arrows pointing out of the image to suggest more steps.]

{{comic discussion}}
[[Category:Analysis]]
[[Category:Flowcharts]]
[[Category:Math]]

Talk:2968: University Age

2024-08-07T01:15:31Z

108.162.245.100: save from main

[[1477]] anyone? [[Special:Contributions/162.158.41.131|162.158.41.131]] 03:29, 6 August 2024 (UTC)
:Not relevant in my opinion. In [[1477: Star Wars]] Cueball is surprised by how fast time seems to flow past him. Not surprised that everything ages at the same rate as the Cueball in this comic. There are more than 40 comics relating to [[:Category:Time|time]] on explain xkcd... [[User:Kynde|Kynde]] ([[User talk:Kynde|talk]]) 06:50, 6 August 2024 (UTC)

Was there a USA political debate this week that I didn't hear about? About. Hear of it's existence. I Assuredly didn't hear it. [[Special:Contributions/172.70.175.121|172.70.175.121]] 06:27, 6 August 2024 (UTC)

On a related note, today I set a new personal best for time spent continuously being alive. I'm still well behind the world records, but I have beaten pretty much all of the current collegeate athletes. [[User:RegularSizedGuy|RegularSizedGuy]] ([[User talk:RegularSizedGuy|talk]]) 07:08, 6 August 2024 (UTC)
:Happy birthday :) [[Special:Contributions/172.70.86.16|172.70.86.16]] 12:59, 6 August 2024 (UTC)

"'''Unfortunately, I have terrible news'''" kind of feels to me like — almost — there is a pop culture reference here that I am missing (I'd say I feel 33% of the way towards that feeling; certainly somewhere less likely than even). It just seems like a little too strong of a statement to be what President Cueball would say in the moment. That said, I can't find any support for such a half-baked theory. Although there is an "I have terrible news" (without "Unfortunately") from [https://www.imdb.com/title/tt2572734/characters/nm0278979 The Office, Season 9, "Vandalism"] ([https://www.youtube.com/watch?v=Eo8RzH-6Tfw&t=614s video]), but it doesn't strike me as particularly significant. 
¶ Apart from that, I sort of expected more emphasis on obscure schemes, like (a) increasing the average altitude of the rival university to increase its velocity for the tiny relativistic effect, e.g. by funding construction of an observatory for its astronomy department; (b) attempting to measure the founding dates more precisely, e.g. what if it were only 1-year-183-days older; (c) merger with the rival, ala the proposed MIT/Harvard mergers of the early 20th century; &c. [[User:JohnHawkinson|JohnHawkinson]] ([[User talk:JohnHawkinson|talk]]) 09:51, 6 August 2024 (UTC)

: I thought the terrible new was that, after sending the rival university on a round trip at relativistic speed, due to an unfortuneate miscalculation, the rival university would collide with Earth while still going too fast, causing the destruction of the planet. --[[User:Itub|Itub]] ([[User talk:Itub|talk]]) 11:51, 6 August 2024 (UTC)

: I think it's more a commentary on the futility of university administration plans and goals, given that actual success for a prestigious university involves mostly decisions entirely out of administration hands. [[Special:Contributions/162.158.90.109|162.158.90.109]] 20:55, 6 August 2024 (UTC)
:: What do you think the universities alluded to were? I'm guessing MIT in 1861 and UMass (originally Amherst) in 1863. [[Special:Contributions/162.158.186.20|162.158.186.20]] 21:39, 6 August 2024 (UTC)
:::From the deletions at [https://www.explainxkcd.com/wiki/index.php?title=2968:_University_Age&diff=prev&oldid=348220], "The particular years referenced in the comic (218 years before 2024 would require a university founded in 1806) do not appear to match with any institution of higher learning in the United States. However, there is a very close near-twin that this might refer to - Miami University was founded in 1809, making it 215 years old, and Ohio University was founded in 1804, making it 220 years old." [[Special:Contributions/108.162.245.100|108.162.245.100]] 01:15, 7 August 2024 (UTC)

