Editing 2034: Equations

{{comic
| number    = 2034
| date      = August 17, 2018
| title     = Equations
| image     = equations.png
| titletext = All electromagnetic equations: The same as all fluid dynamics equations, but with the 8 and 23 replaced with the permittivity and permeability of free space, respectively.
}}
==Explanation==
{{incomplete|TODO: some simplified explanations. Do NOT delete this tag too soon.}}

This comic gives a set of equations supposedly from different areas of science in mathematics, physics, and chemistry. To anyone not familiar with the field in question they look pretty similar to what you might find in research papers or on the relevant Wikipedia pages. To someone who knows even a little about the topic, they are clearly very wrong and only seem even worse the more you look at them.  In many disciplines, the mathematical description of a large area is summed up in a small number of equations, such as Maxwell's equations for electromagnetism.  In similar fashion, the equations here purport to encompass the whole of their given field.

===Simplified Explanations===

;All kinematics equations
Kinematics is the study of the motion of objects. More specifically, it describes how the location, velocity, and acceleration of an object vary over time. The equation shown contains two of these standard kinematic variables, velocity ''v'' and time ''t'', in addition to several quantities (''E'', ''K<sub>0</sub>'', and ''&rho;'') that are completely unrelated to kinematics.

;All number theory equations
Number theory is a branch of mathematics concerned primarily with the study of integers. However, the equation shown contains the non-integer Euler's constant ''e'' (approximately 2.718). It also uses the Greek letter &pi; as an integer-valued variable, even though the symbol &pi; is used in mathematics almost exclusively to denote the well-known ''non''-integer circle constant (approximately 3.14159).  Even with &pi; treated as a variable here, one of its uses in the equation is still unusual, to say the least. <math>\pi-\infty</math> is <math>-\infty</math>, so <math>e^{\pi-\infty}</math> is 0, making the <math>\pi</math> unnecessary.

;All chemistry equations
Randall implies that all chemistry is just combustion of chemicals, demonstrated with an incorrect form of a common example chemistry equation of burning Methane and Oxygen (with added heat), to form water and carbon dioxide. However, in this form "HEAT" is an actual molecule, rather than simply indicating the presence of heat to start the reaction. Thus the equation is modified to incorporate the fictional "HEAT" into the reaction. While the H in "HEAT" is the chemical symbol of the element hydrogen, none of the letters E, A, or T are symbols of any actual elements. Also, to account for the second hydrogen in "H(2)EAT" on the products side, the oxygen gas on the reactants side has been altered to be hydroxide, a strong base that would not facilitate traditional combustion.

TODO: other simplified explanations.

===Technical Explanations===
;All kinematics equations
:<math>E = K_0t + \frac{1}{2}\rho vt^2</math>
{{w|Kinematics}} describes the motion of objects without considering mass or forces. The latter is described by {{w|Kinetics (physics)|kinetics}}. The two fields get frequently confused due to the similarity of words.

This equation here literally states: "Energy equals a constant <math>K_0</math> multiplied by time, plus half of density multiplied by speed multiplied by time squared". 

The first term here is hard to interpret: it could be correct if <math>K_0</math> is a constant power applied to the system, but this symbol would more normally be used to denote an initial energy, in which case multiplying by <math>t</math> would be wrong. Alternatively, the term is similar to <math>k_B T</math> (sometimes written as ''kT''), a term that often appears in {{w|Statistical_mechanics|statistical mechanics}} equations, where ''k<sub>B</sub>'' (or ''k'') is {{w|Boltzmann_constant|the Boltzmann constant}}, and ''T'' is the {{w|Thermodynamic_temperature|absolute temperature}}. In this latter case, the term would have units of energy, consistent with the left side of the equation.

The second term looks similar to the kinetic energy term <math> \frac{1}{2}\rho v^2 </math> in [http://hyperphysics.phy-astr.gsu.edu/hbase/pber.html the Bernoulli equation] for fluids. (More properly, this is the kinetic energy ''density'' in the fluid). 

The whole equation appears to be a play on the kinematics formula: <math>s = v_0t + \frac{1}{2}\ at^2</math>, where distance travelled (''s'') by a constantly accelerating object is determined by initial velocity (''v<sub>0</sub>''), time (''t''), and acceleration (''a'')

Kinematics is often one of the first topics covered in an introductory physics course, both at the high school and freshman college levels. As such, mixing in material from more advanced topics like statistical mechanics and the Bernoulli equation, even if done correctly, would be very confusing for a typical student learning kinematics.

;All number theory equations
:<math>K_n = \sum_{i=0}^{\infty}\sum_{\pi=0}^{\infty}(n-\pi)(i+e^{\pi-\infty})</math>
{{w|Number theory}} is a branch of mathematics primarily studying the properties of integers.

Said in English, the equation can be read: "The ''n''th K-number is equal to the sum for all ''i'' from 0 to infinity of the sum for all &pi; from 0 to infinity of ''n'' minus &pi;, multiplied by ''i'' plus ''e'' raised to the power of &pi; minus infinity." (''i'' here is an iteration variable, not the imaginary number constant; ''e'' is Euler's number, approximately 2.718). A twofold misconception can be seen here. The first is the use of &pi; as a variable instead of the circle constant (3.14...). This might be a jab at how in number theory letters and numbers are used interchangeably, but where some letters are suddenly fixed constants.

Further confusion comes from the use of unusual mathematical models. While the term <math>e^{\pi-\infty}</math> is 0 when considered in standard ("high school") mathematics, it can be taken to be an infinitesimal in {{w|non-standard analysis}}, a concept unfamiliar to most non-mathematicians and uncommon in number theory. Naively, this would signify that (with the use of &pi; as a variable) the exponent would range from negative infinity to zero. In fact, assuming ''e'' really does mean Euler's constant (or at least a real number strictly greater than 1) the term would be zero for every π&nbsp;<&nbsp;∞. Ultimately, the sum diverges for every ''n''.

