Inference rule list: Physics Derivation Graph

List of Inference Rules

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name	latex	input count	feed count	output count	used in derivation
declare assumption	Eq.~\ref{eq:#1} is an assumption.	0	0	1	upper limit on velocity in condensed matter Lorentz transformation speed of Earth around Sun quantum basics Hermitian operators have realvalued observables radius for satellite in geostationary orbit Newton's Law of Gravitation angle of maximum distance for projectile motion equations of motion in 2D (calculus) Maxwell equations to electric field wave equation escape velocity Kepler's Third Law: period squared propto distance cubed Langmuir Adsorption
indefinite integrate RHS over	Indefinite integral of RHS of Eq.~\ref{eq:#2} over $#1$; yields Eq.~\ref{eq:#3}.	1	1	1	Euler equation proof
indefinite integrate LHS over	Indefinite integral of LHS of Eq.~\ref{eq:#2} over $#1$; yields Eq.~\ref{eq:#3}.	1	1	1
X dot both sides	Take inner product of $#1$ with Eq.~\ref{eq:#2}; yields Eq.~\ref{eq:#3}.	1	1	1
expand magnitude to conjugate	Expand $#1$ in Eq.~\ref{eq:#2} with conjugate; yields Eq.~\ref{eq:#3}.	1	1	1	particle in a 1D box
multiply LHS by unity	Multiply LHS of Eq.~\ref{eq:#2} by 1, which in this case is $#1$; yields Eq.~\ref{eq:#3}	1	1	1
multiply both sides by	Multiply both sides of Eq.~\ref{eq:#2} by $#1$; yields Eq.~\ref{eq:#3}.	1	1	1	hyperbolic trigonometric identities equation of motion for a spring upper limit on velocity in condensed matter Lorentz transformation double intensity when phase is coherent (optics) mass of the Earth first law of thermodynamics velocity at distance r of object dropped from infinity time invariant force conserves energy identity sin(2 x) = 2 sin(x) cos(x) using Euler's equation equations of motion in 1D with constant acceleration - SUVAT (algebra) radius for satellite in geostationary orbit Newton's Law of Gravitation particle in a 1D box angle of maximum distance for projectile motion frequency relations equations of motion in 2D (calculus) derivation of Schrodinger Equation Schwarzschild radius for non-rotating black hole Euler equation: trigonometric relations escape velocity Euler equation proof frequency and period Kepler's Third Law: period squared propto distance cubed
select real parts	Select real parts of Eq.~\ref{eq:#1}; yields Eq.~\ref{eq:#2}.	1	0	1	Euler equation: trig square root
replace scalar with vector	Replace scalar variables in Eq.~\ref{eq:#1} with equivalent vector variables; yields Eq.~\ref{eq:#2}.	1	0	1	derivation of Schrodinger Equation
subtract expr 1 from expr 2	Subtract Eq.~\ref{eq:#1} from Eq.~\ref{eq:#2}; yields Eq.~\ref{eq:#2}.	2	0	1	hyperbolic trigonometric identities time invariant force conserves energy
change three variables in expression	Change of variable $#1$ to $#2$ and $#3$ to $#4$ and $#5$ to $#6$ in Eq.~\ref{eq:#7}; yields Eq.~\ref{eq:#8}.	1	6	1	mass of the Earth speed of Earth around Sun time invariant force conserves energy velocity at distance r of object dropped from infinity radius for satellite in geostationary orbit work and force and energy
declare guess solution	Judicious choice as a guessed solution to Eq.~\ref{eq:#1} is Eq.~\ref{eq:#2},	1	0	1	equation of motion for a spring Newton's Law of Gravitation particle in a 1D box electric field wave equation: from time dependent to time independent
add zero to LHS	Add zero to LHS of Eq.~\ref{eq:#2}, where $0=#1$; yields Eq.~\ref{eq:#3}.	1	1	1
substitute LHS of three expressions into expression	Substitute LHS of Eq.~\ref{eq:#1} and LHS of Eq.~\ref{eq:#2} and LHS of Eq.~\ref{eq:#3} into Eq.~\ref{eq:#4}; yields Eq.~\ref{eq:#5}.	4	0	1	total electrical resistance for circuit with two resistors in parallel total electrical resistance for circuit with two resistors in series
