# Engineer handbook

 Subject classification: this is an engineering resource.

## Chemistry

### Chemical Element

Symbol

mZE

States

Solid . Fe, Cu
Liquid . Ex Mg
Gas . Ex He, H

Periodic table

Table of columns and rows that list characteristics of chemical elements

### Chemical bonding

Fe + O2 --> FeO2
Cu + O2 --> CuO2

#### Ionic bonding

H2O --> 2H+ + O-
2H2O --> 4H+ + 2O-2

### Chemical Compound

One or more than one element combines to form chemical compound

Oxygen O + O --> O2
Ozone O + O + O --> O3
Water 2H + O --> H2O

### Chemical Reaction

Chemical interaction of substances to create new substance . mathematically can be expressed in Chemical reaction equation

Substance1 + Substance2 = New Substance

Example

Cu + H2SO4 = CuSO4 + 2H

## Physics

### Force

 Force Symbol Mathematical Formula Opposing Force ${\displaystyle F_{-}}$ ${\displaystyle -F}$ Gravity Force ${\displaystyle F_{g}}$ ${\displaystyle G{\frac {Mm}{r^{2}}}}$ Motion Force ${\displaystyle F_{a}}$ ${\displaystyle ma=m{\frac {v}{t}}={\frac {p}{t}}}$ Pressure Force ${\displaystyle F_{A}}$ ${\displaystyle {\frac {F}{A}}}$ Elastic Force ${\displaystyle F_{x}}$ ${\displaystyle -kx}$ Circulation Force ${\displaystyle F_{r}}$ ${\displaystyle mvr=pr}$ Centripetal Force ${\displaystyle F_{c}}$ ${\displaystyle m{\frac {v^{2}}{r}}}$ Electrostatic Force ${\displaystyle F_{q}}$ ${\displaystyle {\frac {q_{+}q_{-}}{r^{2}}}}$ Electromotive force ${\displaystyle F_{E}}$ ${\displaystyle qE}$ Electromagnetomotive Force ${\displaystyle F_{B}}$ ${\displaystyle \pm qvB}$ Electromagnetic Force ${\displaystyle F_{EB}}$ ${\displaystyle q(E\pm vB)}$

### Motion

#### Linear Motion

O ----> O
 Distance ${\displaystyle s}$ ${\displaystyle vt}$ Time ${\displaystyle t}$ ${\displaystyle t}$ Speed ${\displaystyle v}$ ${\displaystyle {\frac {s}{t}}}$ Accelleration ${\displaystyle a}$ ${\displaystyle {\frac {v}{t}}}$ Force ${\displaystyle F}$ ${\displaystyle m{\frac {v}{t}}}$ Work ${\displaystyle W}$ ${\displaystyle Fs}$ Energy ${\displaystyle E}$ ${\displaystyle {\frac {W}{t}}}$

O
O
 Distance ${\displaystyle s}$ ${\displaystyle h}$ Time ${\displaystyle t}$ ${\displaystyle t}$ Speed ${\displaystyle v}$ ${\displaystyle {\frac {h}{t}}}$ Accelleration ${\displaystyle a}$ ${\displaystyle {\frac {h}{t^{2}}}}$ Force ${\displaystyle F}$ ${\displaystyle mg}$ Work ${\displaystyle W}$ ${\displaystyle mgh}$ Energy ${\displaystyle E}$ ${\displaystyle {\frac {mgh}{t}}}$
 Distance ${\displaystyle s}$ ${\displaystyle (v_{o}+at)t}$ Time ${\displaystyle t}$ ${\displaystyle t}$ Speed ${\displaystyle v}$ ${\displaystyle v_{o}+at}$ Accelleration ${\displaystyle a}$ ${\displaystyle {\frac {\Delta v}{\Delta t}}}$ Force ${\displaystyle F}$ ${\displaystyle m{\frac {\Delta v}{\Delta t}}}$ Work ${\displaystyle W}$ ${\displaystyle Ft(v_{o}+at)}$ Energy ${\displaystyle E}$ ${\displaystyle F(v_{o}+at)}$

