If an expression is named, then it should be included here.
Page content is from Wikipedia's list of equation and Top Ten Important Equations In Physics and Physics formula list and six equations and 11 Most Beautiful Mathematical Equations and list of Physics equations
| Expression | name | domain | description |
|---|---|---|---|
| \(F = m a\) | Second Law of Motion | Classical Mechanics | |
| \(E = m c^2\) | Energy-mass 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 | |
| \( \vec{\nabla} \times \vec{E} = - \frac{\partial \vec{B}}{\partial t} \) | Maxwell-Faraday 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 anti-matter, 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 | |
| \( \frac{\partial \mathbf{u}}{\partial t} + (\mathbf{u} \cdot \nabla) \mathbf{u} - \nu \,\nabla^2 \mathbf{u} = - \nabla w + \mathbf{g} \) | Navier-Stokes | 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 | ||
| \( \dot\rho=-{i\over\hbar}[H,\rho]+\sum_{n,m = 1}^{N^2-1} h_{nm}\left(A_n\rho A_m^\dagger-\frac{1}{2}\left\{A_m^\dagger A_n, \rho\right\}\right) \) | 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 | ||
| \( \) | Reynolds-averaged 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 | ||
| \( v^2 = GM \left({ 2 \over r} - {1 \over a}\right) \) | Vis-viva 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 |
| \( \frac{1}{f} = (n-1)\left( \frac{1}{R_1} - \frac{1}{R_2} + \frac{(n-1)d}{nR_1R_2} \right) \) | 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 | ||
| \( \left[M\frac{\partial }{\partial M}+\beta(g)\frac{\partial }{\partial g}+n\gamma\right] G^{(n)}(x_1,x_2,\ldots,x_n;M,g)=0 \) | Callan-Symanzik 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 |
|
|
| \( \) | |||
| \( \) | |||
| \( \) | |||
| \( \) | |||
| \( \) | |||
| \( \) | |||
| \( \) | |||
| \( \) |