multiply expr 1 by expr 2	Multiply Eq.~\ref{eq:#1} by Eq.~\ref{eq:#2}; yields Eq.~\ref{eq:#3}.	2	0	1	hyperbolic trigonometric identities double intensity when phase is coherent (optics) identity sin(2 x) = 2 sin(x) cos(x) using Euler's equation Compton's equation for scattering Euler equation: trig square root
factor out X from RHS	Factor $#1$ from the RHS of Eq.~\ref{eq:#2}; yields Eq.~\ref{eq:#3}.	1	1	1	Euler equation proof
swap LHS with RHS	Swap LHS of Eq.~\ref{eq:#1} with RHS; yields Eq.~\ref{eq:#2}.	1	0	1	Lorentz transformation Compton's equation for scattering equations of motion in 1D with constant acceleration - SUVAT (algebra) particle in a 1D box integration by parts equations of motion in 2D (calculus) Euler equation: trigonometric relations escape velocity Euler equation: trig square root Euler equation proof
separate two vector components	Separate two vector components in Eq.~\ref{eq:#1}; yields Eq.~\ref{eq:#2} and Eq.~\ref{eq:#3}	1	0	2	equations of motion in 2D (calculus)
boundary condition	Boundary condition: Eq.~\ref{eq:#2} when Eq.~\ref{eq#1}.	1	0	1	angle of maximum distance for projectile motion
subtract X from both sides	Subtract $#1$ from both sides of Eq.~\ref{eq:#2}; yields Eq.~\ref{eq:#3}.	1	1	1	Lorentz transformation optics: Law of refraction to Brewster's angle Compton's equation for scattering equations of motion in 1D with constant acceleration - SUVAT (algebra) quantum basics orthogonality quadratic equation derivation integration by parts angle of maximum distance for projectile motion Euler equation: trig square root
separate vector into two trigonometric ratios	Separate vector in Eq.~\ref{eq:#2} into components related by angle $#1$; yields Eq.~\ref{eq:#3} and Eq.~\ref{eq:#4}.	1	1	2	equations of motion in 2D (calculus)
declare identity	Eq.~\ref{eq:#1} is an identity.	0	0	1	optics: Law of refraction to Brewster's angle variance relation particle in a 1D box integration by parts curl curl identity Maxwell equations to electric field wave equation Euler equation: trig square root
function is even	$#1$ is even with respect to $#2$, so replace $#1$ with $#3$ in Eq.~\ref{eq:#4}; yields Eq.~\ref{eq:#5}.	1	3	1	Euler equation: trigonometric relations
declare final expression	Eq.~\ref{eq:#1} is one of the final equations.	1	0	0	Newton's Second Law of Motion: vector to scalar hyperbolic trigonometric identities total electrical resistance for circuit with two resistors in parallel equation of motion for a spring upper limit on velocity in condensed matter Lorentz transformation double intensity when phase is coherent (optics) optics: Law of refraction to Brewster's angle speed of Earth around Sun coefficient of isothermal compressibility using the equation of state for an ideal gas Euler equation to exp(i pi) + 1 = 0 identity sin(2 x) = 2 sin(x) cos(x) using Euler's equation velocity at distance r of object dropped from infinity equations of motion in 1D with constant acceleration - SUVAT (algebra) quantum basics orthogonality quantum basics Hermitian operators have realvalued observables quadratic equation derivation particle in a 1D box Newton's Law of Gravitation angle of maximum distance for projectile motion integration by parts frequency relations electric field wave equation: from time dependent to time independent derivation of Schrodinger Equation equations of motion in 2D (calculus) coefficient of thermal expansion using the equation of state for an ideal gas Maxwell equations to electric field wave equation escape velocity Euler equation: trigonometric relations work and force and energy projectile path in 2D is parabolic time invariant force conserves energy total electrical resistance for circuit with two resistors in series Euler equation: trig square root Euler equation proof frequency and period Kepler's Third Law: period squared propto distance cubed Langmuir Adsorption