The way I read the "overtaking our rival by 3" part was that part of the "intensive program" was the sabotage of the other university, effectively ending its aging. This would then be the terrible news to share. [[Special:Contributions/172.71.166.219|172.71.166.219]] 14:02, 6 August 2024 (UTC)

2583: Chorded Keyboard

2023-08-02T22:56:20Z

108.162.245.100: /* an->a*/

{{comic
| number = 2583
| date = February 18, 2022
| title = Chorded Keyboard
| image = chorded_keyboard.png
| titletext = And even though it all went wrong / I'll stand before the lord of song / with nothing on my tongue but 'I don't understand, I swear I backed up my keyboard config before messing with it'
}}

==Explanation==
This strip is a parody of the first verse (and in the title text, the end of the last verse) of {{w|Leonard Cohen}}'s "{{w|Hallelujah (Leonard Cohen song)|Hallelujah}}", which has become a distinctive and popular song of which {{w|Cover version|covers}} and versions exist. Written as a {{w|Sentimental ballad|ballad}}, it is partly based upon the allegory of a mystical {{w|Chord (music)|musical chord}} of several musical notes, that the words and tune both describe and illustrate by example.

Here is the verse from the song (see the lyrics [https://genius.com/Leonard-cohen-hallelujah-lyrics here]):
:Now I've heard there was a secret chord
:That David played, and it pleased the Lord
:But you don't really care for music, do ya?
:It goes like this, the fourth, the fifth
:The minor fall, the major lift
:The baffled king composing "Hallelujah"

[[Cueball]] is {{w|Filk music|filking}} upon this theme, but in his case he has somehow set up his computer so that, upon pressing a certain combination of multiple keys on his keyboard, the system will automatically type out the word "hallelujah" (xkcd's all-caps lettering makes it unclear how the word is capitalized). In his description of the process, in both the comic proper and the title text, he uses adapted lyrics that again both describe and illustrate by example. Most of the initial lyrics are floating 'thoughts'. The punchline "hallelujah", however, is 'spoken' out of his computer monitor - typical of how on-screen text is indirectly shown in this comic series. It partially continues as a song parody through the title text but then trails off into a typical computer-complaint that he perhaps [[:Category:Cueball Computer Problems|may often have cause to make]].

The original lyrics rely upon typically nuanced rhymes, such as "do you" (or "do ya'") with "Hallelujah", and "fifth" with "lift", but fairly reliably rhymes "chord" with "Lord". In [[Randall]]'s version, it starts with "chord" and "word" which ''look'' like they should rhyme, but would be /kɔɹd/ {{w|International Phonetic Alphabet|vs.}} /wɝd/ in a typical US accent. Similarly "shift" and "left" might be considered not a {{w|Perfect and imperfect rhymes|perfect rhyme}} when read as prose, but should still be possible to meaningfully sing.

Technically, a {{w|Chorded_keyboard|chorded keyboard}} is one in which (nearly) all inputs are made by simultaneous pressing of a given combination of a limited number of keys, such as a literal handful of non-alphabetic keys, that the user learns to combine to represent the key-presses of more standard keyboards or (in some cases) signify entire phonemes or words. The workings of such a keyboard tends to be handled internally, sending to the computer the signal(s) that ''would'' have been sent from its larger cousin.

A big thing among Xennial hackers like Randall and his original audience was customising keyboard uses. The linux operating system was originally designed and used for personal customisation, and people move their configurations from system to system, often customising how things respond to such a degree that other users struggle to make use of their system at all. The first two major text editors, vim and emacs, were composed of different camps of how to efficiently type. The emacs camp believed it was more effective to hit many keys at once to accomplish a large task, but both editors were designed to be highly customisable. It's {{w|QWERTY#Properties|erroneously believed}} that the traditional qwerty keyboard was specifically designed to make typing inefficient so as to reduce engineering burden in making old typewriters responsive and reliable. Given the prevalence of them, it has been common among hackers to remap a keyboard to something they may personally consider more efficient, such as to use a {{w|dvorak keyboard layout|dvorak layout}} layout rather than a qwerty layout. Chorded configurations are an order of magnitude more efficient than the dvorak layout, but are more complex to configure because the result is not at all a one-to-one mapping. The {{w|Stenotype|traditional court reporting device}} is a chorded keyboard, to keep up with human speech.

