david

David F. Miller

    Important Equations and Formulas (skip this page)

Galileo:
               $d \propto t^2$

Newton:

               $\vec F = \frac{d\vec p}{dt} \, = \, \frac{d}{dt} (m \vec v) \, = \, \vec v \, \frac{dm}{dt} + m \, \frac{d\vec v}{dt} \,.$
               $F = G \frac{m_1 m_2}{r^2}$

Lagrange:
               $\mathcal{S} [\varphi_i] = \int{\mathcal{L} [\varphi_i (x)]\, \mathrm{d}^4x}$
            $\frac{\delta\mathcal{S}}{\delta\varphi}=-\partial_\mu \left(\frac{\partial\mathcal{L}}{\partial(\partial_\mu\varphi)}\right)+ \frac{\partial\mathcal{L}}{\partial\varphi}=0.$

Hamilton:

               $H = \sum_i p_i {\dot q_i} - L$
         $\frac{\mathrm dq}{\mathrm dt}(t) =~~\frac{\partial H}{\partial p}(p(t),q(t),t)$
            $\frac{\mathrm dq}{\mathrm dt}(t) =~~\frac{\partial H}{\partial p}(p(t),q(t),t)$

Noether:
               $\partial_\mu\left[\frac{\partial\mathcal{L}}{\partial(\partial_\mu\phi)}Q[\phi]-f^\mu\right]=0.$

Maxwell-Gauss-Faraday-Ampère:


                  $\nabla \cdot \varepsilon \mathbf{E} = \rho$
             $\nabla \cdot \mu \mathbf{H} = 0$
               $\nabla \times \mathbf{E} = - \mu \frac{\partial \mathbf{H}} {\partial t}$
               $\nabla \times \mathbf{H} = \mathbf{J} + \varepsilon \frac{\partial \mathbf{E}} {\partial t}$
Lorentz:
               $\begin{align}t' &= \gamma \left( t - \frac{v x}{c^{2}} \right) \\ x' &= \gamma \left( x - v t \right)\\ y' &= y \\ z' &= z \end{align}$

Einsten:
             $R_{\mu \nu} - {\textstyle 1 \over 2}R\,g_{\mu \nu} = -\kappa T_{\mu \nu} = -{8 \pi G \over c^4} T_{\mu \nu}.$
               $\frac{\mathrm{d}^2 x^a}{\mathrm{d}\tau^2} + \Gamma^a_{bc} \, \frac{\mathrm{d} x^b}{\mathrm{d}\tau} \,\frac{\mathrm{d} x^c}{\mathrm{d}\tau} = 0$
               $E = \sqrt{p^2c^2+m^2c^4} = \gamma mc^2$
               $\gamma = \frac{1}{\sqrt{1-(\frac{v}{c})^2}} \approx \left( 1+ \frac{1}{2} \left(\frac{v}{c} \right)^2 \right)$

              $E_\mathrm{kinetic} \approx \frac{1}{2} \left(\frac{v}{c} \right)^2 m_0 c^2 =\frac{1}{2} m_0 v^2$

Boltzmann:

                 $S = k_B \ln \Omega \!$
                $\frac{\partial f}{\partial t}+ v \frac{\partial f}{\partial x}+ \frac{F}{m} \frac{\partial f}{\partial v} = \frac{\partial f}{\partial t}\left.{\!\!\frac{}{}}\right|_\mathrm{collision}$

Helmholtz:

                 $A=U-TS\,$

Maxwell-Boltzmann:

                 $d S = {1 \over T} (d U + P d V - \mu d N)$

Navier-Stokes:

                 $\rho \left(\frac{\partial \mathbf{v}}{\partial t} + \left(\mathbf{v} \cdot \nabla\right) \mathbf{v}\right) = -\nabla p + \mu \nabla^2 \mathbf{v} + \mathbf{f}$
                 $\frac{\partial \rho}{\partial t} + \nabla \cdot (\rho \mathbf{v}) = 0$

Planck:

              $I(\nu,T) =\frac{2 h\nu^{3}}{c^2}\frac{1}{ e^{\frac{h\nu}{kT}}-1}.$
Stefan-Boltzmann:

             $j^{\star} = \epsilon\sigma T^{4}$

Bohr:

             $E_n = \frac{-m_e q_e^4}{8 h^2 \epsilon_{0}^2} \frac{1}{n^2} \,$
Rydberg:

               $\frac{1}{\lambda}=\frac{m_e q_e^4}{8 c h^3 \epsilon_{0}^2} \left( \frac{1}{n_{f}^2} - \frac{1}{n_{i}^2} \right). \,$

Heisenberg:
         $\frac{d}{dt}A=(i\hbar)^{-1}[A,H]+\left(\frac{\partial A}{\partial t}\right)_\mathrm{classical}.$
               $\Delta x \Delta p \ge \frac{\hbar}{2}$
           $\Delta E \Delta t \ge \frac{\hbar}{2}$ ,   .
Schrödinger:

             $\left[-\frac{\hbar^2}{2 m} \nabla^2 + U(\mathbf{r}) \right] \psi (\mathbf{r}) = E \psi (\mathbf{r}),$

Dirac:
           $\left(\alpha_0 mc^2 + \sum_{j = 1}^3 \alpha_j p_j \, c\right) \psi (\mathbf{x},t) = i \hbar \frac{\partial\psi}{\partial t}(\mathbf{x},t)$

Fermi:

           $n^0 \rightarrow p^+ + e^- + \bar{\nu}_e$

Hawking:
           $T={\hbar\,c^3\over8\pi G M k}$

Yang-Mills:
          $\ \mathcal{L}_\mathrm{gf} = - \frac{1}{4} \operatorname{Tr}(F^{\mu \nu} F_{\mu \nu})$
          $\ F_{\mu \nu} = [D_\mu, D_\nu]$
Feynman:

Shannon:

          $\begin{matrix} H(X) = \operatorname{E}( I(X) ) & = & \displaystyle{\sum_{i=1}^np(x_i)\log_2 \left(1/p(x_i)\right)} \\ & = & - \displaystyle{\sum_{i=1}^np(x_i)\log_2 p(x_i)} \qquad \end{matrix}$
         $R < C \,$

           (Channel Capacity Theorem)

Hamming:

X <= d + 1

     Two example distances: 100->011 has distance 3 (red path);
     010->111 has distance 2 (blue path)

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