Formulas

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Conservation of angular momentum

$ \mathbf{\tau} = \frac{\mathrm{d}\mathbf{L}}{\mathrm{d}t} = \frac{\mathrm{d}\mathbf{r}}{\mathrm{d}t} \times \mathbf{p} + \mathbf{r} \times \frac{\mathrm{d}\mathbf{p}}{\mathrm{d}t} = 0 + \mathbf{r} \times \mathbf{F} = \mathbf{r} \times \mathbf{F} $

Standard Model

$ \mathcal{L_{SM}} = \underbrace {\frac{1}{4}W_{\mu\nu}\cdot W^{\mu\nu}-\frac{1}{4}B_{\mu\nu}B^{\mu\nu}-\frac{1}{4}G^a_{\mu\nu}G^{{\mu\nu}}_a } $

Kinetic energies and self-interactions of the gauge bosons

+ $ \underbrace {\overline{L}\gamma^\mu(i\partial_\mu-\frac{1}{2}g\tau\cdot W_\mu-\frac{1}{2}g'YB_\mu)L + \overline{R}\gamma^\mu(i\partial_\mu-\frac{1}{2}g'Y B_\mu)R } $

Kinetic energies and electroweak interactions of fermions

+ $ \underbrace {\frac{1}{2} \vert ( i\partial_\mu-\frac{1}{2}g\tau\cdot W_\mu-\frac{1}{2}g'YB_\mu)\phi \vert ^2 - V(\phi) } $

W ± Z, y, and Higgs masses and couplings

+ $ \underbrace { g''(\overline{q}\gamma^\mu T_aq)G^a_\mu } $ + $ \underbrace { (G_1\overline{L}\phi+G_2\overline{L}\phi_c R+h.c.) } $

W ± Z, y, and Higgs masses and couplings
+ fennion masses and couplings to Higgs

Einstein's theory of relativity

Mass–energy equivalence

$ E = mc^2 $

Special Relativity

$ t' = t \frac{1}{\sqrt{1 - \frac{v^2}{c^2}}} $

General Relativity (1915)

$ G_{\mu \nu} + \Lambda g_{\mu \nu}= {8\pi G\over c^4} T_{\mu \nu} $

Einstein field equations (EFE)

$ R_{\mu \nu} - {1 \over 2}g_{\mu \nu}\,R + g_{\mu \nu} \Lambda = {8 \pi G \over c^4} T_{\mu \nu} $
$ G_{\mu\nu} = 8 \pi G ( T_{\mu\nu} + \rho \Lambda g_{\mu\nu}) $
$ G_{\mu\nu}\equiv R_{\mu\nu} - {\textstyle 1 \over 2}R\,g_{\mu\nu} = {8 \pi G \over c^4}   T_{\mu\nu}.\, $

Isaac Newton

Newton's laws of motion

Are three physical laws that together laid the foundation for classical mechanics. They describe the relationship between a body and the forces acting upon it, and its motion in response to said forces. They have been expressed in several different ways over nearly three centuries,[1] and can be summarized as follows.

  1. Law I: Every body persists in its state of being at rest or of moving uniformly straight forward, except insofar as it is compelled to change its state by force impressed.
  2. Law II: The alteration of motion is ever proportional to the motive force impress'd; and is made in the direction of the right line in which that force is impress'd.
  3. Law III: To every action there is always opposed an equal reaction: or the mutual actions of two bodies upon each other are always equal, and directed to contrary parts.

Newton's first law

When viewed in an inertial reference frame, an object either remains at rest or continues to move at a constant velocity, unless acted upon by an external force.

$ \sum \mathbf{F} = 0\; \Leftrightarrow\; \frac{\mathrm{d} \mathbf{v} }{\mathrm{d}t} = 0. $

Newton's second law

The vector sum of the external forces F on an object is equal to the mass m of that object multiplied by the acceleration vector a of the object: F=ma.

$ \vec{F} = m\vec{a} $

The second law states that the net force on an object is equal to the rate of change (that is, the derivative) of its linear momentum p in an inertial reference frame:

$ \mathbf{F} = \frac{\mathrm{d}\mathbf{p}}{\mathrm{d}t} = \frac{\mathrm{d}(m\mathbf v)}{\mathrm{d}t}. $

Thus,

$ \mathbf{F} = m\,\frac{\mathrm{d}\mathbf{v}}{\mathrm{d}t} = m\mathbf{a}, $

where F is the net force applied, m is the mass of the body, and a is the body's acceleration.

Newton's third law

When one body exerts a force on a second body, the second body simultaneously exerts a force equal in magnitude and opposite in direction on the first body.

$ \vec{F}_{12} = -\vec{F}_{21} $

Newton's law of universal gravitation

In modern language, the law states the following: Every point mass attracts every single other point mass by a force pointing along the line intersecting both points. The force is proportional to the product of the two masses and inversely proportional to the square of the distance between them.

$ F = G {m_1 m_2 \over d^2} $

where:

  • F is the force between the masses;
  • G is the gravitational constant (6.673×10−11 N · (m/kg)2);
  • m1 is the first mass;
  • m2 is the second mass;
  • r is the distance between the centers of the masses.

Vector Form

$ \mathbf{F}_{12} = - G {m_1 m_2 \over {\vert \mathbf{r}_{12} \vert}^2} \, \mathbf{\hat{r}}_{12} $ where

F12 is the force applied on object 2 due to object 1,
G is the gravitational constant,
m1 and m2 are respectively the masses of objects 1 and 2,
|r12| = |r2r1| is the distance between objects 1 and 2, and
$ \mathbf{\hat{r}}_{12} \ \stackrel{\mathrm{def}}{=}\ \frac{\mathbf{r}_2 - \mathbf{r}_1}{\vert\mathbf{r}_2 - \mathbf{r}_1\vert} $ is the unit vector from object 1 to 2.

