explain xkcd - User contributions [en]

Talk:2495: Universal Seat Belt

2021-07-30T02:22:11Z

172.68.133.156:

Looks like Randall has started a new series: Cursed Connectors. [[Special:Contributions/172.69.34.171|172.69.34.171]] 01:51, 29 July 2021 (UTC)
:I now await the 10Base2 connector with ''actual'' bayonet blade attached... [[Special:Contributions/141.101.99.183|141.101.99.183]] 03:27, 29 July 2021 (UTC)
::I have now created the series category: [[:Category:Cursed Connectors]]. Looking forward to see how many and for how long he will continue this series. The Bad Map Projection series continued recently after a long break. --[[User:Kynde|Kynde]] ([[User talk:Kynde|talk]]) 08:05, 29 July 2021 (UTC)

Dangit! Its the wrong way around... Wait, it doesn't fit this way either... [[Special:Contributions/172.70.51.134|172.70.51.134]] 01:56, 29 July 2021 (UTC)
: Oh noes. [[User:"iLB"|"iLB"]] ([[User talk:"iLB"|talk]]) 03:50, 29 July 2021 (UTC)
: Yeah, this is a nightmare. If you forget (or ignore) your seatbelt, it'll take 3 tries (with flipping) to get it to connect. You'll either have crashed or be ticketed by then. [[Special:Contributions/162.158.126.147|162.158.126.147]] 04:58, 29 July 2021 (UTC)
:: It teaches you to buckle up _before_ starting the engine. I don't see a problem with that. [[User:Elektrizikekswerk|Elektrizikekswerk]] ([[User talk:Elektrizikekswerk|talk]]) 08:23, 29 July 2021 (UTC)
::: If you insert the belt before starting up, it gets detected (or not) but without fuss. Only if you insert after you're running do you get "Device inserted", "Device not recognised" or occasionally that sickening mid-point between the two where you get neither because it recognises as a drive, but the drive bit behind it is now RAW instead of FAT/whatever. [[Special:Contributions/141.101.99.29|141.101.99.29]] 09:10, 29 July 2021 (UTC)
:::: Or, possibly, a [https://www.youtube.com/watch?v=IW7Rqwwth84 BSOD]. [[User:Dansiman|Dansiman]] ([[User talk:Dansiman|talk]]) 20:50, 29 July 2021 (UTC)
::: And creates a new market for "bypass chips". I'll sell you one for the low, low price of $100 :-) [[Special:Contributions/172.68.133.156|172.68.133.156]] 02:22, 30 July 2021 (UTC)

2492: Commonly Mispronounced Equations

2021-07-22T01:55:08Z

172.68.133.156: /* Transcript */ math

{{comic
| number = 2492
| date = July 21, 2021
| title = Commonly Mispronounced Equations
| image = commonly_mispronounced_equations.png
| titletext = "Epsihootamoo doopsiquorps" --the Schrödinger equation for the hydrogen atom
}}

==Explanation==
{{incomplete|Created by a MISSAID EQUATION. Please mention here why this explanation isn't complete. Do NOT delete this tag too soon.}}

==Transcript==
{{incomplete transcript|Do NOT delete this tag too soon.}}

{{comic discussion}}
[[Category:Physics]]
[[Category:Math]]
[[Category: Language]]

2491: Immune Factory

2021-07-20T07:14:02Z

172.68.133.156:

{{comic
| number = 2491
| date = July 20, 2021
| title = Immune Factory
| image = immune_factory.png
| titletext = In the final vote, the doubters were won over by the strength of the name IMMUNION.
}}

==Explanation==
{{incomplete|Created by an IMMUNION. Please mention here why this explanation isn't complete. Do NOT delete this tag too soon.}}

[[Hairy]] has received the first dose of a COVID-19 vaccine, and is now feeling unwell. He and [[Cueball]] make comments that metaphorically compare Hairy's body to a workplace.

Vaccines in general work by giving the body's immune system a chance to respond to a pathogen without actually being infected. The immune system responds by producing antibodies, proteins customised to attach to the pathogen, either disabling it directly or marking it for attack by immune cells. After the vaccine (or after an actual illness), the {{w|Immunological memory|immune system remembers}} how to make the antibodies and can more quickly respond to future infections. This is why Hairy describes his body as an "antibody factory".