The close proximity of the letters i, e and π also evokes {{w|Euler's identity}} <math>e^{i\pi}+1=0</math> (also written <math>e^{i\pi}=-1</math>), without actually using it, especially since both π and i are used as variables here.

;All fluid dynamics equations
:<math>\frac{\partial}{\partial t}\nabla\cdot \rho = \frac{8}{23}
\int\!\!\!\!\!\!\!\!\!\;\;\bigcirc\!\!\!\!\!\!\!\!\!\;\;\int
\rho\,ds\,dt\cdot \rho\frac{\partial}{\partial\nabla}
</math>
{{w|Fluid dynamics}} describes the movement of non-solid material. In particular for gases, the density <math>\rho</math> is often the most interesting quantity (for liquids, this is often just constant). A unique feature of fluid-dynamic equations is the presence of {{w|Advection|advection terms}}, which take the form of often strange-looking spatial derivatives. This equation turns this up to a new level by differentiating with respect to a differential operator <math>\nabla</math>, which does not make any sense at all. Also it has a contour integral which seems reminiscent to a closed-circle process like in a piston engine, but this does not really fit in the context (differential description of a gas), and it has a pair of {{w|Magic number (programming)|unexplained numbers}} <math>8</math> and <math>23</math>, probably alluding to the {{w|Heat capacity ratio|specific heat ratio}} which is often written out as the fraction <math>\tfrac{7}{5}</math>, whereas most other physics equations [[899: Number Line|avoid including any plain numbers higher than 4]].

The title text stating that the electromagnetism equation is the same as the fluid dynamics equation, but with the arbitrary 8 and 23 replaced with the permittivity and permeability of free space is likely because electromagnetism equations often have relations to fluid dynamics, and because those two constants appear in the vast majority of electromagnetism equations.

;All quantum mechanics equations
:<math>|\psi_{x,y}\rangle = A(\psi) A(|x\rangle \otimes |y\rangle)</math>
{{w|Quantum mechanics}} is a fundamental theory in physics which describes nature at scales of atoms and below. It typically uses the {{w|Bra–ket notation|bra–ket notation}} in its formulae.

This equation takes a state psi in the dimensions of x and y and equates it to an operator A performed on psi multiplied by the same operator performed on the tensor product of x and y. Since the state psi is already the tensor product of the states x and y, this is equivalent to performing the same unknown operator twice on psi, and unless this operator is the identity or is its own inverse such as a bit-flip or Hermitian operator, this equation is therefore incorrect.

;All chemistry equations
:<math>\mathrm{CH}_4 + \mathrm{OH} + \mathrm{HEAT} \rightarrow \mathrm{H}_2\mathrm{O} + \mathrm{CH}_2 + \mathrm{H}_2 \mathrm{EAT}</math>
A {{w|Chemical equation|chemical equation}} represents a chemical reaction as a formula, with the reactant entities on the left-hand side, and the product entities on the right-hand side. The number of each element on the left side must match those on the right side. The energy produced or absorbed in this process is not included in that formula.

This is a modification of the combustion of methane. The correct form is often taught and a good example problem but obviously there are more chemistry problems.<math>\mathrm{HEAT}</math> is normally shorthand for {{w|activation energy}}, but in Randall's version it's jokingly used as a chemical ingredient and becomes <math>\mathrm{H}_2\mathrm{EAT}</math>, taking the hydrogen atom freed by the combustion equation shown. The proper methane combustion equation would be: <math>\mathrm{CH}_4 + 2 \mathrm{O}_2 \rightarrow 2 \mathrm{H}_2\mathrm{O} + \mathrm{CO}_2</math>

While <math>\mathrm{OH}</math> often appears in chemical equations in the form of a negatively charged hydroxide group (<math>\mathrm{OH}^-</math>), the left side of the equation involves a bare <math>\mathrm{OH}</math>, possibly the highly unstable hydroxyl radical (although this would typically be written with a leading dot, e.g. <math>\bullet\mathrm{OH}</math>). Similarly, the right side contains an unstable methylene radical which would generally only appear as an intermediate rather than a product.

;All quantum gravity equations
:<math>\mathrm{SU}(2)\mathrm{U}(1) \times \mathrm{SU}(\mathrm{U}(2))</math>
This is more similar to expressions which appear in {{w|Grand_Unified_Theory|Grand Unified Theory}} (GUT) than general quantum gravity. Unlike some of the other equations, this one has no interpretation which could make it mathematically correct. This is similar to the notations used to describe the symmetry group of a particular phenomena in terms of mathematical {{w|Lie_Group|Lie Groups}}. A real example would be the Standard Model of particle physics which has symmetry according to <math>\rm{SU(3)\times SU(2) \times U(1)}</math>. Here, <math>\rm{SU}</math> and <math>\rm{U}</math> denote the special unitary and unitary groups respectively with the numbers indicating the dimension of the group. Loosely, the three terms correspond to the symmetries of the strong force, weak force and electromagnetism although the exact correspondence is muddied by symmetry breaking and the Higgs mechanism.

Of course, an expression missing an "=" sign, is difficult to interpret as an "equation", because equations normally express an "equality" of some kind. Nobody knows whether Randal refers to a horse, zebra, donkey or other equine here. 

Randall's version clearly involves some similar groups although without the <math>\times</math> symbol it is hard to work out what might be happening. A term like <math>\rm{SU(U(2))}</math> has no current interpretation in mathematics, if anyone thinks otherwise and possibly has a solution to the quantum gravity problem they should probably get in touch with someone about that.

