explain xkcd - User contributions [en]

2142: Dangerous Fields

2019-04-28T08:57:04Z

Vog: Add examples for countries with a high number of executions

{{comic
| number = 2142
| date = April 26, 2019
| title = Dangerous Fields
| image = dangerous_fields.png
| titletext = Eventually, every epidemiologist becomes another statistic, a dedication to record-keeping which their colleagues sincerely appreciate.
}}

==Explanation==
{{incomplete|Created by an INEXORABLE PROCESS. Percentages needed to be added (like [[1895: Worrying Scientist Interviews]]). Do NOT delete this tag too soon.}}

This is a chart of "fields of study by danger", with mathematics being the least dangerous and gerontology being the most. Gerontology is shown as multiple times more dangerous than the other fields, so it is far on the right side of the graph. Generally speaking, the "study of ageing" does not seem likely to kill you, but approaching it philosophically, ageing is a cause of death.

This comic was posted the day after {{w|Joe Biden}} entered the race for the 2020 U.S. Presidential election, which is shaping up to feature the [https://www.cbc.ca/news/thenational/national-today-newsletter-american-politics-scarecrow-1.5107181 oldest set of candidates] in history.

===Fields===
*{{W|Mathematics}} is such a pure non-physical field that the probability of it being the direct cause of death is extremely low, barring workplace disputes or absent-mindedly wandering in front of traffic while pondering (as in xkcd [[356: Nerd Sniping]]).

*{{W|Astronomy}} mostly deals with extremely far-away things, so assuming there isn't a meteor impact, astronomy is probably not going to kill you. Astronomy is slightly more dangerous than mathematics though, since it studies physical objects instead of abstract concepts. In addition to meteor or asteroid impacts, astronomical phenomena that might cause death include nearby supernovas, distant magnetar quakes, a solar flare or solar nova (the likelihood of which will increase over the next billion-odd years), perturbations in earth's orbit, increased or decreased solar radiation, alien invasion, etc. Given that the density of magnetars and potentially hostile alien civilizations in the potentially lethal radius is (like the radius itself) completely unknown, and not all past mass extinctions are explained, this one might be misplaced a bit. The lethal stroke may be unlikely, in absolute terms, but most cut quite a broad swath.

*{{W|Economics}} is the study of markets, which through recessions and scarcity can kill you in any way that capitalism or other economic systems can affect the availability of goods and services you need to survive.

*{{W|Law}} in this context refers to the rules people have to follow in society, and given the nature of laws (civil and criminal), the odds that your death is related to law is low. Possible causes of death more-or-less directly related to the study of law would include attacks by someone you are prosecuting or defending, prosecution for a capital crime, persecution under legal authority (such as being shot or strangled by an officer of the law), attack by a guard or fellow prisoner, or for lack of medical treatment, while incarcerated, or death by exposure after expulsion from one's repossessed or otherwise legally confiscated home. Perhaps most ironically, a lawyer who committed a capital crime, lives in a country where capital punishment is not yet abolished (especially China or Iran), and was executed for it would be directly killed by the thing s/he studies.

*{{W|Criminology}} is very similar to law, but is the study of crime, meaning it's more dangerous than just "law." Criminologists may be directly involved with criminals in the course of their studies, increasing their exposure to potentially life-threatening behavior.

*{{W|Meteorology}} is the study of weather, and encountering powerful weather events such as hurricanes, tornadoes, and thunderstorms brings distinct possibility of injury and death. Curiosity to see a storm in person, or (if working for television news) exposing yourself to the weather event in order to file a report, may expose you to lightning, wind-blown projectiles, cold, etc. any of which can negatively affect your survival.

*{{W|Chemistry}} is the study of chemicals and reactions of those chemicals. Since everything in existence is made up of chemicals (and chemists often use especially reactive or dangerous chemicals), the likelihood of a chemist's death being caused by chemistry (e.g., explosions, poisoning, chemical burns, suffocation...) is not insignificant.

*{{W|Marine Biology}} is the study of marine life. Many marine creatures are venomous, many are very large, many are very hungry. Death could result from exposure to pathogenic bacteria, toxins (such as those produced by cone snails, and "red tide" dinoflagellates), allergies to shellfish, drowning (e.g. in strong ocean currents), scuba accidents, or water pollution, in addition to such perhaps more obvious (but overwhelmingly rarer) risks as shark attacks.