#### Non Linear Motion

 Distance ${\displaystyle s(t)}$ ${\displaystyle \int v(t)dt}$ Time ${\displaystyle t}$ ${\displaystyle t}$ Speed ${\displaystyle v(t)}$ ${\displaystyle v(t)}$ Acceleration ${\displaystyle a(t)}$ ${\displaystyle {\frac {d}{dt}}v(t)}$ Force ${\displaystyle F(t)}$ ${\displaystyle m{\frac {d}{dt}}v(t)}$ Work ${\displaystyle W(t)}$ ${\displaystyle F\int v(t)dt}$ Energy ${\displaystyle E(t)}$ ${\displaystyle {\frac {F}{t}}\int v(t)dt}$

#### Circular Motion

 Distance ${\displaystyle s}$ ${\displaystyle 2\pi r}$ Time ${\displaystyle t}$ ${\displaystyle t}$ Speed ${\displaystyle v}$ ${\displaystyle r{\frac {2\pi }{t}}=r\omega }$ Acceleration ${\displaystyle a}$ ${\displaystyle {\frac {r\omega }{t}}}$ Angular spped ${\displaystyle \omega }$ ${\displaystyle {\frac {2\pi }{t}}=2\pi f={\frac {v}{r}}}$ Frequency ${\displaystyle f}$ ${\displaystyle {\frac {1}{t}}}$ Force ${\displaystyle F}$ ${\displaystyle m{\frac {r\omega }{t}}}$ Work ${\displaystyle W}$ ${\displaystyle pr\omega }$ Energy ${\displaystyle E}$ ${\displaystyle {\frac {pr\omega }{t}}}$
 Distance ${\displaystyle s}$ ${\displaystyle r\theta }$ Time ${\displaystyle t}$ ${\displaystyle t}$ Speed ${\displaystyle v}$ ${\displaystyle r{\frac {\theta }{t}}}$ Acceleration ${\displaystyle a}$ ${\displaystyle {\frac {r\theta }{t^{2}}}}$ Force ${\displaystyle F}$ ${\displaystyle m{\frac {r\theta }{t^{2}}}}$ Work ${\displaystyle W}$ ${\displaystyle pr{\frac {\theta }{t}}}$ Energy ${\displaystyle E}$ ${\displaystyle {\frac {pr\theta }{t^{2}}}}$

### Momentum

#### Characteristics

 Mass ${\displaystyle m}$ ${\displaystyle m}$ Speed ${\displaystyle v}$ ${\displaystyle v}$ Moment ${\displaystyle p}$ ${\displaystyle mv}$ Force ${\displaystyle F}$ ${\displaystyle ma=m{\frac {v}{t}}={\frac {p}{t}}}$ Work ${\displaystyle W}$ ${\displaystyle Fs={\frac {p}{t}}s=pv}$ Energy ${\displaystyle E}$ ${\displaystyle {\frac {Fs}{t}}={\frac {pv}{t}}=pa}$

#### Momentum of a mass

 Speed ${\displaystyle v}$ ${\displaystyle v}$ Mass ${\displaystyle m}$ ${\displaystyle m}$ Momentum ${\displaystyle p}$ ${\displaystyle mv=Ft}$ Force ${\displaystyle F}$ ${\displaystyle ma=m{\frac {v}{t}}={\frac {p}{t}}}$ Work ${\displaystyle W}$ ${\displaystyle Fs=Fvt=pv}$ Energy ${\displaystyle E}$ ${\displaystyle Fv=Fat=pa}$

#### Momentum of a relativistic mass

 Speed ${\displaystyle v}$ ${\displaystyle \gamma ={\sqrt {1-{\frac {v^{2}}{C^{2}}}}}}$ Mass ${\displaystyle m}$ ${\displaystyle m_{o}(\gamma -1)}$ Momentum ${\displaystyle p}$ ${\displaystyle mv}$ Energy ${\displaystyle E}$ ${\displaystyle pv}$

