Applied mathematics

Function

f(x)=y

Function's Graphs

Linear function	$f(x)=Ax$
Parabolic function	$f(x)=Ax^{2}$

Mathematical Operation on Functions

Differentiation

f'(x)={\frac {d}{dt}}f(t)=\sum \lim _{\Delta x\to 0}{\frac {f(x+\Delta x)-f(x)}{\Delta x}}

Integration

F(t)=\int f(t)dt=\sum \lim _{\Delta x\to 0}\Delta x[{f(x)+{\frac {\Delta f(x)}{2}}}]

Transformations

Transformations	Laplace transform	Fourier transform
$f(t)$	$F(s)=\int f(t)e^{-st}$	$F(j\omega )=\int f(t)e^{-j\omega t}$
${\frac {d}{dt}}$	$s$	$j\omega$
$\int dt$	${\frac {1}{s}}$	${\frac {1}{j\omega }}$
$L{\frac {d}{dt}}$	$sL$	$j\omega L$
${\frac {1}{L}}\int dt$	${\frac {1}{sL}}$	${\frac {1}{j\omega L}}$

Partial Differentiation

Vector	${\frac {\partial f}{\partial x}}{\vec {x}}={\vec {f}}_{x}$	${\frac {\vec {f}}{\vec {x}}}={\frac {\partial f}{\partial x}}$	${\vec {x}}={\frac {\vec {f}}{\frac {\partial f}{\partial x}}}$
Scalar	${\frac {\partial f}{\partial x}}=f_{x}$
Unit vector	${\vec {x}}=1$

Laplace notation

Scalar operation	$\nabla ={\frac {\partial }{\partial x}}$	$\nabla R={\frac {\partial R}{\partial x}}$
Vector operation	$\nabla \cdot {\vec {R}}$ $\nabla \times {\vec {R}}$	$\nabla \cdot {\vec {R}}=X(\theta )=RCos\theta$ $\nabla \times {\vec {R}}=Y(\theta )=RSin\theta$

Solving equations

Polynomial equations

Equation	Equation	equation's root
1st ordered polynomial	$Ax+B=0$	$x=-{\frac {B}{A}}$
2nd ordered polynomial	$Ax^{2}+Bx+C=0$	$x=$

Differential equations

Electric circuit	Equation	Function	s	ω
Decay equation of series RL and RC circuit	$f^{'}(t)=-sf(t)$	Gives decay function $f(t)=Ae^{-st}$	$s={\frac {1}{T}}$
Oscillation Wave of seires LC circuit	$f^{''}(t)=-sf(t)$	Gives wave function $f(t)=Ae^{(\pm j{\sqrt {s}})t}=Ae^{\pm j\omega t}=ASin\omega t$	$s={\frac {1}{T}}$	$\pm j{\sqrt {s}}$

Simultaneous equations

Equation	Equation	equation's root
Linear equation of 2 variables	$a_{11}x+a_{12}y=a_{1n}$ $a_{21}x+a_{22}y=a_{2n}$
Linear equation of n variables

Coordination

Real Number Coordination

A point in XY co ordinate can be presented as ( $X,Y$ ) and ( $R,\theta$ ) in R θ co ordinate

A , (

X,Y

) , (

R,\theta

)

Scalar maths	Vector Maths
$R\angle \theta ={\sqrt {X^{2}+Y^{2}}}\angle Tan^{-1}{\frac {Y}{X}}$ $R={\sqrt {X^{2}+Y^{2}}}$ $\theta =\angle Tan^{-1}{\frac {Y}{X}}$	$X(\theta )=RCos\theta$ $Y(\theta )=RSin\theta$ $R(\theta )=X(\theta )+Y(\theta )=R(Cos\theta +Sin\theta )$ $\nabla \cdot R(\theta )=X(\theta )=RCos\theta$ $\nabla \times R(\theta )=Y(\theta )=RSin\theta$

Complex Number Coordination

A point in XY co ordinate can be presented as (X,Y) and (Z,θ) in Z θ co ordinate

Z.(X,jY),(Z,\theta )

Z^{*}.(X,-jY),(Z,-\theta )

