Science

Physics
Planck units	Planck length	$l_{\text{P}}={\color {Melon}{\sqrt {\hbar G/c^{3}}}}$	The shortest length in current physics
	Planck time	$t_{\text{P}}={\color {Melon}{\sqrt {\hbar G/c^{5}}}}$	The shortest time in current physics
	Planck mass	$m_{\text{P}}={\color {Melon}{\sqrt {\hbar c/G}}}$	The smallest mass in current physics
	Planck charge	$q_{\text{P}}={\color {Melon}{\sqrt {4\pi \varepsilon _{0}\hbar c}}}$	The smallest electric charge in current physics
	Planck temperature	$T_{\text{P}}={\color {Melon}{\sqrt {\hbar c^{5}/Gk_{\text{B}}^{2}}}}$	The highest temperature in current physics
	Natural units	${\color {Melon}c}={\color {Melon}G}={\color {Melon}\hbar }={\color {Melon}{\frac {1}{4\pi \varepsilon _{0}}}}={\color {Melon}k_{\text{B}}}=1$	Five universal constants normalize to 1 when using natural units
Classical mechanics
Continuity equations		${\frac {\partial \rho }{\partial t}}=-\nabla \cdot \mathbf {J}$	A conserved quantity cannot be created or destroyed, its rate of change equals to its flow of quantity transport.
Principle of least action		$\delta {\mathcal {S}}=\delta \int _{t_{1}}^{t_{2}}{\mathcal {L}}(\mathbf {q} ,\mathbf {\dot {q}} ,t)dt=0$	The path taken by the system is the one for which the action is stationary to first order.
Newton's laws of motion	First law	$\mathbf {F} =0\leftrightarrow {\frac {\mathrm {d} \mathbf {v} }{\mathrm {d} t}}=0$	An object either remains at rest or continues to move at a constant velocity, unless acted upon by a force.
	Second law	$\mathbf {F} ={\frac {\mathrm {d} \mathbf {p} }{\mathrm {d} t}}=m\mathbf {a}$	The rate of change of momentum of an object is proportional to the net force applied.
	Third law	$\mathbf {F} _{ij}=-\mathbf {F} _{ji}$	The reaction force is equal in magnitude and opposite in direction with the action force.
Analytical mechanics	Euler–Lagrange equation	${\frac {\mathrm {d} }{\mathrm {d} t}}{\frac {\partial {\mathcal {L}}}{\partial {\dot {\mathbf {q} }}}}={\frac {\partial {\mathcal {L}}}{\partial \mathbf {q} }}$	Partial differential equation for the time evolution of a physical system in Lagrangian mechanics.
	Hamilton's equations	${\dot {\mathbf {p} }}=-{\dfrac {\partial {\mathcal {H}}}{\partial \mathbf {q} }},{\dot {\mathbf {q} }}={\dfrac {\partial {\mathcal {H}}}{\partial \mathbf {p} }}$	Partial differential equations for the time evolution of a physical system in Hamiltonian mechanics.
	Hamilton–Jacobi equation	${\mathcal {H}}=-{\frac {\partial {\mathcal {S}}}{\partial t}}$	Partial differential equation for the time evolution of a physical system in Hamilton–Jacobi mechanics.
Inverse-square laws	Newton's law of universal gravitation	$\mathbf {F} =-{\color {Melon}G}m_{1}m_{2}{\frac {\mathbf {\hat {r}} }{\left\vert \mathbf {r} \right\vert ^{2}}}$	The gravitational force between two point masses is proportional to the masses and inversely proportional to the square of their distance.
	Coulomb's law	$\mathbf {F} ={\color {Melon}{\frac {1}{4\pi \epsilon _{0}}}}q_{1}q_{2}{\frac {\mathbf {\hat {r}} }{\left\vert \mathbf {r} \right\vert ^{2}}}$	The electrostatic force between two point charges is proportional to the charges and inversely proportional to the square of their distance.
	Biot–Savart law	$\mathbf {B} ={\color {Melon}{\frac {\mu _{0}}{4\pi }}}\oint {\frac {Id\mathbf {l} \times \mathbf {\hat {r}} }{\left\vert \mathbf {r} \right\vert ^{2}}}$	The magnetic field at a point generated by a steady electric current is proportional to the current and inversely proportional to the square of their distance.
Electromagnetism
Maxwell's equations	Gauss's law	$\nabla \cdot \mathbf {E} =\rho {\color {Melon}{\cancelto {4\pi }{/\varepsilon _{0}}}}$	The electric flux leaving a volume is proportional to the charge inside.
	Gauss's law for magnetism	$\nabla \cdot \mathbf {B} =0$	There are no magnetic monopoles; the total magnetic flux through a closed surface is zero.
	Faraday's law of induction (Maxwell–Faraday equation)	$\nabla \times \mathbf {E} =-{\frac {\partial \mathbf {B} }{\partial t}}$	The voltage induced in a closed loop is proportional to the rate of change of the magnetic flux that the loop encloses.