#### Momentum of a massless quanta

 Speed ${\displaystyle v}$ ${\displaystyle C=\lambda f}$ Mass ${\displaystyle m}$ ${\displaystyle h=p\lambda }$ Energy ${\displaystyle E}$ ${\displaystyle pv=pC=p\lambda f=hf}$ Momentum ${\displaystyle p}$ ${\displaystyle {\frac {h}{\lambda }}}$ Wavelength ${\displaystyle \lambda }$ ${\displaystyle {\frac {h}{p}}={\frac {C}{f}}}$

#### Momentum of an electric charge in circle

 Equilibrium ${\displaystyle QvB=p{\frac {v}{r}}}$ Speed ${\displaystyle v={\frac {Q}{m}}Br}$ Radius ${\displaystyle r={\frac {p}{QB}}}$

#### Momentum of atom's free electron

 Equilibrium ${\displaystyle hf=hf_{o}+{\frac {1}{2}}mv^{2}}$ Speed ${\displaystyle v={\sqrt {{\frac {2}{m}}(hf-hf_{o})}}={\sqrt {{\frac {2}{m}}(nhf_{o})}}}$With${\displaystyle f>f_{o}=nf_{o}}$ to have ${\displaystyle v>0}$

#### Momentum of a bind electron

 Equilibrium ${\displaystyle nhf=mvr2\pi }$ Speed ${\displaystyle v={\frac {1}{2\pi }}{\frac {nhf}{mr}}}$ Radius ${\displaystyle r={\frac {1}{2\pi }}{\frac {nhf}{mv}}}$ Potential Energy Level n ${\displaystyle n=2\pi {\frac {mv}{hf}}}$

### Wave

#### AC electrical sinusoidal wave generator

An interaction of 2 electromagnets creates an AC electricity that has amplitude varies sinusoidally

${\displaystyle v(t)=ASin(\omega t+\theta )}$

#### LC sinusoidal wave generator

Series LC operates at equilibrium satisfy wave equation

${\displaystyle {\frac {d^{2}}{dt^{2}}}i(t)=-{\frac {1}{T}}i(t)}$

that has root of a sinusoidal wave function

${\displaystyle i(t)=ASin\omega t}$
${\displaystyle \omega ={\sqrt {\frac {1}{T}}}}$
${\displaystyle T=LC}$

#### Electromagnetic sinusoidal wave generator

A coil of N turns operates at equilibrium satisfy wave equation

${\displaystyle \nabla ^{2}E(t)=-\omega E(t)}$
${\displaystyle \nabla ^{2}B(t)=-\omega B(t)}$

that has root of a sinusoidal wave function

${\displaystyle E(t)=ASin\omega t}$
${\displaystyle B(t)=ASin\omega t}$
${\displaystyle \omega ={\sqrt {\frac {1}{T}}}=C=\lambda f}$
${\displaystyle T=\mu \epsilon }$

#### Wave Characteristics

 Distance ${\displaystyle s}$ ${\displaystyle \lambda }$ Time ${\displaystyle t}$ ${\displaystyle t}$ Speed ${\displaystyle v}$ ${\displaystyle {\frac {\lambda }{t}}}$ Angular speed ${\displaystyle \omega }$ ${\displaystyle 2\pi f}$ Frequency ${\displaystyle f}$ ${\displaystyle {\frac {1}{t}}}$ Sinusoidal wave equation ${\displaystyle f^{''}(t)}$ ${\displaystyle -\omega f(t)}$ Sinusoidal wave function ${\displaystyle f(t)}$ ${\displaystyle ASin\omega t}$

#### Wave types

 Wave 2 dimensional sinusoidal wave 3 dimensional sinusoidal plane wave Wave oscillation equation ${\displaystyle {\frac {d^{2}}{dt^{2}}}f(t)=-\omega f(t)}$ Wave function ${\displaystyle f(t)=ASin\omega t}$ ${\displaystyle E(t)=ASin\omega t}$${\displaystyle B(t)=ASin\omega t}$${\displaystyle \omega ={\sqrt {\frac {1}{T}}}=C=\lambda f}$${\displaystyle T=\mu \epsilon }$