X(\theta )=ZCos\theta

jY(\theta )=jZSin\theta

-jY(\theta )=-jZSin\theta

Z\angle \theta ={\sqrt {X^{2}+Y^{2}}}\angle Tan^{-1}{\frac {Y}{X}}

Z\angle -\theta ={\sqrt {X^{2}+Y^{2}}}\angle -Tan^{-1}{\frac {Y}{X}}

Z={\sqrt {X^{2}+Y^{2}}}

\theta =\angle Tan^{-1}{\frac {Y}{X}}

Z(\theta )=X(\theta )+jY(\theta )=Z(Cos\theta +jSin\theta )

\nabla \cdot Z(\theta )=X(\theta )=ZCos\theta

\nabla \times Z(\theta )=jY(\theta )=jZSin\theta

Z^{*}(\theta )=X(\theta )-jY(\theta )=Z(Cos\theta -jSin\theta )

\nabla \cdot Z(\theta )=X(\theta )=Z^{*}Cos\theta

\nabla \times Z(\theta )=-jY(\theta )=jZ^{*}Sin\theta

Cos\theta ={\frac {Z(\theta )+Z^{*}(\theta )}{2}}

Sin\theta ={\frac {Z(\theta )-Z^{*}(\theta )}{2j}}

-Sin\theta ={\frac {Z^{*}(\theta )-Z^{(}\theta )}{2j}}

Vector

Real coordination

For any vector in real coordination

R(\theta )=X(\theta )+Y(\theta )=R(Cos\theta +Sin\theta )=R\angle \theta

The horizontal component vector

\nabla \cdot R(\theta )=X(\theta )=RCos\theta

The vertical component vector

\nabla \times R(\theta )=Y(\theta )=RSin\theta

Complex coordination

Vector in complex coordination	Horizontal Vector Component	Vertical Vector Component
For any vector in complex coordination $Z(\theta )=X(\theta )+jY(\theta )=Z\angle \theta =Ze^{j\theta }=Z(Cos\theta +jSin\theta )$	$\nabla \cdot Z(\theta )=X(\theta )$	$\nabla \times Z(\theta )=jY(\theta )$
For any conjugate vector in complex coordination $Z^{}(\theta )=X(\theta )-jY(\theta )=Z^{}\angle -\theta =Z^{}e^{-j\theta }=Z^{}(Cos\theta -jSin\theta )$	$\nabla \cdot Z^{*}(\theta )=X(\theta )$	$\nabla \times Z^{*}(\theta )=-jY(\theta )$

Motion Formulas

Types of motion	$s$	$t$	$v$	$a$	$F$	$W$	$E$
Uniform Horizontal Linear Motion	$s$	$t$	${\frac {s}{t}}$	${\frac {v}{t}}$	$ma$	$Fs$	${\frac {W}{t}}$
Uniform Vertical Linear Motion	$h$	$t$	${\frac {h}{t}}$	${\frac {h}{t^{2}}}$	$mg$	$mgh$	${\frac {mgh}{t}}$
Non Uniform Linear Motion	$(v_{o}+at)t$	$t$	$v_{o}+at$	${\frac {\Delta v}{\Delta t}}$	$m{\frac {\Delta v}{\Delta t}}$	$Ft(v_{o}+at)$	$F(v_{o}+at)$
Non Linear Motion	$\int v(t)dt$	$t$	$v(t)$	${\frac {d}{dt}}v(t)$	$m{\frac {d}{dt}}v(t)$	$F\int v(t)dt$	${\frac {F}{t}}\int v(t)dt$
Complete Circle's Motion	$2\pi r$	$t$	${\frac {\omega r}{t}}=\omega r$	${\frac {\omega r}{t}}$	$m{\frac {\omega r}{t}}$	$p\omega r$	${\frac {p\omega r}{t}}$
Circle's Arc	$\theta r$	$t$	${\frac {\theta r}{t}}$	${\frac {\theta r}{t^{2}}}$	$m{\frac {\theta r}{t^{2}}}$	$p{\frac {\theta r}{t}}$	$p{\frac {\theta r}{t^{2}}}$
Wave	$\lambda$	$t$	${\frac {\lambda }{t}}$	${\frac {\lambda }{t^{2}}}$	$m{\frac {\lambda }{t^{2}}}$	$p{\frac {\lambda }{t}}$	$p{\frac {\lambda }{t^{2}}}$
Momentum			$v$	${\frac {v}{t}}$	${\frac {p}{t}}$	$pv$	$pa$