indefinite integral over	Indefinite integral of both sides of Eq.~\ref{eq:#2} over $#1$; yields Eq.~\ref{eq:#3}.	1	1	1
claim LHS equals RHS	Thus we see that LHS of Eq.~\ref{eq:#1} is equal to RHS.	1	0	0	variance relation particle in a 1D box curl curl identity
substitute LHS of five expressions into expression	Substitute LHS of Eq.~\ref{eq:#1} and LHS of Eq.~\ref{eq:#2} and LHS of Eq.~\ref{eq:#3} and LHS of Eq.~\ref{eq:#4} and LHS of Eq.~\ref{eq:#5} into Eq.~\ref{eq:#6}; yields Eq.~\ref{eq:#7}.	6	0	1
LHS of expr 1 equals LHS of expr 2	LHS of Eq.~\ref{eq:#1} is equal to LHS of Eq.~\ref{eq:#2}; yields Eq.~\ref{eq:#3}.	2	0	1	hyperbolic trigonometric identities equation of motion for a spring mass of the Earth quantum basics orthogonality equations of motion in 1D with constant acceleration - SUVAT (algebra) quadratic equation derivation radius for satellite in geostationary orbit particle in a 1D box angle of maximum distance for projectile motion derivation of Schrodinger Equation Euler equation: trig square root
both sides cross X	Take cross product of Eq.~\ref{eq:#2} and $#1$; yields Eq.~\ref{eq:#3}	1	1	1
sum exponents	Sum exponents on LHS and RHS of Eq.~\ref{eq:#1}; yields Eq.~\ref{eq:#2}.	1	0	1
substitute LHS of six expressions into expression	Substitute LHS of Eq.~\ref{eq:#1} and LHS of Eq.~\ref{eq:#2} and LHS of Eq.~\ref{eq:#3} and LHS of Eq.~\ref{eq:#4} and LHS of Eq.~\ref{eq:#5} and LHS of Eq.~\ref{eq:#6} into Eq.~\ref{eq:#7}; yields Eq.~\ref{eq:#8}.	7	0	1
apply operator to bra	Apply operator in Eq.~\ref{eq:#1} to bra; yields Eq.~\ref{eq:#2}.	1	0	1	quantum basics orthogonality
integrate	Integrate Eq.~ref{eq:#1}; yields Eq.~ref{eq:#2}.	1	0	1	escape velocity work and force and energy
sum exponents RHS	Sum exponents on RHS of Eq.~\ref{eq:#1}; yields Eq.~\ref{eq:#2}.	1	0	1
change five variables in expression	Change of variable $#1$ to $#2$ and $#3$ to $#4$ and $#5$ to $#6$ and $#7$ to $#8$ and $#9$ to $#10$ in Eq.~\ref{eq:#11}; yields Eq.~\ref{eq:#12}.	1	10	1
divide expr 1 by expr 2	Divide Eq.~\ref{eq:#1} by Eq.~\ref{eq:#2}; yields Eq.~\ref{eq:#3}.	2	0	1	double intensity when phase is coherent (optics)
factor out X	Factor $#1$ from Eq.~\ref{eq:#2}; yields Eq.~\ref{eq:#3}.	1	1	1	Langmuir Adsorption
simplify	Simplify Eq.~\ref{eq:#1}; yields Eq.~\ref{eq:#2}.	1	0	1	time invariant force conserves energy hyperbolic trigonometric identities upper limit on velocity in condensed matter Lorentz transformation double intensity when phase is coherent (optics) mass of the Earth first law of thermodynamics speed of Earth around Sun coefficient of isothermal compressibility using the equation of state for an ideal gas Kepler's Third Law: period squared propto distance cubed work and force and energy escape velocity curl curl identity coefficient of thermal expansion using the equation of state for an ideal gas derivation of Schrodinger Equation equations of motion in 2D (calculus) electric field wave equation: from time dependent to time independent angle of maximum distance for projectile motion particle in a 1D box Newton's Law of Gravitation quadratic equation derivation equations of motion in 1D with constant acceleration - SUVAT (algebra) quantum basics orthogonality variance relation velocity at distance r of object dropped from infinity identity sin(2 x) = 2 sin(x) cos(x) using Euler's equation Euler equation to exp(i pi) + 1 = 0