### Electricity

#### Electricity types

 Electricity types Definition Mathematical Formula Source DC Electricity Electricity that provides constant voltage over time ${\displaystyle v(t)=V}$ Electrolysis, Electrochemcial Cell, PhotonVoltaic AC Electricity Electricity that provides sinusoidal changing voltage over changing time ${\displaystyle v(t)=VSin\omega t}$ Electromagnetic induction

#### DC & AC Response

 Characteristics DC AC Voltage ${\displaystyle V=IR}$ ${\displaystyle v=iR}$ Current ${\displaystyle I={\frac {V}{R}}}$ ${\displaystyle i={\frac {v}{R}}}$ Resistance ${\displaystyle R={\frac {V}{I}}}$ ${\displaystyle R={\frac {v}{i}}}$ Power provided ${\displaystyle P_{V}=IV}$ ${\displaystyle P_{V}=iv}$ Power Loss ${\displaystyle P_{R}=I^{2}R(T)={\frac {V^{2}}{R(T)}}}$ ${\displaystyle P_{R}=i^{2}R(T)={\frac {v^{2}}{R(T)}}}$ Power delivered ${\displaystyle P=P_{V}-P_{R}}$ Reactance ${\displaystyle X_{R}=0}$ Impedance ${\displaystyle Z_{R}=X_{R}+R=R}$ Phase ${\displaystyle 0}$
 Characteristics DC AC Voltage ${\displaystyle V=QC={\frac {W}{Q}}}$ ${\displaystyle v={\frac {1}{C}}\int idt}$ Charge ${\displaystyle Q={\frac {V}{C}}}$ Capacitance ${\displaystyle C={\frac {V}{Q}}}$ Current ${\displaystyle I={\frac {Q}{t}}}$ ${\displaystyle i=C{\frac {dv}{dt}}}$ Power provided ${\displaystyle P_{V}=IV=({\frac {Q}{t}})({\frac {W}{Q}})={\frac {W}{t}}}$ ${\displaystyle p={\frac {1}{2}}Cv^{2}}$ Reactance ${\displaystyle X_{C}(t)={\frac {v}{i}}}$ ${\displaystyle X_{C}(j\omega )={\frac {1}{j\omega C}}}$ ${\displaystyle X_{C}(\omega \theta )={\frac {1}{\omega C}}\angle -90}$ Impedance ${\displaystyle Z_{C}(t)=X_{C}+R_{C}}$ ${\displaystyle X_{C}(j\omega )={\frac {1}{j\omega C}}+R_{C}}$${\displaystyle X_{C}(\omega \theta )={\frac {1}{\omega C}}\angle -90+R\angle 0}$ Phase ${\displaystyle Tan\theta ={\frac {1}{\omega T}}}$ Time Constant ${\displaystyle T=CR_{C}}$
 Characteristics DC AC Magnetic Field Strength ${\displaystyle B=LI}$ Current ${\displaystyle I={\frac {B}{L}}}$ Inductance ${\displaystyle C={\frac {B}{I}}}$ Current ${\displaystyle I={\frac {Q}{t}}}$ Power provided ${\displaystyle P_{V}=IV=({\frac {B}{l}})({\frac {W}{Q}})={\frac {W}{t}}}$ Reactance ${\displaystyle X_{L}(t)={\frac {v}{i}}}$ ${\displaystyle X_{L}(j\omega )=j\omega L}$ ${\displaystyle X_{L}(\omega \theta )=\omega L\angle 90}$ Impedance ${\displaystyle Z_{C}(t)=X_{C}+R_{C}}$ ${\displaystyle X_{C}(j\omega )=j\omega L+R_{L}}$${\displaystyle X_{C}(\omega \theta )=\omega L\angle 90+R\angle 0}$ Phase ${\displaystyle Tan\theta =\omega T}$ Time Constant ${\displaystyle T={\frac {L}{R_{L}}}}$
 ${\displaystyle {\frac {d}{dt}}i=-{\frac {1}{T}}i}$ ${\displaystyle i=Ae^{-{\frac {1}{T}}t}}$ ${\displaystyle T={\frac {L}{R}}}$ ${\displaystyle {\frac {d}{dt}}v=-{\frac {1}{T}}v}$ ${\displaystyle v=Ae^{-{\frac {1}{T}}t}}$ ${\displaystyle T=RC}$
 Modes of Oscillation Oscillation equation Wave Function Angular Speed Oscillation Time Constant Oscilation Constant Decay Constant ' Decay Time Constant Electric decay current sinusoidal wave oscillation ${\displaystyle {\frac {d^{2}}{dt^{2}}}i=-2\alpha {\frac {d}{dt}}i-\beta i}$ ${\displaystyle i=A(\alpha )Sin\omega t}$ ${\displaystyle \omega ={\sqrt {\beta -\alpha }}}$ ${\displaystyle T=LC}$ ${\displaystyle \beta ={\frac {1}{T}}}$ ${\displaystyle \alpha =\beta \gamma }$ ${\displaystyle \gamma =RC}$ Electric peak current sinusoidal wave oscillation ${\displaystyle Z_{L}=-Z_{C}}$${\displaystyle Z_{t}=R}$ ${\displaystyle i(\omega =0)=0}$${\displaystyle i(\omega =\omega _{o})={\frac {v}{2}}}$${\displaystyle i(\omega =00)=0}$ ${\displaystyle \omega _{o}={\sqrt {\frac {1}{T}}}}$ ${\displaystyle T=LC}$ Electric current sinusoidal wave oscillation ${\displaystyle {\frac {d^{2}}{dt^{2}}}i=-{\frac {1}{T}}i}$ ${\displaystyle i=ASin\omega t}$ ${\displaystyle \omega ={\sqrt {\frac {1}{T}}}}$ ${\displaystyle T=LC}$ Electric current sinusoidal standing wave oscillation ${\displaystyle Z_{L}=-Z_{C}}$ ${\displaystyle V_{L}=-V_{C}}$ ${\displaystyle \omega _{o}={\sqrt {\frac {1}{T}}}}$ ${\displaystyle T=LC}$