;All gauge theory equations
:[[File:All gauge theory equations.png]]
In physics, a {{w|Gauge theory|gauge theory}} is a type of field theory which is invariant to local transformations. The term gauge refers to any specific mathematical formalism to regulate redundant degrees of freedom.

This equation looks broadly similar to the sorts of things which appear in gauge theory such as the equations which define {{w|Yang–Mills_theory#Quantization|Yang-Mills Theory}}. By the time physics has got this far in, people have normally run out of regular symbols making a lot of the equations look very daunting. The actual equations in this field rarely go far beyond the Greek alphabet though and no-one has yet to try putting hats on brackets. The appearance of many sub- and superscripts is normal (this links to the group theory origins of these equations) and for the layperson it can be impossible to determine which additions are labels on the symbols and which are indices for an {{w|Einstein_notation|Einstein Sum}}.

The left-hand side <math>S_g</math> is the symbol for some {{w|Action_(physics)|action}}, in Yang-Mills theory this is actually used for a so-called "ghost action". On the right-hand side we have a large number of terms, most of which are hard to interpret without knowing Randall's thought processes (this is why real research papers should all label their equations thoroughly). The <math>\frac{1}{2\bar{\varepsilon}}</math> looks like a constant of proportionality which often appears in gauge theories. The factor of <math>i = \sqrt{-1}</math> is not unusual as many of these equations use complex numbers. The <math>\eth</math> symbol looks similar to a <math>\partial</math> partial derivative symbol especially as the {{w|Dirac_equation#Covariant_form_and_relativistic_invariance|Dirac Equation}} uses a slashed version as a convenient shorthand. 

The rest of the equation cannot be mathematically correct as the choice of indices used does not match that on the left-hand side (which has none). In particle physics subscripts (or superscripts) of greek letters (usually <math>\mu</math> or <math>\nu</math>) indicate terms which transform nicely under Lorentz transformations (special relativity). Roman indices from the beginning of the alphabet relate to various gauge transformation propetries, the triple index seen on <math>p^{abc}_v</math> would likely come from some <math>\rm{SU(3)}</math> transformation (related to the strong nuclear force). Since <math>S_g</math> has none of these (and is thus a scalar which remains constant under these operations), we would need the right-hand side to behave in the same way. Most of the indices which appear are unpaired and so will not result in a scalar making the equation very wrong. For those not familiar with this type of equation, this is similar to the mistake of messing up units, for instance setting a distance equal to a mass.

;All cosmology equations
:<math>H(t) + \Omega + G \cdot \Lambda \, \dots \begin{cases} \dots > 0 & \text{(Hubble model)} \\ \dots = 0 & \text{(Flat sphere model)} \\ \dots < 0  & \text{(Bright dark matter model)} \end{cases}
</math>
This is a parody of equations defining the {{w|Hubble's_law#Derivation_of_the_Hubble_parameter|Hubble Parameter}} <math>H(t)</math> although it looks like Randall has become bored and not bothered to finish his equation. Such equations usually have several <math>\Omega</math> terms representing the contributions of different substances to the energy-density of the Universe (matter, radiation, dark energy etc.). In this context <math>G</math> could be Newton's constant and <math>\Lambda</math> is the cosmological constant (energy density of empty space) although seeing them appear multiplied and on the same footing as <math>H</math> is unusual (the dot is entirely unnecessary). Choosing to make <math>H</math> a function of time <math>t</math> and not of redshift <math>z</math> is also unusual.

The second section looks like the inequalities used to show how the equation varies with the shape of the Universe, based on the value of the curvature parameter <math>\Omega_k</math>. A value of 0 indicates a flat Universe (this is more or less what we observe) while a positive /negative value indicates an open /closed curved Universe. Randall's choice of labels further makes fun of the field as both a flat sphere and bright dark matter are oxymoronic terms which would involve some rather strange model universes.

;All truly deep physics equations
:[[File:All truly deep physics equations.png]]
<math>\hat H</math> is the Hamiltonian operator, which when applied to a system returns the total energy. In this context, U would usually be the potential energy. However, there is also a subscript 0 and a diacritic marking indicating some other variable. Much of physics is based on Lagrangian and Hamiltonian mechanics. The Lagrangian is defined as <math>\hat L = \hat K - \hat U </math> with K being the kinetic energy and U the potential. Hamiltonian mechanics uses the equation <math>\hat H = \hat K + \hat U </math>. The Hamiltonian must be conserved so taking the time derivative and setting it equal to zero is a powerful tool. The "principle of least action" allows most modern physics to be derived by setting the time derivative of the Lagrangian to zero.

==Transcript==
:[Nine equations are listed, three in the top row and two in each of the next three rows. Below each equation there are labels:]

:E = K<sub>0</sub>t + 1/2 &rho;vt<sup>2</sup>
:All kinematics equations

:K<sub>n</sub> = &sum;<sub>i=0</sub><sup>&infin;</sup>&sum;<sub>&pi;=0</sub><sup>&infin;</sup>(n-&pi;)(i-e<sup>&pi;-&infin;</sup>)
:All number theory equations

:&#x2202;/&#x2202;t &nabla; &sdot; &rho; = 8/23 (&#x222F; &rho; ds dt &sdot; &rho; &#x2202;/&#x2202;&nabla;)
:All fluid dynamics equations

:|&psi;<sub>x,y</sub>&#x232a; = A(&psi;) A(|x&#x232a;&#x2297; |y&#x232a;)
:All quantum mechanics equations