Oscillation Formulas

Oscillation	Wave Equation	Wave function	Angular speed	Time constant
Spring's Up and Down Motion	${\frac {d^{2}}{dt^{2}}}y(t)=-\omega y(t)$	$y(t)=ASin\omega t$	$\omega ={\sqrt {\frac {k}{m}}}$
Spring's Side to Side Motion	${\frac {d^{2}}{dt^{2}}}x(t)=-\omega x(t)$	$x(t)=ASin\omega t$	$\omega ={\sqrt {\frac {k}{m}}}$
Series RLC at equilibrium	${\frac {d^{2}}{dt^{2}}}i(t)=-2\alpha {\frac {d}{dt}}i(t)-\beta i(t)$	$i(t)=A(\alpha )Sin\omega t$	$\omega ={\sqrt {\beta -\alpha }}$	$T=LC$
Series RLC at resonsnce
Series LC at equilibrium	${\frac {d^{2}}{dt^{2}}}i(t)=-\omega i(t)$	$i(t)=ASin\omega t$	$\omega ={\sqrt {\frac {1}{T}}}$	$T=LC$
Series LC at resonsnce
Electromagnetic oscillation	$\nabla ^{2}E(t)=-\omega E(t)$ $\nabla ^{2}B(t)=-\omega B(t)$	$E(t)=ASin\omega t$ $B(t)=ASin\omega t$	$\omega ={\sqrt {\frac {1}{T}}}$	$T=\mu \epsilon$

Momentum Formulas

Momentum	Mass	Speed	Moment	Work
Mometum of a mass in motion	$m$	$v$	$mv=Ft$	$pv$
Mometum of a relativistic mass in motion	$m_{o}(\gamma -1)$	$\gamma ={\sqrt {1-{\frac {v^{2}}{C^{2}}}}}$	$mv$	$pv$
Mometum of a massless quanta in motion	$h=p\lambda$	$C=\lambda f$	${\frac {h}{\lambda }}$	$hf$

Electricity Formulas

Electriciy can be generated from the following sources

Electrolysis	DC	$v(t)=V$
Electrochemical Cell	DC	$v(t)=V$
Photon Electricity	DC	$v(t)=V$
Electromagnetic Induction	AC	$v(t)=VSin\omega t$

Electric Conductor

Matter that interacts with electricity are divided into three types Conductor, Semi Conductor, and Non Conductor

Conductor . All matter allow current to flow easily like Metal Ferrite, Copper
Semi Conductor . All matter does not allow current to flow easily like Semi Conductor Silicon, Germanium
Non Conductor . All matter does not allow current to flow like Rubber

Resistor

Resistor is a electric component thats exhibit resistance to the current flow . Resistance has a symbol R measured in ohm unit Ω .

Resistor is made from straight wire line conductor

DC Response	$V=IR$ $R={\frac {V}{I}}$ $I={\frac {V}{R}}$ $P=IV=I(IR)=I^{2}R$
AC Response	$Z_{R}=R+X_{R}=R+0=R$ $i={\frac {v}{Z_{R}}}$ $v=iZ_{R}$ $p=iv=i(iZ_{R})=i^{2}R$

Capacitor

DC response'

AC response

Q={\frac {V}{C}}

V={\frac {Q}{C}}

C={\frac {V}{Q}}

v_{C}={\frac {1}{C}}\int idt

i_{C}=C{\frac {dv}{dt}}

p_{C}={\frac {1}{2}}Cv^{2}

X_{C}={\frac {v_{C}}{i_{C}}}

X_{C}(\theta )={\frac {1}{\omega C}}\angle -90

$X_{C}(j\omega )={\frac {1}{j\omega C}}$

$Z_{C}=R_{C}+X_{C}$
$Z_{C}=R_{C}+{\frac {1}{j\omega C}}$
$Z_{C}={\frac {j\omega T+1}{j\omega C}}$
$Z_{C}=R_{C}\angle 0+{\frac {1}{\omega C}}\angle -90$
$Z_{C}={\sqrt {R_{C}^{2}+({\frac {1}{\omega T}})^{2}}}\angle -{\frac {1}{\omega T}}$