### Electromagneticism

#### Electric charge

 Charge acquired process Electric charge Charge quantity Electric field Magnetic field Matter + e- - -Q -->E<-- B ↓ Matter - e- + +Q <--E--> B ↑

#### Electromagnetic force

 Force Electric charge Charge quantity Electrostatic Force ${\displaystyle F_{q}}$ ${\displaystyle K{\frac {q_{+}q_{-}}{r^{2}}}}$ Electromotive Force ${\displaystyle F_{E}}$ ${\displaystyle qE}$ Electromagnetomotive Force ${\displaystyle F_{B}}$ ${\displaystyle \pm qvB}$ Electromagnetic Force ${\displaystyle F_{EB}}$ ${\displaystyle qE\pm qvB=q(E\pm B)}$

#### Electromagnetic Field Intensity

 Configuration Symbol Mathematical Formulas For any configuration ${\displaystyle B}$ ${\displaystyle B=LI}$ For straight line conductor ${\displaystyle B}$ ${\displaystyle B=LI={\frac {\mu }{2\pi r}}I}$ Circular B field For circular loop conductor ${\displaystyle B}$ ${\displaystyle B=LI={\frac {\mu }{2r}}I}$ Circular B field around a point charge For coil of N circular loops conductor ${\displaystyle B}$ ${\displaystyle B=LI={\frac {N\mu }{l}}I}$ Eleptic B field around the coil