However, many common symptoms of illness (such as fever, soreness, diarrhea and nausea) are actually caused by the body's immune response rather than the infection itself. As a result, vaccines can result in similar symptoms to an illness, albeit milder and of shorter duration.

Hairy extends the "body as factory" metaphor by complaining that, since he feels unwell, the factory must be violating {{w|Occupational Safety and Health Administration|OSHA}} regulations—that is, rules that protect workers from unsafe work conditions. Hairy says his {{w|lymphatic system}} (a major component of the immune system) is protesting the "brutal" work of responding to the vaccine, as human workers might protest a dangerous workplace.

In real workplaces, one possible response to worker dissatisfaction is for them to {{w|Trade union|unionize}}, forming an organization that can use their solidarity to bargain for improvements to working conditions. Hairy says that this is what his immune cells have done. It is not clear whether this corresponds to any actual part of the immune response, or whether it is simply a humorous expansion on the "factory" metaphor.

Cueball uses the "union" statement to set up a pun on two meanings of the word "scab". If unions make demands that an employer refuses, their workers may {{w|Strike action|strike}}, or refuse to work. Employers may keep the workplace running by hiring {{w|strikebreaker}}s, non-union workers (or union workers who break ranks with their colleagues). Union members may refer to strikebreakers by the pejorative term "scabs".

Another meaning of "scab" is the hard coating the body produces to cover a wound while it heals. {{w|Smallpox}} is a dangerous illness that causes ulcers on the skin, leading to many small scabs forming as the ulcers heal. Prior to modern vaccination techniques, people were sometimes deliberately infected with smallpox—typically from a person with a mild case—while they were healthy. This process, now called {{w|variolation}} (after ''Variola'', the virus that causes smallpox), could be done in various ways, but one was to {{w|Insufflation (medicine)|insufflate}} (blow up their nose) the powdered scabs of a person who had been sick.

The pun therefore is that members of the immune system union would not like ''either'' kind of scab.

==Transcript==
{{incomplete transcript|Do NOT delete this tag too soon.}}

:[Cueball walks in from the left, into a room where Hairy is sitting in a chair facing away, sick. Hairy is wrapped in a blanket and holding a steaming mug.]
:Cueball: I guess the first shot made your body build defenses, and now it's ramping up production.
:Hairy: So I've become an antibody factory.

:[In the next panel, Cueball is now facing Hairy on the right.]
:Hairy: I don't feel great. I think my factory has some OSHA violations.
:Hairy: My lymphatic system is protesting brutal working conditions.

:[In a frame-less panel, Cueball continues to stand in front of Hairy; Hairy's mug is steaming less.]
:Hairy: Update: my immune cells have unionized.
:Cueball: Common side effect. Helps maintain a healthy balance.

:[In a panel with a frame, Hairy's mug is no longer steaming; Cueball has his hand raised and Hairy is pointing in Cueball's direction]
:Cueball: Immune system unions are actually why we stopped doing variolation.
:Hairy: Oh? Why?
:Cueball: They don't like scabs.
:Hairy: Ugh. ''Leave.''

{{comic discussion}}

[[Category:Comics featuring Cueball]]
[[Category:Comics featuring Hairy]]
[[Category:COVID-19 vaccine]]

2117: Differentiation and Integration

2021-07-17T04:52:33Z

172.68.133.156: /* Explanation */

{{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 <math> f(x)=g(x)^a </math> and <math> \frac{d}{dx}g(x)=g'(x) </math>, it follows that <math> \frac{d}{dx}f(x)=a\cdot g(x)^{a-1}\cdot g'(x) </math>.

'''{{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 <math> d(f(u)) = (df(u))du</math>, it follows that <math>f(u) = \int df(u) du</math>. By finding appropriate values for functions <math>f, u</math> such that your problem is in the form <math>\int df(u) du</math> 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. <math>\sum_{i=0}^{n-1} f(t_i) \left(x_{i+1}-x_i\right).</math> Elementary textbooks on calculus sometimes present finding a definite integral as a process of approximating an area by strips of equal width 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 <math> x^2 \frac{dy^2}{dx^2}+x \frac{dy}{dx}+(x^2-n^2)*y=0</math>, where n is the order of 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. 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 that the negative consequences of a persecution can only be avoided by destroying the evidence.

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

Symbolic algebra 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==
{{incomplete transcript|Do NOT delete this tag too soon.}}
:[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]]