apply divergence	Apply divergence to both sides of Eq.~\ref{eq:#1}; yields Eq.~\ref{eq:#2}.	1	0	1	derivation of Schrodinger Equation
X cross both sides by	Take cross product of $#1$ and Eq.~\ref{eq:#2}; yields Eq.~\ref{eq:#3}.	1	1	1
change six variables in expression	Change of variable $#1$ to $#2$ and $#3$ to $#4$ and $#5$ to $#6$ and $#7$ to $#8$ and $#9$ to $#10$ and $#11$ to $#12$ in Eq.~\ref{eq:#13}; yields Eq.~\ref{eq:#14}.	1	12	1
distribute conjugate to factors	Distribute conjugate to factors in Eq.~\ref{eq:#1}; yields Eq.~\ref{eq:#2}.	1	0	1	double intensity when phase is coherent (optics)
raise both sides to power	Raise both sides of Eq.~\ref{eq:#2} to $#1$; yields Eq.~\ref{eq:#3}.	1	1	1	equations of motion in 1D with constant acceleration - SUVAT (algebra) radius for satellite in geostationary orbit derivation of Schrodinger Equation Schwarzschild radius for non-rotating black hole Kepler's Third Law: period squared propto distance cubed Langmuir Adsorption
conjugate both sides	Conjugate both sides of Eq.~\ref{eq:#1}; yields Eq.~\ref{eq:#2}.	1	0	1	double intensity when phase is coherent (optics)
apply function to both sides of expression	Apply function $#1$ with argument $#2$ to Eq.~\ref{eq:#3}; yields Eq.~\ref{eq:#4}	1	2	1	optics: Law of refraction to Brewster's angle
normalization condition	Normalization condition is Eq.~\ref{eq:#1}.	0	0	1	particle in a 1D box
function is odd	$#1$ is odd with respect to $#2$, so replace $#1$ with $#3$ in Eq.~\ref{eq:#4}; yields Eq.~\ref{eq:#5}.	1	3	1	Euler equation: trigonometric relations
square root both sides	Take the square root of both sides of Eq.~\ref{eq:#1}; yields Eq.~\ref{eq:#2} and Eq.~\ref{eq:#3}.	1	0	2	equation of motion for a spring Lorentz transformation velocity at distance r of object dropped from infinity quadratic equation derivation particle in a 1D box escape velocity
add X to both sides	Add $#1$ to both sides of Eq.~\ref{eq:#2}; yields Eq.~\ref{eq:#3}.	1	1	1	Lorentz transformation Euler equation to exp(i pi) + 1 = 0 equations of motion in 1D with constant acceleration - SUVAT (algebra) quadratic equation derivation angle of maximum distance for projectile motion equations of motion in 2D (calculus) escape velocity time invariant force conserves energy Euler equation: trig square root
apply gradient to scalar function	Apply gradient to both sides of Eq.~\ref{eq:#1}; yields Eq.~\ref{eq:#2}.	1	0	1	derivation of Schrodinger Equation
solve for X	Solve Eq.~\ref{eq:#2} for $#1$; yields Eq.~\ref{eq:#3}.	1	1	1
expand RHS	Expand the RHS of Eq.~\ref{eq:#1}; yields Eq.~\ref{eq:#2}.	1	0	1	Euler equation: trig square root
claim expr 1 equals expr 2	Thus we see that Eq.~\ref{eq:#1} is equivalent to Eq.~\ref{eq:#2}.	2	0	0	Euler's equation from MacLaurin series
separate three vector components	Separate three vector components in Eq.~\ref{eq:#1}; yields Eq.~\ref{eq:#2} and Eq.~\ref{eq:#3} and Eq.~\ref{eq:#4}.	1	0	3
substitute LHS of expr 1 into expr 2	Substitute LHS of Eq.~\ref{eq:#1} into Eq.~\ref{eq:#2}; yields Eq.~\ref{eq:#3}.	2	0	1	velocity at distance r of object dropped from infinity hyperbolic trigonometric identities equation of motion for a spring upper limit on velocity in condensed matter Lorentz transformation double intensity when phase is coherent (optics) optics: Law of refraction to Brewster's angle first law of thermodynamics speed of Earth around Sun coefficient of isothermal compressibility using the equation of state for an ideal gas Langmuir Adsorption Kepler's Third Law: period squared propto distance cubed projectile path in 2D is parabolic work and force and energy escape velocity Maxwell equations to electric field wave equation coefficient of thermal expansion using the equation of state for an ideal gas equations of motion in 2D (calculus) derivation of Schrodinger Equation electric field wave equation: from time dependent to time independent angle of maximum distance for projectile motion particle in a 1D box Newton's Law of Gravitation radius for satellite in geostationary orbit Compton's equation for scattering