#### Electromagnetism

 Electromagnetic field intencity ${\displaystyle B}$ ${\displaystyle LI}$ Induced electromagnetic field intencity ${\displaystyle \phi }$ ${\displaystyle -NB=-NLI}$ Electromagnetization field intencity ${\displaystyle H}$ ${\displaystyle {\frac {B}{\mu }}={\frac {\phi }{N\mu }}}$ Eletromagnetization ${\displaystyle \nabla \cdot D=\rho }$${\displaystyle \nabla \times E=-\nabla B}$${\displaystyle \nabla \cdot B=0}$${\displaystyle \nabla \times H=J+\nabla B}$

#### Electromagnetic induction

 Coil's voltage induction intencity ${\displaystyle V}$ ${\displaystyle V={\frac {dB}{dt}}=L{\frac {dI}{dt}}}$ Turn's induced voltage intencity ${\displaystyle \epsilon }$ ${\displaystyle -{\frac {d\phi }{dt}}=-N{\frac {d\phi }{dt}}=-NL{\frac {d}{dt}}I}$

 Electromagnetic oscillation ${\displaystyle \nabla \cdot E=0}$${\displaystyle \nabla \times E={\frac {1}{T}}}$${\displaystyle \nabla \cdot B=0}$${\displaystyle \nabla \times B={\frac {1}{T}}}$ Electromagnetic wave ${\displaystyle \nabla ^{2}E=-\omega E}$${\displaystyle \nabla ^{2}B=-\omega B}$${\displaystyle E=ASin\omega t}$${\displaystyle B=ASin\omega t}$${\displaystyle \omega ={\sqrt {\frac {1}{T}}}=C=\lambda f}$${\displaystyle T=\mu \epsilon }$ Electromagnetic wave radiation ${\displaystyle v=\omega ={\sqrt {\frac {1}{\mu \epsilon }}}=C=\lambda f}$${\displaystyle E=pv=pC=p\lambda f=hf}$${\displaystyle h=p\lambda }$${\displaystyle p={\frac {h}{\lambda }}}$${\displaystyle \lambda ={\frac {h}{p}}={\frac {C}{f}}}$

### Photon

Mathematical formula of Photon

${\displaystyle E=hf=\hbar \omega }$

Photon exist in 2 states

Radiant Photon ${\displaystyle E=hf_{o}=\hbar \omega _{o}}$
Non Radiant Photon ${\displaystyle E=hf=\hbar \omega }$ . With f>fo

Photon can only exist in one state at a time . Heiseinberg's uncertainty principle of successfully finding photon

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

#### Photon and matter

Photon and matter interacts to create Heat transfer through 3 phases Heat conduction, Heat convection and Heat radiation

 Heat transfer Heat conduction Matter changes it's temperature while absorb heat energy ${\displaystyle \Delta T=T_{1}-T_{0}}$${\displaystyle E=mC\Delta T}$ Heat convection Matter conducts heat energy to the max at Threshold frequency fo ${\displaystyle f_{o}={\frac {C}{f_{o}}}}$${\displaystyle E=hf_{o}}$ Heat radiation Matter uses excess energy above maximum absorbing energy to release electron off atom ${\displaystyle hf-hf_{o}={\frac {1}{2}}mv^{2}}$. When v>0 ${\displaystyle v={\sqrt {{\frac {2}{m}}nhf_{o}}}}$ with f > fo

Causes matter to decay through 3 kinds of decays

 Mattter decay Reaction Alpha decay Ur--> Th - He + Alpha radiation Beta decay C-->N + Beta radiation Gamma decay ee) + Gamma radiation

Experienment has shown that , electron of an atom can be freed or binded from absorbing or releasing photon's energy

 Atom decay Absorbing photon energy Releasing photon energy Equilibrium ${\displaystyle hf=hf_{o}+{\frac {1}{2}}mv^{2}}$ ${\displaystyle nhf=mvr2\pi }$ v ${\displaystyle {\sqrt {{\frac {2}{m}}(hf-hf_{o})}}={\sqrt {{\frac {2}{m}}(nf_{o})}}}$ ${\displaystyle {\frac {1}{2\pi }}{\frac {nhf}{mr}}}$ r ${\displaystyle {\frac {1}{2\pi }}{\frac {nhf}{mv}}}$ n ${\displaystyle 2\pi {\frac {mv}{hf}}}$