	Ampère's circuital law with Maxwell's extension	$\nabla \times \mathbf {B} ={\color {Melon}\mu _{0}\varepsilon _{0}}{\frac {\partial \mathbf {D} }{\partial t}}+{\color {Melon}{\cancelto {4\pi }{\mu _{0}}}}\mathbf {J} _{\text{e}}$	The magnetic field induced around a closed loop is proportional to the displacement current plus the electric current that the loop encloses.
Relativity
Mass–energy equivalence		$E=m{\color {Melon}c^{2}}$	Mass and energy are equivalent and interconvertable.
Lorentz transformations	Length contraction	$l={\frac {l_{0}}{\gamma }}$	Length appears to be shorter along the direction of motion when observed from a moving observer.
	Time dilation	$t=\gamma t_{0}$	Time appears to be slower when observed from a moving observer.
	Relativistic mass	$m=\gamma m_{0}$	Mass appears to be heavier when observed from a moving observer.
	Lorentz factor	$\gamma ={\frac {1}{\sqrt {1-v^{2}/{\color {Melon}c^{2}}}}}$	Factor of length contration / time dilation / mass increament approaches infinity when speed approaches the speed of light.
Einstein field equations		$G_{\mu \nu }+\Lambda g_{\mu \nu }=8\pi {\color {Melon}{\frac {G}{c^{4}}}}T_{\mu \nu }$	The curvature of spacetime (with cosmological constant term) is determined by its matter/energy content.
Quantum mechanics
Planck–Einstein relation		$E=h\nu ={\color {Melon}\hbar }\omega$	The energy of photon is proportional to its frequency.
Dirac equation		$(i{\color {Melon}\hbar }{\partial \!\!\!/}-m{\color {Melon}c})\psi =0$	Partial differential equation for the quantum fields corresponding to spin-1/2 particles; predicts the existence of antimatter.
Schrödinger equation		$i{\color {Melon}\hbar }{\frac {\partial }{\partial t}}\Psi ={\hat {H}}\Psi$	Partial differential equation for the time evolution of the wave function of a physical system in which quantum effects are significant; predicts the quantization of certain properties.
Heisenberg's uncertainty principle		$\Delta x\Delta p\geq {\color {Melon}\hbar }/2$ $\Delta E\Delta t\geq {\color {Melon}\hbar }/2$	The more precisely the position of a particle / the time interval of a state is determined, the less precisely its momentum / energy can be known, and vice versa.
Astronomy and Particle physics
Kepler's laws of planetary motion	1. Law of orbits	$R=a(1\pm \varepsilon )$	All planets move in elliptical orbits with the sun at one focus.
	2. Law of areas	${\frac {\mathrm {d} A}{\mathrm {d} t}}={\frac {L}{2m}}$	A line joining a planet to the sun sweeps out equal areas in equal times.
	3. Law of periods	$T^{2}\propto a^{3}$	The square of orbital period is proportional to the cube of semimajor axis.
Lagrangian of the Standard model		${\begin{aligned}{\mathcal {L}}=&-{\frac {1}{4}}F_{\mu \nu }F^{\mu \nu }\\&+i{\bar {\psi }}{D}\!\!\!\!/\ \psi +h.c.\\&+\psi _{i}y_{ij}\psi _{j}\phi +h.c.\\&+\left\vert D_{\mu }\phi \right\vert ^{2}-V(\phi )\end{aligned}}$	Simplfied equation of the standard model describing the fundamental forces; how these forces act on the fundamental particles; how these particles obtain their masses from the Higgs boson; and the Higgs mechanism.
Thermodynamics and Information theory
Entropy	Boltzmann and Gibbs entropy	$S={\color {Melon}k_{\text{B}}}\ln W=-{\color {Melon}k_{\text{B}}}\sum p_{i}\ln p_{i}$	The statistical thermodynamic entropy of an equilibrium ensemble is the logarithm of the number of microstates, or the sum of probability-weighted log probabilities of the microstates.
	Shannon and Hartley entropy	$H=log_{b}\left\|M\right\|=-\sum p_{i}\log _{b}p_{i}$	The information entropy of a message space is the logarithm of the number of messages, or the sum of probability-weighted log probabilities of the messages.
	Bekenstein–Hawking entropy	$S={\color {Melon}{\frac {k_{\text{B}}c^{3}}{G\hbar }}}{\frac {A}{4}}$	The black hole entropy is proportional to the area of its event horizon; related to the holographic principle.
Laws of thermodynamics	Zeroth law	$T_{A}=T_{B}\,,T_{B}=T_{C}\Rightarrow T_{A}=T_{C}$	If two systems are in thermal equilibrium with a third system, they are in thermal equilibrium with each other.
	First law	$\mathrm {d} U=\delta Q-\delta W$	The increase in internal energy of a closed system is equal to the total energy added to the system.
	Second law	$\mathrm {d} S\geq \delta Q/T$	The total entropy of an isolated system can only increase over time.
	Third law	$\lim _{T\to 0}S=0$	The entropy of a perfect crystal at absolute zero is exactly equal to zero.