expand LHS	Expand the LHS of Eq.~\ref{eq:#1}; yields Eq.~\ref{eq:#2}.	1	0	1
sum exponents LHS	Sum exponents on LHS of Eq.~\ref{eq:#1}; yields Eq.~\ref{eq:#2}.	1	0	1
both sides dot X	Take inner product of Eq.~\ref{eq:#2} with $#1$; yields Eq.~\ref{eq:#3}	1	1	1
expand integrand	Expand integrand of Eq.~\ref{eq:#1}; yields Eq.~\ref{eq:#2}.	1	0	1	particle in a 1D box
indefinite integration	Indefinite integral of both sides of Eq.~\ref{eq:#1}; yields Eq.~\ref{eq:#2}.	1	0	1	integration by parts equations of motion in 2D (calculus)
factor out X from LHS	Factor $#1$ from the LHS of Eq.~\ref{eq:#2}; yields Eq.~\ref{eq:#3}.	1	1	1	Lorentz transformation
substitute RHS of expr 1 into expr 2	Substitute RHS of Eq.~\ref{eq:#1} into Eq.~\ref{eq:#2}; yields Eq.~\ref{eq:#3}.	2	0	1	equation of motion for a spring upper limit on velocity in condensed matter time invariant force conserves energy equations of motion in 1D with constant acceleration - SUVAT (algebra) quantum basics Hermitian operators have realvalued observables particle in a 1D box frequency relations derivation of Schrodinger Equation coefficient of thermal expansion using the equation of state for an ideal gas curl curl identity Euler equation proof Kepler's Third Law: period squared propto distance cubed Langmuir Adsorption
multiply RHS by unity	Multiply RHS of Eq.~\ref{eq:#2} by 1, which in this case is $#1$; yields Eq.~\ref{eq:#3}	1	1	1	speed of Earth around Sun derivation of Schrodinger Equation Kepler's Third Law: period squared propto distance cubed
differentiate with respect to	Differentiate Eq.~\ref{eq:#2} with respect to $#1$; yields Eq.~\ref{eq:#3}.	1	1	1	equation of motion for a spring electric field wave equation: from time dependent to time independent Euler equation proof
evaluate definite integral	Evaluate definite integral Eq.~\ref{eq:#1}; yields Eq.~\ref{eq:#2}.	1	0	1	velocity at distance r of object dropped from infinity escape velocity work and force and energy
partially differentiate with respect to	Partially differentiate Eq.~\ref{eq:#2} with respect to $#1$; yields Eq.~\ref{eq:#3}.	1	1	1	derivation of Schrodinger Equation Maxwell equations to electric field wave equation
conjugate transpose both sides	Conjugate transpose of both sides of Eq.~\ref{eq:#1}; yields Eq.~\ref{eq:#2}.	1	0	1	quantum basics Hermitian operators have realvalued observables
expr 1 is true under condition expr 2	Eq.~\ref{eq:#1} is valid when Eq.~\ref{eq:#2} occurs; yields Eq.~\ref{eq:#3}.	2	0	1	particle in a 1D box
replace constant with value	Replace constant $#1$ with value $#2$ and units $#3$ in Eq.~\ref{eq:#4}; yields Eq.~\ref{eq:#5}	1	3	1	mass of the Earth
add zero to RHS	Add zero to RHS of Eq.~\ref{eq:#2}, where $0=#1$; yields Eq.~\ref{eq:#3}.	1	1	1
make expr power	Make Eq.~\ref{eq:#2} the power of $#1$; yields Eq.~\ref{eq:#3}.	1	1	1	Euler equation proof
combine like terms	Combine like terms in Eq.~\ref{eq:#1}; yields Eq.~\ref{eq:#2}.	1	0	1	quantum basics orthogonality
substitute LHS of two expressions into expression	Substitute LHS of Eq.~\ref{eq:#1} and LHS of Eq.~\ref{eq:#2} into Eq.~\ref{eq:#3}; yields Eq.~\ref{eq:#4}.	3	0	1	Newton's Second Law of Motion: vector to scalar first law of thermodynamics radius for satellite in geostationary orbit Newton's Law of Gravitation escape velocity work and force and energy