*{{w|Volcanology}} involves the study of {{w|volcanoes}}, {{w|lava}}, and {{w|magma}}, with obvious risks to the scientists studying them in the field. At least 67 scientists have been killed in volcanic eruptions, as of 2017 ("[https://cosmosmagazine.com/geoscience/volcanologists-lose-their-lives-in-pursuit-of-knowledge Volcanologists lose their lives in pursuit of knowledge]").

*{{w|Gerontology}} involves the study of aging, and of growing old in general. As everyone ages and eventually dies, those who study gerontology are not immune to dying in old age even if they evade all the other possible causes of death - thus making it the most likely among all shown fields. A gerontologist still can die from something else first, but without the inherent risk factors of other professions such as active volcanoes or underwater diving they're more likely to survive to retirement and thus meet their death of old age.

The title text is about {{w|Epidemiology}}: the study of health and disease conditions in populations. In the event of an epidemic, there is a strong chance that epidemiologists in the search for the causation, transmission and treatment will be exposed and become victims of the disease in their own right. However, the title text refers more broadly to the role of epidemiology in maintaining detailed statistical records of diseases and other causes of death, such that eventually any epidemiologist (whatever the cause of death) will become one of his/her own statistics.

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

:[A line chart is shown going from left to right with two arrows on either side. On the line are ten dots spread out unevenly from close to each end. The first four dots are clustered together on the left side. Then follows 5 more dots unevenly spaced, all to the left of center. On the far right of the line, near the end, there is one dot. Beneath each dot there goes a line down to a label written beneath each line. Above the chart there is a big title and below that an explanation. Below that again, there is a small arrow pointing to the right with a label above it.]
:<big>Probability that you'll be killed by the thing you study</big>
:By field

:[Arrow label:]
:More likely

:[Labels for the ten dots from left to right:]
:Mathematics
:Astronomy
:Economics
:Law
:Criminology
:Meteorology
:Chemistry
:Marine Biology
:Volcanology
:Gerontology

{{comic discussion}}

[[Category:Charts]]
[[Category:Rankings]]

2142: Dangerous Fields

2019-04-28T08:52:36Z

Vog: The ability for a lawyer to be directly killed by law depends on the country they live in.

{{comic
| number = 2142
| date = April 26, 2019
| title = Dangerous Fields
| image = dangerous_fields.png
| titletext = Eventually, every epidemiologist becomes another statistic, a dedication to record-keeping which their colleagues sincerely appreciate.
}}

==Explanation==
{{incomplete|Created by an INEXORABLE PROCESS. Percentages needed to be added (like [[1895: Worrying Scientist Interviews]]). Do NOT delete this tag too soon.}}

This is a chart of "fields of study by danger", with mathematics being the least dangerous and gerontology being the most. Gerontology is shown as multiple times more dangerous than the other fields, so it is far on the right side of the graph. Generally speaking, the "study of ageing" does not seem likely to kill you, but approaching it philosophically, ageing is a cause of death.

This comic was posted the day after {{w|Joe Biden}} entered the race for the 2020 U.S. Presidential election, which is shaping up to feature the [https://www.cbc.ca/news/thenational/national-today-newsletter-american-politics-scarecrow-1.5107181 oldest set of candidates] in history.

===Fields===
*{{W|Mathematics}} is such a pure non-physical field that the probability of it being the direct cause of death is extremely low, barring workplace disputes or absent-mindedly wandering in front of traffic while pondering (as in xkcd [[356: Nerd Sniping]]).

*{{W|Astronomy}} mostly deals with extremely far-away things, so assuming there isn't a meteor impact, astronomy is probably not going to kill you. Astronomy is slightly more dangerous than mathematics though, since it studies physical objects instead of abstract concepts. In addition to meteor or asteroid impacts, astronomical phenomena that might cause death include nearby supernovas, distant magnetar quakes, a solar flare or solar nova (the likelihood of which will increase over the next billion-odd years), perturbations in earth's orbit, increased or decreased solar radiation, alien invasion, etc. Given that the density of magnetars and potentially hostile alien civilizations in the potentially lethal radius is (like the radius itself) completely unknown, and not all past mass extinctions are explained, this one might be misplaced a bit. The lethal stroke may be unlikely, in absolute terms, but most cut quite a broad swath.

*{{W|Economics}} is the study of markets, which through recessions and scarcity can kill you in any way that capitalism or other economic systems can affect the availability of goods and services you need to survive.

*{{W|Law}} in this context refers to the rules people have to follow in society, and given the nature of laws (civil and criminal), the odds that your death is related to law is low. Possible causes of death more-or-less directly related to the study of law would include attacks by someone you are prosecuting or defending, prosecution for a capital crime, persecution under legal authority (such as being shot or strangled by an officer of the law), attack by a guard or fellow prisoner, or for lack of medical treatment, while incarcerated, or death by exposure after expulsion from one's repossessed or otherwise legally confiscated home. Perhaps most ironically, a lawyer who committed a capital crime, lives in a country where capital punishment is not yet abolished, and was executed for it would be directly killed by the thing s/he studies.