:CH<sub>4</sub> + OH + HEAT &rarr; H<sub>2</sub>O + CH<sub>2</sub> + H<sub>2</sub>EAT
:All chemistry equations

:SU(2)U(1) &times; SU(U(2))
:All quantum gravity equations

:S<sub>g</sub> = (-1)/(2&epsilon;&#x0304;) i &eth; (&#x302; &xi;<sub>0</sub> +&#x030a; p<sub>&epsilon;</sub> &rho;<sub>v</sub><sup>abc</sup> &eta;<sub>0</sub> )&#x302; f&#x0335;<sub>a</sub><sup>0</sup> &lambda;(&#x0292;&#x0306;) &psi;(0<sub>a</sub>)
:All gauge theory equations

:[There is a brace linking the three cases together.]
:H(t) + &Omega; + G&sdot;&Lambda; ... 
:... > 0 (Hubble model)
:... = 0 (Flat sphere model)
:... < 0 (Bright dark matter model)
:All cosmology equations

:&#x0124; - u&#x0327;<sub>0</sub> = 0
:All truly deep physics equations

{{comic discussion}}

[[Category:Science]]
[[Category:Physics]]
[[Category:Math]]
[[Category:Chemistry]]
[[Category:Astronomy]]
@@ Line 7: / Line 7: @@
 }}
 ==Explanation==
+{{incomplete|TODO: some simplified explanations. Do NOT delete this tag too soon.}}
-This comic gives a set of mock equations. To anyone not familiar with the field in question they look pretty similar to what you might find in research papers or on the relevant Wikipedia pages. Most of the jokes are related to the symbols or "look" of most equations in the given field.
+This comic gives a set of equations supposedly from different areas of science in mathematics, physics, and chemistry. To anyone not familiar with the field in question they look pretty similar to what you might find in research papers or on the relevant Wikipedia pages. To someone who knows even a little about the topic, they are clearly very wrong and only seem even worse the more you look at them.  In many disciplines, the mathematical description of a large area is summed up in a small number of equations, such as Maxwell's equations for electromagnetism.  In similar fashion, the equations here purport to encompass the whole of their given field.
-The comic makes jokes about the fields of kinematics, number theory, fluid dynamics, quantum mechanics, chemistry, quantum gravity, gauge theory, cosmology, and physics equations. Of course, all of the equations listed are not real equations (<math>\pi-\infty</math> and H<sub>2</sub>EAT are clearly jokes and making a mockery of the given field). As always, Randall is just having a laugh.
+===Simplified Explanations===
-:<math>E=K_{0}t+\frac{1}{2}\rho{}vt^2</math>
+;All kinematics equations
-;All {{w|kinematics}} equations
+Kinematics is the study of the motion of objects. More specifically, it describes how the location, velocity, and acceleration of an object vary over time. The equation shown contains two of these standard kinematic variables, velocity ''v'' and time ''t'', in addition to several quantities (''E'', ''K<sub>0</sub>'', and ''&rho;'') that are completely unrelated to kinematics.
-Most kinematics equations tend to make heavy use of constants, addition, powers, and multiplication. This specific equation resembles the actual kinematics equation d = vt + 1/2at^2, but replaces a (acceleration) with v (velocity) times <math>\rho{}</math> (density) and replaces velocity with "K<sub>0</sub>", which is not a term used in kinematics.
-:<math>K_{n}=\sum_{i=0}^{\infty}\sum_{\pi=0}^{\infty}(n-\pi)(i+e^{\pi-\infty})</math>
+;All number theory equations
-;All {{w|number theory}} equations
+Number theory is a branch of mathematics concerned primarily with the study of integers. However, the equation shown contains the non-integer Euler's constant ''e'' (approximately 2.718). It also uses the Greek letter &pi; as an integer-valued variable, even though the symbol &pi; is used in mathematics almost exclusively to denote the well-known ''non''-integer circle constant (approximately 3.14159).  Even with &pi; treated as a variable here, one of its uses in the equation is still unusual, to say the least. <math>\pi-\infty</math> is <math>-\infty</math>, so <math>e^{\pi-\infty}</math> is 0, making the <math>\pi</math> unnecessary.
-Randall jokes about how number theory often involves the use of summations. The use of ''&pi;'' as an integer variable in the double summation is a joke, as ''&pi;'' is essentially always used for the well-known constant 3.14159..., not a variable. The use of ''i'' as a summation variable '''is''' common, though it can also be confused with the imaginary unit &radic;-1. The constants ''e'', ''i'', and ''&pi;'', as well as the theoretical upper bound <math>\infty</math>, often appear in number theory equations.
-:<math>\frac{\partial}{\partial{t}}\nabla\!\cdot\!\rho=\frac{8}{23}\int\!\!\!\!\int\!\!\!\!\!\!\!\!\!\subset\!\!\supset\rho\,{ds}\,{dt}\cdot{}\rho\frac{\partial}{\partial\nabla}</math>
+;All chemistry equations
-;All {{w|fluid dynamics}} equations
+Randall implies that all chemistry is just combustion of chemicals, demonstrated with an incorrect form of a common example chemistry equation of burning Methane and Oxygen (with added heat), to form water and carbon dioxide. However, in this form "HEAT" is an actual molecule, rather than simply indicating the presence of heat to start the reaction. Thus the equation is modified to incorporate the fictional "HEAT" into the reaction. While the H in "HEAT" is the chemical symbol of the element hydrogen, none of the letters E, A, or T are symbols of any actual elements. Also, to account for the second hydrogen in "H(2)EAT" on the products side, the oxygen gas on the reactants side has been altered to be hydroxide, a strong base that would not facilitate traditional combustion.