expr 1 is equivalent to expr 2 under the condition	Eq.~\ref{eq:#1} is equivalent to Eq.~\ref{eq:#2} under the condition in Eq.~\ref{eq:#3}.	2	0	1	Lorentz transformation
maximum of expression	The maximum of Eq.~\ref{eq:#2} with respect to $#1$ is Eq.~\ref{eq:#3}	1	1	1	upper limit on velocity in condensed matter angle of maximum distance for projectile motion
take curl of both sides	Apply curl to both sides of Eq.~\ref{eq:#1}; yields Eq.~\ref{eq:#2}.	1	0	1	Maxwell equations to electric field wave equation
change four variables in expression	Change of variable $#1$ to $#2$ and $#3$ to $#4$ and $#5$ to $#6$ and $#7$ to $#8$ in Eq.~\ref{eq:#9}; yields Eq.~\ref{eq:#10}.	1	8	1	radius for satellite in geostationary orbit
drop non-dominant term	Based on the assumption $#1$, drop non-dominant term in Eq.~\ref{#2}; yeilds Eq.~\ref{#3}	1	1	1	upper limit on velocity in condensed matter
assume N dimensions	Assume $#1$ dimensions; decompose vector to be Eq.~\ref{eq:#2}.	0	1	1	equations of motion in 2D (calculus)
substitute LHS of four expressions into expression	Substitute LHS of Eq.~\ref{eq:#1} and LHS of Eq.~\ref{eq:#2} and LHS of Eq.~\ref{eq:#3} and LHS of Eq.~\ref{eq:#4} into Eq.~\ref{eq:#5}; yields Eq.~\ref{eq:#6}.	5	0	1	Lorentz transformation double intensity when phase is coherent (optics)
boundary condition for expression	A boundary condition for Eq.~\ref{eq:#1} is Eq.~\ref{eq:#2}	1	0	1	particle in a 1D box
RHS of expr 1 equals RHS of expr 2	RHS of Eq.~\ref{eq:#1} is equal to RHS of Eq.~\ref{eq:#2}; yields Eq.~\ref{eq:#3}.	2	0	1	identity sin(2 x) = 2 sin(x) cos(x) using Euler's equation
change variable X to Y	Change variable $#1$ to $#2$ in Eq.~\ref{eq:#3}; yields Eq.~\ref{eq:#4}.	1	2	1	hyperbolic trigonometric identities double intensity when phase is coherent (optics) mass of the Earth optics: Law of refraction to Brewster's angle speed of Earth around Sun Euler equation to exp(i pi) + 1 = 0 identity sin(2 x) = 2 sin(x) cos(x) using Euler's equation velocity at distance r of object dropped from infinity quadratic equation derivation radius for satellite in geostationary orbit Newton's Law of Gravitation particle in a 1D box angle of maximum distance for projectile motion Euler equation: trigonometric relations Euler equation: trig square root Langmuir Adsorption
distribute conjugate transpose to factors	Distribute conjugate transpose to factors in Eq.~\ref{eq:#1}; yields Eq.~\ref{eq:#2}.	1	0	1	quantum basics Hermitian operators have realvalued observables
select imaginary parts	Select imaginary parts of Eq.~\ref{eq:#1}; yields Eq.~\ref{eq:#2}.	1	0	1
replace summation notation with vector notation	Replace summation notation in Eq.~\ref{eq:#1} with vector notation; yields Eq.~\ref{eq:#2}.	1	0	1	curl curl identity
replace curl with LeviCevita summation contravariant	Replace curl in Eq.~\ref{eq:#1} with Levi-Cevita contravariant; yields Eq.~\ref{eq:#2}.	1	0	1	curl curl identity
integrate over from to	Integrate Eq.~\ref{eq:#4} over $#1$ from lower limit $#2$ to upper limit $#3$; yields Eq.~\ref{eq:#5}.	1	3	1
apply operator to ket	Apply operator in Eq.~\ref{eq:#1} to ket; yields Eq.~\ref{eq:#2}.	1	0	1	quantum basics orthogonality
