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Expression  name  domain  description 

\(F = m a\)  Second Law of Motion  Classical Mechanics  
\(E = m c^2\)  Energymass equivalence  special theory of relativity 
wikidata:Q35875 academic.microsoft.com/topic/185321693 
\( \Delta x \Delta \rho \geq \frac{\hbar}{2} \)  Uncertainty principle  Quantum Mechanics 

\( i \hbar \frac{\partial}{\partial t}  \psi(t) \rangle = \hat{H}  \psi(t) \rangle \)  Schrödinger equation  Quantum Mechanics  
\( \vec{\nabla} \times \vec{E} =  \frac{\partial \vec{B}}{\partial t} \)  MaxwellFaraday equation  Electrodynamics  
\( (i \partial\!\!\!\big / m) \psi = 0 \)  Dirac equation  Partical Physics  Dirac developed an equation that explained spin number as a consequence of the union of quantum mechanics and special relativity. The equation also predicted the existence of antimatter, previously unsuspected and unobserved, and which was experimentally discovered in 1932. 
\( \Delta S \geq 0 \) where \( S = k_{Boltzmann} \ln W \)  Law of entropy  Thermodynamics  when energy changes from one form to another form, or when matter moves freely, the disorder in a closed system increases. 
\( G_{\mu \nu} + \Lambda g_{\mu \nu} = \frac{8 \pi G}{c^4} T_{\mu \nu} \)  Einstein field equations  The expression on the left hand side of the equation represents the curvature of spacetime. The expression on the right is the energy density of spacetime. The equation dictates how energy determines he curvature of space and time. 

\( \frac{\partial^2 u}{\partial t^2} = \frac{\partial^2 u}{\partial x^2} c^2 \)  wave equation  
\( E = h f \)  Planck's equation  quantum mechanics  
\( \frac{\partial \mathbf{u}}{\partial t} + (\mathbf{u} \cdot \nabla) \mathbf{u}  \nu \,\nabla^2 \mathbf{u} =  \nabla w + \mathbf{g} \)  NavierStokes  fluid dynamics  describes the motion of viscous fluid substances. mathematically
expresses conservation of momentum and conservation of mass for Newtonian fluids.

\( \vec{\nabla} \times \vec{B} = \mu_0 \vec{J} \) and \( \vec{\nabla} \times \vec{H} = \vec{J}_f \)  Ampère's circuital law 


\( P_1 + \frac{1}{2} \rho v_1^2 + \rho g h_1 = P_2 + \frac{1}{2} \rho v_2^2 + \rho g h_2 \)  Bernoulli's equation  
\( \)  Bogoliubov–Born–Green–Kirkwood–Yvon hierarchy of equations  
\( \)  Bessel's differential equation  
\( \frac{\partial f}{\partial t} + \frac{\mathbf{p}}{m}\cdot\nabla f + \mathbf{F} \cdot \frac{\partial f}{\partial \mathbf{p}} = \left(\frac{\partial f}{\partial t} \right)_\mathrm{coll} \)  Boltzmann equation  
\( \Delta E = \xi \frac{1}{2} \rho \left( v_1  v_2 \right)^2 \)  Borda–Carnot equation  
\( \frac{\partial u}{\partial t} + u \frac{\partial u}{\partial x} = \nu\frac{\partial^2 u}{\partial x^2} \)  Burgers' equation 


\( \frac{\Delta p}{L} = f_{\mathrm{D}} \cdot \frac{\rho }{2} \cdot \frac{\langle v \rangle^2}{D} \)  Darcy–Weisbach equation  
\( f = \left( \frac{c \pm v_{\rm{receiver}}}{ c \pm v_{\rm{source}} } \right) f_0 \)  Doppler equations  
\( N = R_{*} f_{p} n_e f_1 f_i f_c L \)  Drake equation (aka Green Bank equation)  
\( \)  Euler equations (fluid dynamics)  
\( \)  Euler's equations (rigid body dynamics)  
\( T^{\mu\nu} \, = (e+p)u^\mu u^\nu+p g^{\mu\nu} \)  Relativistic Euler equations  
\( \frac{d}{dt} \left( \frac{\partial L}{\partial \dot{\mathbf{q}}} \right) = \frac{\partial L}{\partial \mathbf{q}} \)  Euler–Lagrange equation  
\( \mathcal{E} = \frac{\mathrm{d}\Phi_B}{\mathrm{d}t} \)  Faraday's law of induction 


\( \frac{\partial}{\partial t} p(x, t) = \frac{\partial}{\partial x}\left[\mu(x, t) p(x, t)\right] + \frac{\partial^2}{\partial x^2}\left[D(x, t) p(x, t)\right] \)  Fokker–Planck equation  describes the time evolution of the probability density function of the velocity of a particle under the influence of drag forces and random forces, as in Brownian motion 

