Identities and Formulas Sheet

Geometry

Arc Length \begin{equation} s=\int_a^b \sqrt{1+f'(x)^2} dx \end{equation} \begin{equation} s=\int_a^b \sqrt{\rho^2+(\frac{d\rho}{d\theta} )^2} d\theta \ \ \ \ \ \ (Polar\ coordinates) \end{equation}

Formulas

The mean of a function over an interval T is given by \begin{equation} \langle f(t) \rangle = \frac{1}{T} \int_0^{T} f(t) dt \end{equation} The Taylor Series is given by \begin{equation} f(x)= \sum_0^\infty \frac{f^{(n)} (x_0)}{n !} (x-x_0)^n) \end{equation} If $x_o=0$, then (Maclaurin series) \begin{equation} f(x)= \sum_0^\infty \frac{f^{(n)} (0)}{n !} x^n) \end{equation}

Trigonometric Identities

\begin{equation} \sin{x}^2 + \cos{x}^2 = 1 \end{equation} \begin{equation} \sin(x \pm y) = \sin{x}\cos{y} \pm \sin{y}\cos{x} \end{equation} \begin{equation} \cos(x \pm y) = \cos{x}\cos{y} \pm \sin{x}\sin{y} \end{equation} \begin{equation} e^{\pm i \theta} =\cos(\theta) \pm i \sin(\theta) \end{equation}

Vector Calculus Identities

\begin{equation} \vec{A} \cdot \vec{B} \times \vec{C} = \vec{A} \times \vec{B} \cdot \vec{C} \end{equation} \begin{equation} \vec{A} \times( \vec{B} \times \vec{C}) = \vec{B}( \vec{A} \cdot \vec{C}) - \vec{C}( \vec{A} \cdot \vec{B}) \end{equation} \begin{equation} \vec{\nabla} \times \vec{\nabla} \Phi =0 \end{equation} \begin{equation} \vec{\nabla} \cdot \vec{\nabla} \times \vec{V} =0 \end{equation}

Optics

Velocity $v$ \begin{equation} v=\frac{\omega}{k}=\frac{c}{n} \end{equation} Frequency $\nu$ \begin{equation} \nu=\frac{\omega}{2 \pi}=\frac{1}{T}=\frac{v}{\lambda} \end{equation} Wavelength $\lambda$ \begin{equation} \lambda=v \frac{2 \pi}{\omega}=v T \end{equation} \begin{equation} \lambda_0=c T=n \lambda \end{equation} Wavenumber $k$ \begin{equation} k=\frac{2 \pi}{\lambda}=\frac{n \omega}{c}=\frac{\omega}{v} \end{equation} \begin{equation} k_0=\frac{k}{n}=\frac{2 \pi}{\lambda_0}=\frac{\omega}{c} \end{equation}

SI Base Units

Amount of substance	(mole)	$Mol$
Electric current	(ampere)	$A$
Length	(meter)	$m$
Luminous intensity	(candela)	$Cd$
Mass	(kilogram)	$kg$
Time	(second)	$s$
Temperature	(Kelvin)	$K$

SI Derived Units

Frequency	(Hertz)	$Hz=\frac{1}{s}$
Force	(Newton)	$N=\frac{m\cdot kg}{s^2}$
Pressure,stress	(Pascal)	$Pa=\frac{N}{m^2} =\frac{J}{m^3}=\frac{kg}{m\cdot s^2}$
Energy,Work,quantity of heat	(Joule)	$J=N\cdot m= Pa\cdot m^3=W\cdot s=C\cdot V=\frac{kg \cdot m^2}{s^2}$
Power	(Watt)	$W=\frac{J}{s}=\frac{N\cdot m}{s} = V\cdot A =\frac{V^2}{\Omega}=\frac{kg \cdot m^2}{s^3}$
Electric charge	(Coulomb)	$C=A\cdot s=F\cdot V$
Electric potential difference	(Volt)	$V=\frac{W}{A}=\frac{J}{C}=A\cdot \Omega =\frac{eV}{e}=\frac{kg \cdot m^2}{A\cdot s^3}$
Electric resistance	(Ohm)	$\Omega=\frac{V}{A}=\frac{W}{A^2}=\frac{V^2}{W}=\frac{s}{F}=\frac{J\cdot s}{C^2}=\frac{J}{s\cdot A^2}=\frac{kg\cdot m^2}{s\cdot C^2} =\frac{kg \cdot m^2}{A^2 \cdot s^3}$
Electric capacitance	(Farads)	$F=\frac{C}{V}=\frac{A\cdot s}{V}=\frac{J}{V^2}=\frac{W\cdot s}{V^2}=\frac{N\cdot m}{V^2}=\frac{C^2}{J}=\frac{C^2}{N\cdot m}=\frac{s}{\Omega}=\frac{1}{\Omega \cdot Hz}=\frac{s^2}{H}=\frac{A^2\cdot s^4}{kg\cdot m^2}=\frac{C^2\cdot s^2}{kg\cdot m^2}$
Electrical Inductance	(Henry)	$H=\frac{kg\cdot m^2}{s^2 \cdot A^2}$
Magnetic Flux	(Weber)	$Wb=\frac{kg\cdot m^2}{s^2 \cdot A}$
Magnetic Flux density	(Tesla)	$T=\frac{Wb}{m^2}=\frac{kg}{s^2 \cdot A}$

Physical Constants

$\alpha = 0.00729927$		fine structure constant
$K = 1.38064852\times 10^{-23}\ J/K = 8.6173324\times 10^{-5}\ eV/K$		boltzman constant
$Na = 6.022140857\times 10^{23}\ mol^{-1} $		avogadro constant
$R_{gas} = 8.3144598\ J/(K\ mol)$		gas constant
$1_{atm} = 1.013\times 10^{5} N/m^2$		1 atmosphere
$bar = 1\times 10^{5}\ N/m^2$		1 bar
$\sigma = 5.670367\times 10^{-8}\ \frac{W}{m^2K^4}$		stefan constant
$M_s = 1.98855\times 10^{30}\ kg$		solar mass
$L_s = 3.828\times 10^{26}\ W$		solar luminosity
$R_s = 6.957\times 10^{8}\ m$		solar radius
$c = 2.99792458\times 10^{8}\ m/s$		speed of light
$\pi = 3.14159265359$		pi
$h = 6.62607004\times 10^{-34}\ Js = 4.135667662\times 10^{-15}\ eV s$		planck constant
$\hbar = 1.054571800\times 10^{-34}\ Js = 6.582119514\times 10^{-16}\ eV s$		reduced planck constant
$hc=1.98644568\times 10^{-25}\ Jm = 1.23984193\times 10^{-6}\ eVm$		pi
$e = 1.60217662\times 10^{-19}\ C$		electron charge
$eV = e\ [eV/J]$		electron volts/joule
$m_e = 9.10938356\times 10^{-31}\ kg = 0.5109989461\ MeV/c^2$		electron mass
$m_p = 1.6726219\times 10^{-27}\ kg$		proton mass
$\epsilon_0 = 8.854187817\times 10^{-12}\ \frac{A^2s^4}{m^{3}kg}$		vacuum permittivity [F/m]
$G_{grav} = 6.67408\times 10^{-11}\ Nm^2/kg^2$		newton gravitational constant