-Fluid dynamics equations often involve copious integrals, especially those over closed contours as done here, which are often the main telling factors of those equations to an outsider. The time derivative and gradient operator <math>\nabla</math> are common in fluid dynamics, mostly via the Navier-Stokes equation, and the fluid density <math>\rho</math> is one of the functions of central importance. The fraction 8/23 is a comically weird choice, but various unexpected fractions do pop up in fluid dynamics. The ds and dt go with the double contour integral (s is probably distance, t is time), but the derivative with respect to <math>\nabla</math> at the end is very much not allowed.
-:<math>|\psi_{x,y}\rangle=A(\psi)A(|x\rangle\otimes|y\rangle)</math>
+TODO: other simplified explanations.
-;All {{w|quantum mechanics}} equations
-Quantum mechanics often involves some of the foreign-looking symbols listed, including {{w|Bra–ket notation|bra-ket notation}}, the {{w|Tensor product|tensor product}}, and the Greek letter Psi for a quantum state. Specifically, the left side of the equation is a ket state labeled Psi that depends on x and y (probably positions), while the right-hand side may be an operator A that depends on the state Psi (it is very unusual to have such a dependence) acting on what looks like another copy of that operator which depends on the outer product of states labeled by x and y (again strange). A charitable interpretation could be that the second A is the eigenfunction A of the operator A. Normally this is clearly indicated by giving the operator a “hat” (^ symbol) or making the eigenfunction into a ket eigenstate, but since the equation is intentionally nonsense both A’s are left ambiguous. Also note that the bra-ket math is inconsistent here, as the left side is a ket, but the right side is just two A’s, which are either operators or functions but are definitely not kets.
-:<math>CH_4+OH+HEAT\rightarrow{}H_2O+CH_{2}+H_2EAT</math>
+===Technical Explanations===
-;All {{w|chemistry}} equations
+;All kinematics equations
-Chemistry equations use formulas of chemical compounds to describe a chemical reaction. Such equations show the starting chemicals on the left side and the resulting products on the right side, as displayed. Sometimes such an equation might optionally indicate that an {{w|activation energy}} is required, for the reaction to take place in a sensible timeframe, e.g. by heating. A reaction requiring heating is usually indicated by a Greek capital letter Delta (''&Delta;'') or a specified temperature above the reaction arrow, however this comic uses the "+ HEAT" term on the left side instead. The joke is that Randall interprets "HEAT" to be another chemical, possibly the nonsensical helium-monastatide, which reacts with Hydrogen (H) to H<sub>2</sub>EAT, which is nonsensical, as heat is transferred energy here, not added matter. Regardless of this, Randall gets the {{w|stoichiometry}} of this equation correct, with the same number of all types of 'atoms' on each side of the equation.
+:<math>E = K_0t + \frac{1}{2}\rho vt^2</math>
+{{w|Kinematics}} describes the motion of objects without considering mass or forces. The latter is described by {{w|Kinetics (physics)|kinetics}}. The two fields get frequently confused due to the similarity of words.
-:<math>SU(2)U(1)\times{}SU(U(2))</math>
+This equation here literally states: "Energy equals a constant <math>K_0</math> multiplied by time, plus half of density multiplied by speed multiplied by time squared".
-;All {{w|quantum gravity}} equations
-Quantum gravity uses mathematical {{w|Group (mathematics)|groups}} denoted by uppercase letters, as shown. {{w|Special unitary group|SU(2)}}, {{w|Unitary group|U(1)}}, and {{w|Unitary group|U(2)}} are all well-studied groups, though 'SU(U(2))' makes no sense.  The lack of relator means this expression isn't an equation.
+The first term here is hard to interpret: it could be correct if <math>K_0</math> is a constant power applied to the system, but this symbol would more normally be used to denote an initial energy, in which case multiplying by <math>t</math> would be wrong. Alternatively, the term is similar to <math>k_B T</math> (sometimes written as ''kT''), a term that often appears in {{w|Statistical_mechanics|statistical mechanics}} equations, where ''k<sub>B</sub>'' (or ''k'') is {{w|Boltzmann_constant|the Boltzmann constant}}, and ''T'' is the {{w|Thermodynamic_temperature|absolute temperature}}. In this latter case, the term would have units of energy, consistent with the left side of the equation.
+The second term looks similar to the kinetic energy term <math> \frac{1}{2}\rho v^2 </math> in [http://hyperphysics.phy-astr.gsu.edu/hbase/pber.html the Bernoulli equation] for fluids. (More properly, this is the kinetic energy ''density'' in the fluid).
+The whole equation appears to be a play on the kinematics formula: <math>s = v_0t + \frac{1}{2}\ at^2</math>, where distance travelled (''s'') by a constantly accelerating object is determined by initial velocity (''v<sub>0</sub>''), time (''t''), and acceleration (''a'')
+Kinematics is often one of the first topics covered in an introductory physics course, both at the high school and freshman college levels. As such, mixing in material from more advanced topics like statistical mechanics and the Bernoulli equation, even if done correctly, would be very confusing for a typical student learning kinematics.
+;All number theory equations
+:<math>K_n = \sum_{i=0}^{\infty}\sum_{\pi=0}^{\infty}(n-\pi)(i+e^{\pi-\infty})</math>
+{{w|Number theory}} is a branch of mathematics primarily studying the properties of integers.