\( \)  Fresnel equations  
\( \)  Friedmann equations  
\( \Phi_E = \frac{Q}{\varepsilon_0} \)  Gauss's law for electricity  Electrodynamics  The net electric flux through any hypothetical closed surface is equal to \( \frac{1}{\varepsilon _{0}} \) times the net electric charge within that closed surface. 
\(\vec{\nabla} \cdot \vec{g} = 4 \pi G \rho \)  Gauss's law for gravity  equivalent to Newton's law of universal gravitation


\( \vec{\nabla} \cdot \vec{B} = 0 \)  Gauss's law for magnetism  
\( \left( \frac{\partial \left( \frac{G} {T} \right) } {\partial T} \right)_p =  \frac {H} {T^2} \)  Gibbs–Helmholtz equation  
\( \left(\frac{\hbar^2}{2m}{\partial^2\over\partial\mathbf{r}^2} + V(\mathbf{r}) + {4\pi\hbar^2a_s\over m}\vert\psi(\mathbf{r})\vert^2\right)\psi(\mathbf{r})=\mu\psi(\mathbf{r}) \)  Gross–Pitaevskii equation  
\( \)  Hamilton–Jacobi–Bellman equation  
\( \nabla^2 f = k^2 f \)  Helmholtz equation  
\( J(\phi) = A \cos^2 \phi + B \cos\,\phi + C \)  Karplus equation  
\( M = E  e \sin E \)  Kepler's equation 


\( \)  Kepler's laws of planetary motion  
\( U(P) = \frac{1}{4\pi} \int_{S} \left[ U \frac{\partial}{\partial n} \left( \frac{e^{iks}}{s} \right)  \frac{e^{iks}}{s} \frac{\partial U}{\partial n} \right]dS \)  Kirchhoff's diffraction formula  
\( \left( \nabla^2  \frac{m^2 c^2}{\hbar^2} \right) \psi(\vec{r}) = 0 \)  Klein–Gordon equation  a relativistic wave equation, related to the Schrödinger equation 

\( \partial_t \phi + \partial^3_x \phi  6\, \phi\, \partial_x \phi =0\ \)  Korteweg–de Vries equation  
\( \frac{d \vec{M}}{d t}=\gamma \left(\vec{M} \times \vec{H}_{\mathrm{eff}}  \eta \vec{M}\times\frac{d \vec{M}}{d t}\right) \)  Landau–Lifshitz–Gilbert equation 


\( \frac{1}{\xi^2} \frac{d}{d\xi} \left({\xi^2 \frac{d\theta}{d\xi}}\right) + \theta^n = 0 \)  Lane–Emden equation  
\( m \frac{d \vec{v}}{dt} = \lambda \vec{v} + \eta(t) \)  Langevin equation 


\( \frac{\mathbf{d}\varepsilon_1}{\sigma'_1}=\frac{\mathbf{d}\varepsilon_2} {\sigma'_2}=\frac{\mathbf{d}\varepsilon_3}{\sigma'_3}=\mathbf{d}\lambda \)  Levy–Mises equations  
\( \dot\rho={i\over\hbar}[H,\rho]+\sum_{n,m = 1}^{N^21} h_{nm}\left(A_n\rho A_m^\dagger\frac{1}{2}\left\{A_m^\dagger A_n, \rho\right\}\right) \)  Lindblad equation  
\( \vec{F} = q \vec{E} + q\vec{v}\times \vec{B} \)  Lorentz equation  The electromagnetic force on a charge \(q\) is a combination of a force in the direction of the electric field \vec{E} proportional to the magnitude of the field and the quantity of charge, and a force at right angles to the magnetic field \vec{B} and the velocity \vec{v} of the charge, proportional to the magnitude of the field, the charge, and the velocity. 

\( \)  Maxwell's equations  
\( \)  Maxwell's relations  
\( \)  Newton's laws of motion  
\( \)  Reynoldsaveraged Navier–Stokes equations  
\( \)  Prandtl–Reuss equations  
\( h_f = \frac{L}{D} (aV + bV^2) \)  Prony equation  
\( \)  Rankine–Hugoniot equation  
\( \hat{F} \hat{C} = \hat{S} \hat{C} \hat{\epsilon} \)  Roothaan equations  
\( \)  Saha ionization equation  
\( \frac{S}{k_{\rm B} N} = \ln \left[ \frac VN \left(\frac{4\pi m}{3h^2}\frac UN\right)^{3/2}\right]+ {\frac 52} \)  Sackur–Tetrode equation  
\( \left[ \Delta  \lambda^2 \right] u(\vec{r}) =  f(\vec{r}) \)  screened Poisson equation  
\( \)  Schwinger–Dyson equation  
\( \)  Sellmeier equation  
\( \)  Stokes–Einstein relation  
\( \delta v = v_e \ln \frac{m_0}{m_f} = I_{\rm{sp}} g_0 \ln \frac{m_0}{m_f} \)  Tsiolkovsky rocket equation 