$T=CR_{C}$

Inductor

DC Reponse

Inductance . Inductance has a symbol L measured in Henry unit H

L={\frac {B}{I}}

Magnetic Intensity . Measurement of magnetic strength

B=LI

Current

I={\frac {B}{L}}

AC Response

Voltage . $v_{L}=L{\frac {di_{L}}{dt}}$

Current . $i_{L}={\frac {1}{L}}\int {dv_{L}}{dt}$

Reactance . Resistance to the AC current flow

X_{C}={\frac {v_{L}}{i_{L}}}

In frequency domain

X_{L}(j\omega )=j\omega L

In phasor domain

X_{L}(\omega \theta )=\omega L\angle 90

Impedance . Resistance to the AC current flow

Z_{L}=R_{L}+X_{L}

In frequency domain

Z_{L}=R_{L}+j\omega L

In phasor domain

Z_{L}=R_{L}\angle 0+j\omega L\angle 90

Z_{L}={\sqrt {R_{L}^{2}+(j\omega L)^{2}}}\angle \omega T

Time Constant

T={\frac {L}{R_{L}}}

Power . Power of the capacitor

p={\frac {1}{2}}Li^{2}

Straight line conductor

Entities	Symbol	Mathematical Formula	Unit
Voltage	$V$	$IR$	Volt . v
Current	$I$	${\frac {V}{R}}$	Ampere . a
Resistance	$R$	${\frac {V}{I}}$	Ohom . Ω
Conductance	$G$	${\frac {I}{V}}$	1/Ω
Power provided	$P_{V}$	$IV$	Watt . W
Power loss as Heat dissipated	$P_{R}$	$I^{2}R(T)$	W
Power transmitted	$P$	$P_{V}-P_{R}$	W
Magnetic field	$B$	$LI={\frac {\mu }{2\pi r}}I$

Electromagneticism Formulas

Entities	Symbol	Mathematical Formula	Unit
North pole	N	+
South pole	S	-
Magnetic field direction	-->	N --> S
Magnetic field	$B$	$LI={\frac {N\mu }{l}}$
Potential difference	$V$	${\frac {d}{dt}}B=L{\frac {d}{dt}}I$	Volt . v
Induced magnetic field	$\phi$	$-NB=-NLI$	Volt . v
Induced potential difference	$\epsilon$	$-{\frac {d}{dt}}\phi =-N{\frac {d}{dt}}B=-NL{\frac {d}{dt}}I$	Volt . v
Magnetic force	$F$	$Bl$	Newton . N
Magnetization field	$H$	$H={\frac {B}{\mu }}={\frac {\phi }{N\mu }}$
Magnetization equations		$\nabla \cdot D=\rho$ $\nabla \times E=-\nabla B$ $\nabla \cdot B=0$ $\nabla \times H=J+\nabla B$
Magnetic oscillation equation		$\nabla \cdot E=0$ $\nabla \times E={\frac {1}{T}}E$ $\nabla \cdot B=0$ $\nabla \times B={\frac {1}{T}}E$
Magnetic wave equation		$\nabla ^{2}E=-\omega E$ $\nabla ^{2}B=-\omega B$
Magnetic wave function		$E=ASin\omega t$ $B=ASin\omega t$ $\omega ={\sqrt {\frac {1}{T}}}$ $T=\mu \epsilon$
Magnetic wave radiation		$v=\omega ={\sqrt {\frac {1}{\mu \epsilon }}}=C=\lambda f=hf$ $E=pv=pC=p\lambda f=hf$ $h=p\lambda$ $p={\frac {h}{\lambda }}$ $\lambda ={\frac {h}{p}}={\frac {C}{f}}$

Photon Formulas

Photon

Photon, Massless Quanta's energy of electromagnetic radiation

E_{h}=hf=\hbar \omega

Photons are found in 2 states

Radiant photon at threshold frequency

E=hf_{o}=\hbar \omega _{o}

Non - Radiant photon at frequency greater than threshold frequency

E_{h}=hf=\hbar \omega

Photon can only be found in 1 out of 2 states at a time . Hence, Heiseinberg's uncertainty principle

\Delta p\Delta \lambda =({\frac {1}{2}})({\frac {h}{2\pi }})={\frac {h}{4\pi }}={\frac {\hbar }{2}}

Quanta

Quanta . Massless quantity

h=p\lambda

Process Wave Particle Duality

Wave like .

\lambda ={\frac {h}{p}}

Particle like .

p={\frac {h}{\lambda }}

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