Evolutionary biology
Fundamental theorem of natural selection		${\frac {\mathrm {d} }{\mathrm {d} t}}\phi =\sigma _{f}^{2}\geq 0$	The rate of increase in fitness of any organism at any time is equal to its genetic variance in fitness at that time.

Mathematics

Mathematics
Pythagorean theorem		$a^{2}+b^{2}=c^{2}$	The length squared of the hypotenuse in a triangle is the sum of the lengths squared of the other two sides.
Euler's identity		$e^{i\pi }+1=0$	Starting at e⁰ = 1, travelling at velocity i relative to one's position for the length of time π, and adding 1, arrives at 0. Five fundamental mathematical constants (0, 1, π, e, i) are linked with three basic arithmetic operations (+, ×, ^).
Stokes' theorem (Newton-Leibniz-Gauss-Green-Ostrogradskii-Stokes-Poincaré formula)		$\int _{\partial \Omega }\omega =\int _{\Omega }\mathrm {d} \omega$	The integral of a differential form over the boundary of an orientable manifold is equal to the integral of its exterior derivative over the whole manifold.
Fundamental theorems	Fundamental theorem of arithmatic	$n=\prod _{i=1}^{k}p_{i}^{\alpha _{i}}$	Every natural number is either prime itself or a unique product of prime numbers.
Fundamental theorems	Fundamental theorem of algebra	$\sum _{i=0}^{n}a_{i}x^{i}=a_{n}\prod _{i=i}^{n}(x-k_{i})$	Every polynomial of non-zero degree has at least one complex root. Corollary: Every polynomial of degree n has exactly n solutions (includeing repeated ones).
Logic and Set theory
Gödel's completeness theorem		${\mathcal {T}}\models \phi \Rightarrow {\mathcal {T}}\vdash \phi$ $\forall \phi ({\mathcal {T}}\vdash \phi \lor {\mathcal {T}}\vdash \neg \phi )$	Every logically valid formula can either be proved or disproved.
Gödel's incompleteness theorems	First theorem	$\exists \phi ({\mathcal {F}}\not \vdash \phi \land {\mathcal {F}}\not \vdash \neg \phi )$	Within any consistent formal system there exists true statements which can neither be proved nor disproved.
Gödel's incompleteness theorems	Second theorem	${\mathcal {F}}\not \vdash {\text{Cons}}({\mathcal {F}})$	No consistent formal system can prove its own consistency.
Continuum hypothesis		$2^{\aleph _{0}}=\aleph _{1}$	There is no set with cardinality strictly between the integers and the real numbers.
ZFC Axioms (Zermelo–Fraenkel set theory with the axiom of choice)	1. Axiom of extensionality	$\forall A\forall B(\forall a(a\in A\Leftrightarrow a\in B)\Leftrightarrow A=B)$	Two sets are equal if they have the same elements.
	2. Axiom of foundation / regularity	$\forall A(A\neq \varnothing \Rightarrow \exists B\in A(A\cap B=\varnothing ))$	Any non-empty set contains an element that is disjoint from the set.
	3. Axiom of paring	$\forall A\forall B\exists C\forall c(c\in C\Leftrightarrow (c=A\lor c=B))$ $\quad C=\{A,B\}$	Any two sets have a pair that consists of both of them.
	4. Axiom of union	$\forall A\exists B\forall b(b\in B\Leftrightarrow \exists C(C\in A\land b\in C))$ $\quad B=\cup A$	Any set has a union set that consists of the elements of all its elements.
	5. Axiom of power set	$\forall A\exists B\forall C(C\in B\Leftrightarrow C\subseteq A)$ $\quad B={\mathcal {P}}(A)$	Any set has a power set that consists of all its subsets.
	6. Axiom schema of replacement	${\begin{aligned}&\forall A\forall p[(\forall x\in A\;\exists !y\;\phi (x,y,p))\\&\Rightarrow \exists B\forall y(y\in B\Leftrightarrow \exists x\in A\;\phi (x,y,p))]\end{aligned}}$ $\quad B=F[A]$	The image of a set under any definable function is also a set.
	7a. Axiom schema of specification / separation / subsets / comprehension	$\forall A\forall p\exists B\forall x(x\in B\Leftrightarrow (x\in A\land \phi (x,p)))$ $\quad B=\{x\in A:\phi (x,p)\}$	Any set has a subset that consists of all elements satisfying certain property. (= Axiom schema of replacement + Axiom of empty set)
	7b. Axiom of empty set	$\exists A\forall x(x\notin A)$ $\quad A=\varnothing$	There exists an empty set having no element.
	8. Axiom of infinity	$\exists \mathbf {I} (\varnothing \in \mathbf {I} \land \forall a\in \mathbf {I} (a\cup \{a\}\in \mathbf {I} ))$	There exists an infinite set having infinitely many members.
	9. Axiom of choice	$\forall X(\varnothing \notin X\Rightarrow \exists f\forall A\in X(f(A)\in A))$	Any family of non-empty sets has a choice function.

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