\( PV = nRT \)  Van der Waals equation 


\( \frac{\partial f_{\alpha}}{\partial t} + \mathbf {v}_{\alpha} \cdot \frac{\partial f_{\alpha}}{\partial \mathbf {x}}+ \frac{q_{\alpha}\mathbf {E}}{m_{\alpha}} \cdot \frac{\partial f_{\alpha}}{\partial \mathbf {v}} = 0 \)  Vlasov equation  
\( \vec{v} = \frac{d \vec{x}}{d t} = g(t) \)  Wiener equation  
\( \frac{\partial\psi}{\partial t} +\nabla\cdot\left( \psi{\mathbf u}\right) =0 \)  Advection equation  
\( \)  Barotropic vorticity equation  
\( \frac{\partial \rho}{\partial t} + \nabla \cdot \mathbf{j} = \sigma \)  Continuity equation  
\( \frac{\partial\phi(\mathbf{r},t)}{\partial t} = \nabla \cdot \big[ D(\phi,\mathbf{r}) \ \nabla\phi(\mathbf{r},t) \big] \)  Diffusion equation  
\( F_{\rm d} = \frac{1}{2} \rho\, u^2\, c_{\rm d}\, A \)  Drag equation 


\( \)  Equation of motion  
\( \)  Equation of state  
\( \)  Equation of time  
\( \)  Heat equation  
\( p V = n R T \)  Ideal gas equation 


\( \)  Ideal MHD equations  
\( \)  Mass–energy equivalence equation  
\( \)  Primitive equations  
\( \)  Relativistic wave equations  
\( v^2 = GM \left({ 2 \over r}  {1 \over a}\right) \)  Visviva equation  astrodynamics 

\( \frac{D\boldsymbol\omega}{Dt} = \frac{\partial \boldsymbol \omega}{\partial t} + (\mathbf u \cdot \nabla) \boldsymbol \omega \)  Vorticity equation  
\( \sum_i I_i = 0 \)  Kirchoff’s Current Law  Electrical circuits  At every node of an electrical circuit, current \( I \) sums to zero. 
\( dU = dQ + dW \)  First Law of Thermodynamics  Thermodynamics  
\( \lambda = \sqrt{1  (v^2/c^2)} \)  Lorentz Transformations  
\( \)  Snell's law  optics  
\( \frac{1}{u} + \frac{1}{v} = \frac{1}{f} \)  Gauss Lens Formula, aka thin lens equation  optics  u = object distance; v = image distance; f = Focal length of the lens 
\( \frac{1}{f} = (n1)\left( \frac{1}{R_1}  \frac{1}{R_2} + \frac{(n1)d}{nR_1R_2} \right) \)  Lensmaker's equation  optics  The focal length of a lens in air, where

\( \lambda = h/p \)  De Broglie Wavelength  quantum mechanics 

\( 2 a \sin \theta = n \lambda \)  Bragg’s Law of Diffraction  optics 

\( \mathcal{L}_{SM} = \frac{1}{4} F_{\mu \nu} F^{\mu \nu} + ... \)  Lagrangian for the Standard model  subatomic physics  
\( F = G \frac{m_1m_2}{r^2} \)  Newton's law of universal gravitation  classical mechanics 

\( \hat{f}(\xi) = \int_{\infty}^{\infty} f(x)\ e^{2\pi i x \xi}\,dx \)  Fourier Transform  
\( \left[M\frac{\partial }{\partial M}+\beta(g)\frac{\partial }{\partial g}+n\gamma\right] G^{(n)}(x_1,x_2,\ldots,x_n;M,g)=0 \)  CallanSymanzik equation  Quantum field theory  
\( F_{\rm{spring}} = k x \)  Hooke's Law  Classical Mechanics  
\( \lambda_{\rm{peak}} = \frac{b}{T} \)  Wien's displacement law 


\( j^* = \sigma T^4 \)  Stefan–Boltzmann law  
\( F = K \frac{q_1 q_2}{r^2} \)  Coulomb's law  Electrodynamics  quantifies the amount of force between two stationary, electrically charged particles. 
\( P_1V_1 = P_2V_2 \)  Boyle's law  fluid mechanics  
\( V_1T_2 = V_2T_1 \)  Charles's law 


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