+Said in English, the equation can be read: "The ''n''th K-number is equal to the sum for all ''i'' from 0 to infinity of the sum for all &pi; from 0 to infinity of ''n'' minus &pi;, multiplied by ''i'' plus ''e'' raised to the power of &pi; minus infinity." (''i'' here is an iteration variable, not the imaginary number constant; ''e'' is Euler's number, approximately 2.718). A twofold misconception can be seen here. The first is the use of &pi; as a variable instead of the circle constant (3.14...). This might be a jab at how in number theory letters and numbers are used interchangeably, but where some letters are suddenly fixed constants.
+Further confusion comes from the use of unusual mathematical models. While the term <math>e^{\pi-\infty}</math> is 0 when considered in standard ("high school") mathematics, it can be taken to be an infinitesimal in {{w|non-standard analysis}}, a concept unfamiliar to most non-mathematicians and uncommon in number theory. Naively, this would signify that (with the use of &pi; as a variable) the exponent would range from negative infinity to zero. In fact, assuming ''e'' really does mean Euler's constant (or at least a real number strictly greater than 1) the term would be zero for every π&nbsp;<&nbsp;∞. Ultimately, the sum diverges for every ''n''.
+The close proximity of the letters i, e and π also evokes {{w|Euler's identity}} <math>e^{i\pi}+1=0</math> (also written <math>e^{i\pi}=-1</math>), without actually using it, especially since both π and i are used as variables here.
+;All fluid dynamics equations
+:<math>\frac{\partial}{\partial t}\nabla\cdot \rho = \frac{8}{23}
+\int\!\!\!\!\!\!\!\!\!\;\;\bigcirc\!\!\!\!\!\!\!\!\!\;\;\int
+\rho\,ds\,dt\cdot \rho\frac{\partial}{\partial\nabla}
+</math>
+{{w|Fluid dynamics}} describes the movement of non-solid material. In particular for gases, the density <math>\rho</math> is often the most interesting quantity (for liquids, this is often just constant). A unique feature of fluid-dynamic equations is the presence of {{w|Advection|advection terms}}, which take the form of often strange-looking spatial derivatives. This equation turns this up to a new level by differentiating with respect to a differential operator <math>\nabla</math>, which does not make any sense at all. Also it has a contour integral which seems reminiscent to a closed-circle process like in a piston engine, but this does not really fit in the context (differential description of a gas), and it has a pair of {{w|Magic number (programming)|unexplained numbers}} <math>8</math> and <math>23</math>, probably alluding to the {{w|Heat capacity ratio|specific heat ratio}} which is often written out as the fraction <math>\tfrac{7}{5}</math>, whereas most other physics equations [[899: Number Line|avoid including any plain numbers higher than 4]].
+The title text stating that the electromagnetism equation is the same as the fluid dynamics equation, but with the arbitrary 8 and 23 replaced with the permittivity and permeability of free space is likely because electromagnetism equations often have relations to fluid dynamics, and because those two constants appear in the vast majority of electromagnetism equations.
+;All quantum mechanics equations
+:<math>|\psi_{x,y}\rangle = A(\psi) A(|x\rangle \otimes |y\rangle)</math>
+{{w|Quantum mechanics}} is a fundamental theory in physics which describes nature at scales of atoms and below. It typically uses the {{w|Bra–ket notation|bra–ket notation}} in its formulae.
+This equation takes a state psi in the dimensions of x and y and equates it to an operator A performed on psi multiplied by the same operator performed on the tensor product of x and y. Since the state psi is already the tensor product of the states x and y, this is equivalent to performing the same unknown operator twice on psi, and unless this operator is the identity or is its own inverse such as a bit-flip or Hermitian operator, this equation is therefore incorrect.
+;All chemistry equations
+:<math>\mathrm{CH}_4 + \mathrm{OH} + \mathrm{HEAT} \rightarrow \mathrm{H}_2\mathrm{O} + \mathrm{CH}_2 + \mathrm{H}_2 \mathrm{EAT}</math>
+A {{w|Chemical equation|chemical equation}} represents a chemical reaction as a formula, with the reactant entities on the left-hand side, and the product entities on the right-hand side. The number of each element on the left side must match those on the right side. The energy produced or absorbed in this process is not included in that formula.
+This is a modification of the combustion of methane. The correct form is often taught and a good example problem but obviously there are more chemistry problems.<math>\mathrm{HEAT}</math> is normally shorthand for {{w|activation energy}}, but in Randall's version it's jokingly used as a chemical ingredient and becomes <math>\mathrm{H}_2\mathrm{EAT}</math>, taking the hydrogen atom freed by the combustion equation shown. The proper methane combustion equation would be: <math>\mathrm{CH}_4 + 2 \mathrm{O}_2 \rightarrow 2 \mathrm{H}_2\mathrm{O} + \mathrm{CO}_2</math>
+While <math>\mathrm{OH}</math> often appears in chemical equations in the form of a negatively charged hydroxide group (<math>\mathrm{OH}^-</math>), the left side of the equation involves a bare <math>\mathrm{OH}</math>, possibly the highly unstable hydroxyl radical (although this would typically be written with a leading dot, e.g. <math>\bullet\mathrm{OH}</math>). Similarly, the right side contains an unstable methylene radical which would generally only appear as an intermediate rather than a product.
+;All quantum gravity equations
+:<math>\mathrm{SU}(2)\mathrm{U}(1) \times \mathrm{SU}(\mathrm{U}(2))</math>