Using a combination of normally single-use keys (the 'H' and a cursor) with others, including modifiers ('shift' and 'control'), i.e. 'chording' ''with'' his keyboard, is a kind of key combination found traditionally in emacs and operating system commands (such as pressing ctrl+alt+c, to copy a selection to clipboard). The ballad then comes across as an ode to system customisation and the practice of user-interface hacking, wherein a computer user knows how to rebuild their interface in almost any way they desire.

The chording example goes beyond mainstream use (shift and an alphabetic character changes the character case, whilst ctrl and a character may initiate an editing command) or mainstream multi-modifier combinations (ctrl, alt and the 'e' may result in the 'é', where the keyboard does not otherwise support it) and even goes beyond [[378: Real Programmers|emacs-like command ''sequences'']] which are generally software-specific. It seems likely that a setup such as that depicted in this comic is handled within the computer, either defined within the OS (all mainstream desktop operating systems support alternative keyboard mapping and customisable key-combinations, often for accessibility and international keyboard support), or (as is often the case with specialist configurable gaming keyboards) via the driver installed to mediate such esoteric keyboard combinations as the user has predefined for themselves.

Cueball's combination-keypress may in fact be better termed a 'macro', in some contexts. The single event, somehow triggered by this particular simultaneous multi-key input, invokes the injection of a pre-specified sequence of standard characters into the appropriate text-buffer/-stream, in lieu of manual per-character input.

The title text spoofs the last verse of the (original) song, with "Hallelujah" being replaced by Cueball trailing off musing about having apparently lost the backup of his keyboard configuration, implying that he ended up in a position where he would want to restore said backup (for instance, having tampered with it to the point he is no longer capable of operating the keyboard efficiently, if at all).

Here is the original verse, where the title text spoofs the last three lines:
:I did my best, it wasn't much
:I couldn't feel, so I tried to touch
:I've told the truth, I didn't come to fool ya
:And even though it all went wrong
:I'll stand before the lord of song
:With nothing on my tongue but hallelujah

As added irony, while in the original that verse is hopeful, with the singer being thankful for experiencing joy even from a relationship that ultimately failed, contrarily in the alt text Cueball is apparently expressing regret. Or, if taken literally, it could instead imply that God himself is questioning Cueball about his tampering with software, which could fit with the running gag of [[:Category:Cueball Computer Problems|Cueball's (often self-inflicted) computer problems being hyperbolically atrocious]].

When one modifies one's keyboard config, it can make the system seem unusable (or at least highly unexpected) to things like a boss, a spouse, or an automated maintenance system. When an error is made somewhere in the process, it can make the system seem unusable to the very person who made the changes, making it hard to change them back.

==Transcript==
:[Cueball is sitting in an office chair at his desk, typing on his keyboard as shown by small lines over one hand, while looking at the screen of his stationary computer. The screen is on a raised platform on his desk. Lyrical text is written upon each scene, presumably what Cueball is typing.]
:I heard there was a secret chord
:That David pressed and it typed a word

:[A closeup on Cueball in a slim panel. We see him from the waist up, with his hands on the keyboard just beneath the panels frame.]
:But you don't use a chorded keyboard, do you

:[Same setting as in the first panel, except Cueball's arms have moved and there are movement lines above and below his arms.]
:It goes like this, <control> and <shift>
:The other hand hits H and <left>

:[Slimmer panel but same setting as in the first panel, again the arms have moved a bit, with movement lines above them. The final written word of text is marked as arising directly from the computer.]
:And all at once it types out
:Computer: Hallelujah

{{comic discussion}}

[[Category:Comics featuring Cueball]]
[[Category:Songs]]
[[Category:Cueball Computer Problems]]