# Applied mathematics

## Function

${\displaystyle f(x)=y}$

## Function's Graphs

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

## Mathematical Operation on Functions

### Differentiation

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

### Integration

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

### Transformations

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

### Partial Differentiation

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

Laplace notation

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

## Solving equations

### Polynomial equations

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

### Differential equations

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

### Simultaneous equations

 Equation Equation equation's root Linear equation of 2 variables ${\displaystyle a_{11}x+a_{12}y=a_{1n}}$${\displaystyle 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 (${\displaystyle X,Y}$) and (${\displaystyle R,\theta }$) in R θ co ordinate

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

 ${\displaystyle Z.(X,jY),(Z,\theta )}$${\displaystyle Z^{*}.(X,-jY),(Z,-\theta )}$ ${\displaystyle X(\theta )=ZCos\theta }$${\displaystyle jY(\theta )=jZSin\theta }$${\displaystyle -jY(\theta )=-jZSin\theta }$ ${\displaystyle Z\angle \theta ={\sqrt {X^{2}+Y^{2}}}\angle Tan^{-1}{\frac {Y}{X}}}$${\displaystyle Z\angle -\theta ={\sqrt {X^{2}+Y^{2}}}\angle -Tan^{-1}{\frac {Y}{X}}}$${\displaystyle Z={\sqrt {X^{2}+Y^{2}}}}$${\displaystyle \theta =\angle Tan^{-1}{\frac {Y}{X}}}$ ${\displaystyle Z(\theta )=X(\theta )+jY(\theta )=Z(Cos\theta +jSin\theta )}$${\displaystyle \nabla \cdot Z(\theta )=X(\theta )=ZCos\theta }$${\displaystyle \nabla \times Z(\theta )=jY(\theta )=jZSin\theta }$${\displaystyle Z^{*}(\theta )=X(\theta )-jY(\theta )=Z(Cos\theta -jSin\theta )}$${\displaystyle \nabla \cdot Z(\theta )=X(\theta )=Z^{*}Cos\theta }$${\displaystyle \nabla \times Z(\theta )=-jY(\theta )=jZ^{*}Sin\theta }$${\displaystyle Cos\theta ={\frac {Z(\theta )+Z^{*}(\theta )}{2}}}$${\displaystyle Sin\theta ={\frac {Z(\theta )-Z^{*}(\theta )}{2j}}}$${\displaystyle -Sin\theta ={\frac {Z^{*}(\theta )-Z^{(}\theta )}{2j}}}$

## Vector

### Real coordination

 For any vector in real coordination${\displaystyle R(\theta )=X(\theta )+Y(\theta )=R(Cos\theta +Sin\theta )=R\angle \theta }$ The horizontal component vector${\displaystyle \nabla \cdot R(\theta )=X(\theta )=RCos\theta }$ The vertical component vector${\displaystyle \nabla \times R(\theta )=Y(\theta )=RSin\theta }$

### Complex coordination

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

## Motion Formulas

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

## Oscillation Formulas

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

## Momentum Formulas

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

## Electricity Formulas

Electriciy can be generated from the following sources

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

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

### Capacitor

 DC response' AC response ${\displaystyle Q={\frac {V}{C}}}$${\displaystyle V={\frac {Q}{C}}}$ ${\displaystyle C={\frac {V}{Q}}}$ ${\displaystyle v_{C}={\frac {1}{C}}\int idt}$${\displaystyle i_{C}=C{\frac {dv}{dt}}}$${\displaystyle p_{C}={\frac {1}{2}}Cv^{2}}$${\displaystyle X_{C}={\frac {v_{C}}{i_{C}}}}$ ${\displaystyle X_{C}(\theta )={\frac {1}{\omega C}}\angle -90}$ ${\displaystyle X_{C}(j\omega )={\frac {1}{j\omega C}}}$ ${\displaystyle Z_{C}=R_{C}+X_{C}}$${\displaystyle Z_{C}=R_{C}+{\frac {1}{j\omega C}}}$${\displaystyle Z_{C}={\frac {j\omega T+1}{j\omega C}}}$${\displaystyle Z_{C}=R_{C}\angle 0+{\frac {1}{\omega C}}\angle -90}$${\displaystyle Z_{C}={\sqrt {R_{C}^{2}+({\frac {1}{\omega T}})^{2}}}\angle -{\frac {1}{\omega T}}}$${\displaystyle T=CR_{C}}$

### Inductor

#### DC Reponse

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

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

Magnetic Intensity . Measurement of magnetic strength

${\displaystyle B=LI}$

Current

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

#### AC Response

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

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

Reactance . Resistance to the AC current flow

${\displaystyle X_{C}={\frac {v_{L}}{i_{L}}}}$
In frequency domain
${\displaystyle X_{L}(j\omega )=j\omega L}$
In phasor domain
${\displaystyle X_{L}(\omega \theta )=\omega L\angle 90}$

Impedance . Resistance to the AC current flow

${\displaystyle Z_{L}=R_{L}+X_{L}}$
In frequency domain
${\displaystyle Z_{L}=R_{L}+j\omega L}$

In phasor domain
${\displaystyle Z_{L}=R_{L}\angle 0+j\omega L\angle 90}$
${\displaystyle Z_{L}={\sqrt {R_{L}^{2}+(j\omega L)^{2}}}\angle \omega T}$

Time Constant

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

Power . Power of the capacitor

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

### Straight line conductor

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

## Electromagneticism Formulas

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

## Photon Formulas

### Photon

Photon, Massless Quanta's energy of electromagnetic radiation

${\displaystyle E_{h}=hf=\hbar \omega }$

Photons are found in 2 states

Radiant photon at threshold frequency ${\displaystyle E=hf_{o}=\hbar \omega _{o}}$
Non - Radiant photon at frequency greater than threshold frequency ${\displaystyle E_{h}=hf=\hbar \omega }$

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

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

### Quanta

Quanta . Massless quantity

${\displaystyle h=p\lambda }$

Process Wave Particle Duality

Wave like . ${\displaystyle \lambda ={\frac {h}{p}}}$
Particle like . ${\displaystyle p={\frac {h}{\lambda }}}$