+This is more similar to expressions which appear in {{w|Grand_Unified_Theory|Grand Unified Theory}} (GUT) than general quantum gravity. Unlike some of the other equations, this one has no interpretation which could make it mathematically correct. This is similar to the notations used to describe the symmetry group of a particular phenomena in terms of mathematical {{w|Lie_Group|Lie Groups}}. A real example would be the Standard Model of particle physics which has symmetry according to <math>\rm{SU(3)\times SU(2) \times U(1)}</math>. Here, <math>\rm{SU}</math> and <math>\rm{U}</math> denote the special unitary and unitary groups respectively with the numbers indicating the dimension of the group. Loosely, the three terms correspond to the symmetries of the strong force, weak force and electromagnetism although the exact correspondence is muddied by symmetry breaking and the Higgs mechanism.
+Of course, an expression missing an "=" sign, is difficult to interpret as an "equation", because equations normally express an "equality" of some kind. Nobody knows whether Randal refers to a horse, zebra, donkey or other equine here.
+Randall's version clearly involves some similar groups although without the <math>\times</math> symbol it is hard to work out what might be happening. A term like <math>\rm{SU(U(2))}</math> has no current interpretation in mathematics, if anyone thinks otherwise and possibly has a solution to the quantum gravity problem they should probably get in touch with someone about that.
+;All gauge theory equations
 :[[File:All gauge theory equations.png]]
-;All {{w|gauge theory}} equations
+In physics, a {{w|Gauge theory|gauge theory}} is a type of field theory which is invariant to local transformations. The term gauge refers to any specific mathematical formalism to regulate redundant degrees of freedom.
-Gauge theory is a subset of field theory. Most gauge theory equations appear to have many strange-looking constants and variables with odd labels. However, almost none of the symbols used here are found or applicable to gauge theory.
-:<math>H(t)+\Omega+G\!\cdot\!\Lambda...\begin{cases}...>0\mathrm{\ (Hubble\ model)}\\
+This equation looks broadly similar to the sorts of things which appear in gauge theory such as the equations which define {{w|Yang–Mills_theory#Quantization|Yang-Mills Theory}}. By the time physics has got this far in, people have normally run out of regular symbols making a lot of the equations look very daunting. The actual equations in this field rarely go far beyond the Greek alphabet though and no-one has yet to try putting hats on brackets. The appearance of many sub- and superscripts is normal (this links to the group theory origins of these equations) and for the layperson it can be impossible to determine which additions are labels on the symbols and which are indices for an {{w|Einstein_notation|Einstein Sum}}.
-...=0\mathrm{\ (Flat\ sphere\ model)}\\
-...<0\mathrm{\ (Bright\ dark\ matter\ model)}
-\end{cases}</math>
-;All {{w|cosmology}} equations
-Cosmology is the science of the development and ultimate fate of the universe. The joke here may be pertaining to the different models accepted in the field of cosmology. H is the {{w|Hubble's law#Time-dependence of Hubble parameter|Hubble parameter}}, &Omega; is the universal {{w|Friedmann equations#Density parameter|density parameter}}, G is the {{w|gravitational constant}}, and &Lambda; is the {{w|cosmological constant}}.
+The left-hand side <math>S_g</math> is the symbol for some {{w|Action_(physics)|action}}, in Yang-Mills theory this is actually used for a so-called "ghost action". On the right-hand side we have a large number of terms, most of which are hard to interpret without knowing Randall's thought processes (this is why real research papers should all label their equations thoroughly). The <math>\frac{1}{2\bar{\varepsilon}}</math> looks like a constant of proportionality which often appears in gauge theories. The factor of <math>i = \sqrt{-1}</math> is not unusual as many of these equations use complex numbers. The <math>\eth</math> symbol looks similar to a <math>\partial</math> partial derivative symbol especially as the {{w|Dirac_equation#Covariant_form_and_relativistic_invariance|Dirac Equation}} uses a slashed version as a convenient shorthand.
+The rest of the equation cannot be mathematically correct as the choice of indices used does not match that on the left-hand side (which has none). In particle physics subscripts (or superscripts) of greek letters (usually <math>\mu</math> or <math>\nu</math>) indicate terms which transform nicely under Lorentz transformations (special relativity). Roman indices from the beginning of the alphabet relate to various gauge transformation propetries, the triple index seen on <math>p^{abc}_v</math> would likely come from some <math>\rm{SU(3)}</math> transformation (related to the strong nuclear force). Since <math>S_g</math> has none of these (and is thus a scalar which remains constant under these operations), we would need the right-hand side to behave in the same way. Most of the indices which appear are unpaired and so will not result in a scalar making the equation very wrong. For those not familiar with this type of equation, this is similar to the mistake of messing up units, for instance setting a distance equal to a mass.
+;All cosmology equations
+:<math>H(t) + \Omega + G \cdot \Lambda \, \dots \begin{cases} \dots > 0 & \text{(Hubble model)} \\ \dots = 0 & \text{(Flat sphere model)} \\ \dots < 0  & \text{(Bright dark matter model)} \end{cases}
+</math>
+This is a parody of equations defining the {{w|Hubble's_law#Derivation_of_the_Hubble_parameter|Hubble Parameter}} <math>H(t)</math> although it looks like Randall has become bored and not bothered to finish his equation. Such equations usually have several <math>\Omega</math> terms representing the contributions of different substances to the energy-density of the Universe (matter, radiation, dark energy etc.). In this context <math>G</math> could be Newton's constant and <math>\Lambda</math> is the cosmological constant (energy density of empty space) although seeing them appear multiplied and on the same footing as <math>H</math> is unusual (the dot is entirely unnecessary). Choosing to make <math>H</math> a function of time <math>t</math> and not of redshift <math>z</math> is also unusual.
