MSPS:Constants and Equations Poster

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Table of contents

1 Overview

2 Constants

2.1 Universal
2.2 Electromagnetism
2.3 Atomic/Nuclear
2.4 Thermodynamics

3 Equations

3.1 Vector Calculus
3.2 Maxwell's Equations

3.2.1 Differential Form
3.2.2 Integral Form

3.3 Fourier Expansions
3.4 Taylor Expansions
3.5 Trigonometric Functions
3.6 Tensor Manipulation

3.6.1 Levi-Civita Symbol

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Overview

We are trying to compile a list of the most used physical constants and equations. This list will be on a giant poster in the lounge so that any student doing homework in the lounge who needs to look up the value of a physical constant or an equation can literally just look up.

So we leave it to YOU. Please help us compile a list of the physical constants and equations that you would like on the poster. Preferably with their values.

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Constants

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Universal

$\mu_0 = 4 \pi\ \times \ 10^{-7}\ N/A^2$
- $= 4.135 667 33 \times 10^{-15} eV s$
- $= 6.582 118 99 \times 10^{-16} eV s$
$\mathrm{c} = 2.99 792 458 \times 10^{8} m/s$
$\epsilon_0 = 8.854 187 817... \times 10^{-12} F m^{-1}$
$G = 6.674 28 \times 10^{-11} m^{3} kg^{-1} s^{-2}$

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Electromagnetism

$\mathrm{e} = 1.602 176 487 \times 10^{-19} C$

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Atomic/Nuclear

$\alpha = \mu_0 e^2 c / (2 h) \$
$m_e = 9.109 382 15(45) \times 10^{-31} kg$
$m_p = 1.672 621 637(83) \times 10^{-27} kg$

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Thermodynamics

$k_B=1.380 6505 \times 10^{-23} J \cdot K^{-1}$
$R=8.314 472 J \cdot K^{-1}\cdot mol^{-1}$

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Equations

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Vector Calculus

Green's Theorem

$\int_{C^+}P\,dx+Q\,dy=\iint_D(\frac{\delta Q}{\delta x}-\frac{\delta P}{\delta y})\,dx\,dy$

Green's Theorem (Vector Form)

$\int_{\delta D} F\cdot\,ds = \iint_D \nabla \times F\cdot k\,dA$

Stoke's Theorem

$\int_{\delta S}F\cdot ds=\iint_S (\nabla \times F)\cdot dS$

Gauss' Divergence Theorem

$\iint_{\delta W}F\cdot dS=\iiint_W \nabla \cdot F\,dV$

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Maxwell's Equations

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Differential Form

Gauss' Law for electricity

$\nabla \cdot E = \frac{\rho}{\epsilon_0}$

Gauss' Law for magnetism

$\nabla \cdot B = 0$

Faraday's Law of induction

$\nabla \times E = - \frac{\partial B}{\partial t}$

Ampere's Circuital Law

$\nabla \times B = \mu_0 J + \mu_0 \epsilon_0 \frac{\partial E}{\partial t}$

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Integral Form

Gauss' Law for electricity

$\oint_S E \cdot dA = \frac{Q_s}{\epsilon_0}$

Gauss' Law for magnetism

$\oint_S B \cdot dA = 0$

Faraday's Law of induction

$\oint_{\partial S} E \cdot dl = -\frac{d \Phi_B,S}{dt}$

Ampere's Circuital Law

$\oint_{\partial S} B \cdot dl = \mu_0 I_S + \mu_0 \epsilon_0 \frac{d \Phi_E,S}{dt}$

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Fourier Expansions

$f(x) = \frac{a_0}{2} + \sum_{n=1}^{\infty} a_n \cos{(\frac{n \pi L}{L})} + b_n \sin{(\frac{n \pi L}{L})}$

where:

$a_n = \frac{1}{L} \int_{-L}^{L} f(x) \cos{(\frac{n \pi L}{L})} \, dx$

$b_n = \frac{1}{L} \int_{-L}^{L} f(x) \sin{(\frac{n \pi L}{L})} \, dx$

$a_0 = \frac{1}{L} \int_{-L}^{L} f(x) \, dx$

and $2 L$ your period.

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Taylor Expansions

$\sin(x)=x-\frac{x^3}{3!}+\frac{x^5}{5!}-...$

$\cos(x)=1-\frac{x^2}{2!}+\frac{x^4}{4!}-...$

$\sinh(x)=x+\frac{x^3}{3!}+\frac{x^5}{5!}+...$

$\cosh(x)=1+\frac{x^2}{2!}+\frac{x^4}{4!}+...$

$e^x=1+x+\frac{x^2}{2!}+\frac{x^3}{3!}-...$

Plus others from Complex Variables.

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Trigonometric Functions

First of all, obviously:

$e^{i \theta}= \cos \theta+i \sin{ \theta}\frac{}{}$

By using this formula and by plugging in $( - θ)$ and by making a substitution (that is left to the reader as an exercise), we obtain a pair of equations:

$\cos z = \frac{1}{2}(e^{iz}+e^{-iz})$