*{{W|Criminology}} is very similar to law, but is the study of crime, meaning it's more dangerous than just "law." Criminologists may be directly involved with criminals in the course of their studies, increasing their exposure to potentially life-threatening behavior.

*{{W|Meteorology}} is the study of weather, and encountering powerful weather events such as hurricanes, tornadoes, and thunderstorms brings distinct possibility of injury and death. Curiosity to see a storm in person, or (if working for television news) exposing yourself to the weather event in order to file a report, may expose you to lightning, wind-blown projectiles, cold, etc. any of which can negatively affect your survival.

*{{W|Chemistry}} is the study of chemicals and reactions of those chemicals. Since everything in existence is made up of chemicals (and chemists often use especially reactive or dangerous chemicals), the likelihood of a chemist's death being caused by chemistry (e.g., explosions, poisoning, chemical burns, suffocation...) is not insignificant.

*{{W|Marine Biology}} is the study of marine life. Many marine creatures are venomous, many are very large, many are very hungry. Death could result from exposure to pathogenic bacteria, toxins (such as those produced by cone snails, and "red tide" dinoflagellates), allergies to shellfish, drowning (e.g. in strong ocean currents), scuba accidents, or water pollution, in addition to such perhaps more obvious (but overwhelmingly rarer) risks as shark attacks.

*{{w|Volcanology}} involves the study of {{w|volcanoes}}, {{w|lava}}, and {{w|magma}}, with obvious risks to the scientists studying them in the field. At least 67 scientists have been killed in volcanic eruptions, as of 2017 ("[https://cosmosmagazine.com/geoscience/volcanologists-lose-their-lives-in-pursuit-of-knowledge Volcanologists lose their lives in pursuit of knowledge]").

*{{w|Gerontology}} involves the study of aging, and of growing old in general. As everyone ages and eventually dies, those who study gerontology are not immune to dying in old age even if they evade all the other possible causes of death - thus making it the most likely among all shown fields. A gerontologist still can die from something else first, but without the inherent risk factors of other professions such as active volcanoes or underwater diving they're more likely to survive to retirement and thus meet their death of old age.

The title text is about {{w|Epidemiology}}: the study of health and disease conditions in populations. In the event of an epidemic, there is a strong chance that epidemiologists in the search for the causation, transmission and treatment will be exposed and become victims of the disease in their own right. However, the title text refers more broadly to the role of epidemiology in maintaining detailed statistical records of diseases and other causes of death, such that eventually any epidemiologist (whatever the cause of death) will become one of his/her own statistics.

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

:[A line chart is shown going from left to right with two arrows on either side. On the line are ten dots spread out unevenly from close to each end. The first four dots are clustered together on the left side. Then follows 5 more dots unevenly spaced, all to the left of center. On the far right of the line, near the end, there is one dot. Beneath each dot there goes a line down to a label written beneath each line. Above the chart there is a big title and below that an explanation. Below that again, there is a small arrow pointing to the right with a label above it.]
:<big>Probability that you'll be killed by the thing you study</big>
:By field

:[Arrow label:]
:More likely

:[Labels for the ten dots from left to right:]
:Mathematics
:Astronomy
:Economics
:Law
:Criminology
:Meteorology
:Chemistry
:Marine Biology
:Volcanology
:Gerontology

{{comic discussion}}

[[Category:Charts]]
[[Category:Rankings]]

2117: Differentiation and Integration

2019-03-05T20:50:38Z

Vog: Remove comment "Created by a BESSEL FUNCTION?" which doesn't make any sense for the current article.

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

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-know-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)=x^a </math>, it follows that <math> \frac{d}{dx}f(x)=a\cdot x^{a-1} </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. 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://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" 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 w/ 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]]

2117: Differentiation and Integration

2019-03-04T14:11:36Z

Vog: /* 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==
{{incomplete|Created by a BESSEL FUNCTION? Please mention here why this explanation isn't complete. Do NOT delete this tag too soon.}}

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-know-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))*g'(x)</math>.

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

For any <math> f(x)=x^a </math>, it follows that <math> \frac{d}{dx}f(x)=a*x^{a-1} </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)g(x)-f(x)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)*g(x))=f'(x)*g(x)+f(x)*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. 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://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" 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 w/ 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]]