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==

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.

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.

:<math>E=K_{0}t+\frac{1}{2}\rho{}vt^2</math>
;All {{w|kinematics}} equations
Most kinematics equations are polynomials, usually at most quadratic, and are often integral quantities (corresponding to the 1/2 t^2 term). 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 {{w|number theory}} equations
Number theory is the study of the integers and their properties. Randall jokes about how this often involves the use of summations over integers. The subscripts are indices, but the use of ''&pi;'' as a summation index is a joke — ''&pi;'' is almost always used for the well-known constant 3.14159..., not a variable. The use of ''i'' as a summation variable '''is''' common, but the joke is that when it appears along with the constant ''e'', as it does here it usually represents the imaginary unit &radic;-1. As well, the constant ''e'' is out-of-place in a field like number theory — it's more closely related to fields like calculus/analysis. The use of the symbol <math>\infty</math> as a literal is also a joke.

:<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 {{w|fluid dynamics}} equations
Fluid dynamics equations often involve copious use of vector calculus operators. It's common in field theories like fluid dynamics and electromagnetism to represent equations of motion using multidimensional operators, like the <math>\nabla</math> and closed contour integral ∮ which appear here. 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 a joke and not allowed.

:<math>|\psi_{x,y}\rangle=A(\psi)A(|x\rangle\otimes|y\rangle)</math>
;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>
;All {{w|chemistry}} 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 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>SU(2)U(1)\times{}SU(U(2))</math>
;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, but an expression.

:[[File:All gauge theory equations.png]]
;All {{w|gauge theory}} equations
Gauge theory is a subset of quantum field theory. Because the objects in gauge theory carry a lot of symmetry information, they tend to have a lot of indices and twiddly annotations. Shorthand, such the {{w|Feynman slash}}, is also very common. Although most of the symbols used here do not have specific meanings in gauge theory, &xi; corresponds to the {{w|Gauge fixing#Rξ gauges|Rξ gauges}} in quantum electrodynamics, and the  "abc" superscript is reminiscent of a {{w|structure constant}}.

:<math>H(t)+\Omega+G\!\cdot\!\Lambda...\begin{cases}...>0\mathrm{\ (Hubble\ model)}\\
...=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}}.

:[[File:All truly deep physics equations.png]]
;All truly deep {{w|physics}} equations
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>
: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))]
: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> &#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>)
:All gauge theory equations

:H(t)+&Omega;+G&sdot;&Lambda; ... 
:[There is a brace linking the three cases together.]
:... > 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}}
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