+The second section looks like the inequalities used to show how the equation varies with the shape of the Universe, based on the value of the curvature parameter <math>\Omega_k</math>. A value of 0 indicates a flat Universe (this is more or less what we observe) while a positive /negative value indicates an open /closed curved Universe. Randall's choice of labels further makes fun of the field as both a flat sphere and bright dark matter are oxymoronic terms which would involve some rather strange model universes.
+;All truly deep physics equations
 :[[File:All truly deep physics equations.png]]
-;All truly deep {{w|physics}} equations
+<math>\hat H</math> is the Hamiltonian operator, which when applied to a system returns the total energy. In this context, U would usually be the potential energy. However, there is also a subscript 0 and a diacritic marking indicating some other variable. Much of physics is based on Lagrangian and Hamiltonian mechanics. The Lagrangian is defined as <math>\hat L = \hat K - \hat U </math> with K being the kinetic energy and U the potential. Hamiltonian mechanics uses the equation <math>\hat H = \hat K + \hat U </math>. The Hamiltonian must be conserved so taking the time derivative and setting it equal to zero is a powerful tool. The "principle of least action" allows most modern physics to be derived by setting the time derivative of the Lagrangian to zero.
-The joke about the "truly deep physics equations" is that most of the universal physics equations are simple, almost exceedingly so. In general, many of these equations are types of [https://en.wikipedia.org/wiki/Conservation_law conservation law] equations, which reflect some of the basic truths of the universe. A hallmark of conservation laws is that the total amount of some physical value does not change, and so one side of the equation is zero, as shown in the example. One example is Einstein's ''E = mc²'', which shows conservation of mass-energy. Noether's theorem shows that conservation laws have a one-to-one correspondence with a symmetry of nature, making these equations truly 'deep'.
-The title text is referencing the fact that the {{w|magnetic field|electric and magnetic fields}} are often explained to physics students using an analogy with fluid dynamics, as well as the fact that they do share some similarities (only in terms of mathematical description as three-dimensional vector fields) with fluids. The permittivity constant (represented with ''&epsilon;''<sub>0</sub>) and the permeability constant (represented with ''&mu;''<sub>0</sub>) are coefficients that relate the amount of charge required to cause a specific amount of electric flux in a vacuum and the ability of vacuum to support the formation of magnetic fields, respectively. They appear frequently in Maxwell's equations (the equations that define the electric and magnetic fields in classical mechanics), so Randall is making the joke that any surface integral with them in it automatically is an electromagnetism equation.
 ==Transcript==
 :[Nine equations are listed, three in the top row and two in each of the next three rows. Below each equation there are labels:]
-:E=K<sub>0</sub>t+1/2 &rho;vt<sup>2</sup>
+:E = K<sub>0</sub>t + 1/2 &rho;vt<sup>2</sup>
 :All kinematics equations
-:K<sub>n</sub>=&sum;<sup>&infin;</sup><sub>i=0</sub>&sum;<sup>&infin;</sup><sub>&pi;=0</sub>(n-&pi;)(i-e<sup>&pi;-&infin;</sup>) [K sub n = the summation from i = 0 to infinity of the sum from pi = 0 to infinity of (n - pi) * (i-e^(pi-infinity))]
+:K<sub>n</sub> = &sum;<sub>i=0</sub><sup>&infin;</sup>&sum;<sub>&pi;=0</sub><sup>&infin;</sup>(n-&pi;)(i-e<sup>&pi;-&infin;</sup>)
 :All number theory equations
-:&#x2202;/&#x2202;t &nabla;&sdot;&rho;=8/23 (&#x222F; &rho; ds dt &sdot; &rho; &#x2202;/&#x2202;&nabla;)
+:&#x2202;/&#x2202;t &nabla; &sdot; &rho; = 8/23 (&#x222F; &rho; ds dt &sdot; &rho; &#x2202;/&#x2202;&nabla;)
 :All fluid dynamics equations
-:|&psi;<sub>x,y</sub>&#x232a;=A(&psi;)A(|x&#x232a;&#x2297;|y&#x232a;)
+:|&psi;<sub>x,y</sub>&#x232a; = A(&psi;) A(|x&#x232a;&#x2297; |y&#x232a;)
 :All quantum mechanics equations
-:CH<sub>4</sub>+OH+HEAT&rarr;H<sub>2</sub>O+CH<sub>2</sub>+H<sub>2</sub>EAT
+:CH<sub>4</sub> + OH + HEAT &rarr; H<sub>2</sub>O + CH<sub>2</sub> + H<sub>2</sub>EAT
 :All chemistry equations
-:SU(2)U(1)&times;SU(U(2))
+:SU(2)U(1) &times; SU(U(2))
 :All quantum gravity equations
-:S<sub>g</sub>=(-1)/(2&epsilon;&#x0304;) i&eth;(&#x302; &xi;<sub>0</sub> &#x2a22; p<sub>&epsilon;</sub> &rho;<sub>v</sub><sup>abc</sup>&sdot;&eta;<sub>0</sub>)&#x302; f&#x0335;<sub>a</sub><sup>0</sup> &lambda;(<span style="display:inline-block; -ms-transform:rotate(180deg); -webkit-transform:rotate(180deg); transform:rotate(180deg);">&xi;</span>) &psi;(0<sub>a</sub>)
+:S<sub>g</sub> = (-1)/(2&epsilon;&#x0304;) i &eth; (&#x302; &xi;<sub>0</sub> +&#x030a; p<sub>&epsilon;</sub> &rho;<sub>v</sub><sup>abc</sup> &eta;<sub>0</sub> )&#x302; f&#x0335;<sub>a</sub><sup>0</sup> &lambda;(&#x0292;&#x0306;) &psi;(0<sub>a</sub>)
 :All gauge theory equations
-:H(t)+&Omega;+G&sdot;&Lambda; ...
 :[There is a brace linking the three cases together.]
+:H(t) + &Omega; + G&sdot;&Lambda; ...
 :... > 0 (Hubble model)
 :... = 0 (Flat sphere model)
@@ Line 93: / Line 141: @@
 [[Category:Math]]
 [[Category:Chemistry]]
-[[Category:Cosmology]]
+[[Category:Astronomy]]