Calculus

$ \int_{a}^{b} f(x)\,dx = F(b) - F(a). $

The fundamental theorem of calculus

$ {df \over dt} = \lim_{h\to 0}\frac{f(t+h)-f(t)}{h} $

Pythagorean theorem

$ a^2 + b^2 = c^2 $

1 = 0.999999999…

$ 1 = 0.99999999999999999999…. $

Euler's identity

The Most Beautiful Equation in The World

Euler's equation

$ e^{i \pi} + 1 = 0 $

Euler's equation

$ V - E + F = 2 $ or $ F - E + V = 2 $

Euler–Lagrange equations and Noether's theorem

$ \frac{d}{dt} \left( \frac{\partial L}{\partial \dot{q}_k} \right) = \frac{dp_k}{dt} = 0~, $

Euler line

$ \sin 2A \sin(B - C)x + \sin 2B \sin(C - A)y + \sin 2C \sin(A - B)z = 0.\, $

The Callan-Symanzik equation

$ \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 $

The minimal surface equation

$ A_(u_) = \int_ \Omega ( 1 + |\nabla u\dashv ^2 )^\frac{1}2 dx_1,dx_2,\ldots,dx_n, $

The logarithm and its identities

$ \log_b(xy) = \log_b(x) + \log_b(y) \!\, $

The origin of complex numbers

$ i^2 = -1 $

The normal distribution

$ \Sigma(x) = \frac{1}{\sigma\sqrt{2\pi\sigma}} e^{ -\frac{(x-\mu)^2}{2\sigma^2} } $

The wave equation

$ { \partial^2 u \over \partial t^2 } = c^2 \nabla^2 u $

or

$ { \partial^2 u \over \partial t^2 } = c^2 { \partial^2 u \over \partial x^2 } $

The Fourier transform

$ \hat{f}(\xi) = \int_{-\infty}^\infty f(x)\ e^{- 2\pi i x \xi}\,dx $

The Navier-Stokes equations

$ \rho \left(\frac{\partial \mathbf{v}}{\partial t} + \mathbf{v} \cdot \nabla \mathbf{v} \right) = -\nabla p + \nabla \cdot\boldsymbol{\mathsf{T}} + \mathbf{f} $

Maxwell's equations

Maxwell's equations (1861-62) / Electromagnetism

$ \begin{align} \nabla \cdot \mathbf{E} &= 0 \quad &\nabla \times \mathbf{E} = \ -&\frac{\partial\mathbf B}{\partial t} \\ \nabla \cdot \mathbf{B} &= 0 \quad &\nabla \times \mathbf{B} = \frac{1}{c^2} &\frac{\partial \mathbf E}{\partial t} \end{align} $

Gauss's law

$ \Phi_E = \frac{Q}{\varepsilon_0} $

Gauss's law (Differential form)

$ \nabla \cdot \mathbf{E} = \frac{\rho}{\varepsilon_0} $

Laws of thermodynamics

Zeroth law of thermodynamics

If two systems are in thermal equilibrium respectively with a third system, they must be in thermal equilibrium with each other. This law helps define the notion of temperature.

First law of thermodynamics

The law of conservation of energy states that the total energy of an isolated system is constant; energy can be transformed from one form to another, but cannot be created or destroyed.

$ {\Delta U}system = Q - W $

Conservation of Energy Equation

The law of conservation of energy states that the total energy of an isolated system remains constant—it is said to be conserved over time. Energy can be neither created nor be destroyed, but it transforms from one form to another, for instance chemical energy can be converted to kinetic energy

$ E = K + U $

or

$ mgh = \frac{1}{2} mv^2 $

Second law of thermodynamics

$ dS \geq 0 $

or

$ dS = \frac{\delta Q}{T} \! $

Third law of thermodynamics

The entropy of a perfect crystal at absolute zero is exactly equal to zero.

$ S - S_0 = k_B \ln \, \Omega \ $

where S is entropy, kB is the Boltzmann constant, and $ \Omega $ is the number of microstates consistent with the macroscopic configuration. The counting of states is from the reference state of absolute zero, which corresponds to the entropy of S0.

Schrödinger equation

$ i \hbar \frac{\partial}{\partial t}\Psi = \hat H \Psi $

Shannon's information theory

$ H = - \sum p(x) \log p(x) $

or

$ H(X) = \mathbb{E}_{X} [I(x)] = -\sum_{x \in \mathbb{X}} p(x) \log p(x) $

The logistic model for population growth

$ X_t+1 = kx_t(1-x_t) $

The Black–Scholes model

$ \frac{1}{2}\sigma^2 S^2 \frac{\partial^2 V}{\partial S^2} + rS\frac{\partial V}{\partial S} + \frac{\partial V}{\partial t} - rV = 0 $

Infinite Pi

$ \frac{1}{\pi} = \frac{2\sqrt{2}}{9801} \sum^\infty_{k=0} \frac{(4k)!(1103+26390k)}{(k!)^4 396^{4k}}. $

Pick A Digit, Any Digit (of Pi)

$ \pi = \sum_{k=0}^\infty \frac{1}{16^k} \left( \frac{4}{8k + 1} - \frac{2}{8k + 4} - \frac{1}{8k + 5} - \frac{1}{8k + 6}\right) $

Dividing By Zero

$ \lim_{x \to 0} {\sin{x}\over x} = 1 $

Tupper's self-referential formula

$ {1\over 2} < \left\lfloor \mathrm{mod}\left(\left\lfloor {y \over 17} \right\rfloor 2^{-17 \lfloor x \rfloor - \mathrm{mod}(\lfloor y\rfloor, 17)},2\right)\right\rfloor $

Notes