divide both sides by	Divide both sides of Eq.~\ref{eq:#2} by $#1$; yields Eq.~\ref{eq:#3}.	1	1	1	total electrical resistance for circuit with two resistors in parallel equation of motion for a spring Lorentz transformation mass of the Earth optics: Law of refraction to Brewster's angle first law of thermodynamics coefficient of isothermal compressibility using the equation of state for an ideal gas time invariant force conserves energy equations of motion in 1D with constant acceleration - SUVAT (algebra) quadratic equation derivation radius for satellite in geostationary orbit particle in a 1D box frequency relations angle of maximum distance for projectile motion derivation of Schrodinger Equation coefficient of thermal expansion using the equation of state for an ideal gas Schwarzschild radius for non-rotating black hole Euler equation: trigonometric relations projectile path in 2D is parabolic Euler equation: trig square root total electrical resistance for circuit with two resistors in series Euler equation proof frequency and period Kepler's Third Law: period squared propto distance cubed Langmuir Adsorption
add expr 1 to expr 2	Add Eq.~\ref{eq:#1} to Eq.~\ref{eq:#2}; yields Eq.~\ref{eq:#2}.	2	0	1	hyperbolic trigonometric identities Euler equation: trigonometric relations
declare initial expression	Eq.~\ref{eq:#1} is an initial equation.	0	0	1	angle of maximum distance for projectile motion Euler's equation from MacLaurin series Newton's Second Law of Motion: vector to scalar hyperbolic trigonometric identities total electrical resistance for circuit with two resistors in parallel equation of motion for a spring upper limit on velocity in condensed matter Lorentz transformation double intensity when phase is coherent (optics) mass of the Earth optics: Law of refraction to Brewster's angle first law of thermodynamics speed of Earth around Sun coefficient of isothermal compressibility using the equation of state for an ideal gas time invariant force conserves energy velocity at distance r of object dropped from infinity Euler equation to exp(i pi) + 1 = 0 identity sin(2 x) = 2 sin(x) cos(x) using Euler's equation Compton's equation for scattering equations of motion in 1D with constant acceleration - SUVAT (algebra) quantum basics orthogonality quantum basics Hermitian operators have realvalued observables quadratic equation derivation radius for satellite in geostationary orbit particle in a 1D box Newton's Law of Gravitation frequency relations Langmuir Adsorption Kepler's Third Law: period squared propto distance cubed Euler equation proof frequency and period total electrical resistance for circuit with two resistors in series projectile path in 2D is parabolic Euler equation: trig square root work and force and energy escape velocity Euler equation: trigonometric relations Maxwell equations to electric field wave equation derivation of Schrodinger Equation coefficient of thermal expansion using the equation of state for an ideal gas equations of motion in 2D (calculus) electric field wave equation: from time dependent to time independent
change two variables in expression	Change variable $#1$ to $#2$ and $#3$ to $#4$ in Eq.~\ref{eq:#5}; yields Eq.~\ref{eq:#6}.	1	4	1	total electrical resistance for circuit with two resistors in parallel double intensity when phase is coherent (optics) optics: Law of refraction to Brewster's angle speed of Earth around Sun time invariant force conserves energy radius for satellite in geostationary orbit angle of maximum distance for projectile motion Schwarzschild radius for non-rotating black hole escape velocity total electrical resistance for circuit with two resistors in series Kepler's Third Law: period squared propto distance cubed
conjugate function X	Conjugate $#1$ in Eq.~\ref{eq:#2}; yields Eq.~\ref{eq:#3}.	1	1	1	particle in a 1D box
magnitude of vectors	Absolute value of both sides	1	0	1	Newton's Second Law of Motion: vector to scalar

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