### Quanta

Mathematical formula of Quanta

${\displaystyle h=p\lambda }$

Quanta process Wave particle duality meaning

Sometimes it behaves like wave ${\displaystyle \lambda ={\frac {h}{p}}}$
Sometimes it behaves like particle ${\displaystyle p={\frac {h}{\lambda }}}$

## Mathematics

### Trigonometry

#### Fundamental trigonometry functions

 Trignometry Function Sin Cos Tan Cotan Sec Cosec Definition Opposite side over hypotnuse Adjacent side over hypotnuse Opposite over Adjacent Adjacent over opposite 1 over opposite 1 over adjacent Mathematical Formula ${\displaystyle \sin \theta ={\frac {b}{c}}}$ ${\displaystyle \cos \theta ={\frac {a}{c}}}$ ${\displaystyle \tan \theta ={\frac {b}{a}}}$ ${\displaystyle \cot \theta ={\frac {a}{b}}}$ ${\displaystyle \sec \theta ={\frac {1}{a}}}$ ${\displaystyle \csc \theta ={\frac {1}{b}}}$ Graphs
 Trigonometry function Laplacian operation Complex operation power of e periodic ${\displaystyle Cos\theta }$ ${\displaystyle \nabla \cdot R(\theta )}$ ${\displaystyle {\frac {z+z^{*}}{2}}}$ ${\displaystyle e^{\pm j(\theta +{\frac {\pi }{2}})}}$ ${\displaystyle Cos(\theta +n2\pi )}$ ${\displaystyle Sin\theta =}$ ${\displaystyle \nabla \times R(\theta )}$ ${\displaystyle {\frac {z-z^{*}}{2}}}$ ${\displaystyle e^{\pm j\theta }}$ ${\displaystyle Sin(\theta +n2\pi )}$

### Arithmatic mathematic

#### Trigonometry functions

Relation of Pythagorean's right triangle sides are defined as trigonometry functions below

 Trignometry Function Sin Cos Tan Cotan Sec Cosec Definition Opposite side over hypotnuse Adjacent side over hypotnuse Opposite over Adjacent Adjacent over opposite 1 over opposite 1 over adjacent Mathematical Formula ${\displaystyle \sin \theta ={\frac {b}{c}}}$ ${\displaystyle \cos \theta ={\frac {a}{c}}}$ ${\displaystyle \tan \theta ={\frac {b}{a}}}$ ${\displaystyle \cot \theta ={\frac {a}{b}}}$ ${\displaystyle \sec \theta ={\frac {1}{a}}}$ ${\displaystyle \csc \theta ={\frac {1}{b}}}$ Graphs

#### Sinusoidal wave equation and wave function

 Sinusoidal wave Wave equation wave function Electric sinusoidal wave ${\displaystyle {\frac {d^{2}}{dt^{2}}}f(t)=-\omega f(t)}$ ${\displaystyle f(t)=ASin\omega t}$ Electromagnetic wave ${\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}$

#### Vector

If there exists a vector R(θ) as a sum of vector x(θ) and vector y(θ) where R , x , y are the sides of pythagorean's right triangle

${\displaystyle R(\theta )=x(\theta )+y(\theta )=R\angle \theta }$
${\displaystyle R(\theta )=RCos\theta +RSin\theta =R(Cos\theta +Sin\theta )}$
${\displaystyle R(\theta )=RCos\theta +RSin\theta =Re^{\pm j(\theta +{\frac {\pi }{2}})}+Re^{\pm j\theta }=(j+1)Re^{\pm j\theta }}$

With

${\displaystyle RCos\theta =Re^{\pm j(\theta +{\frac {\pi }{2}})}}$
${\displaystyle RSin\theta =Re^{\pm j\theta }}$

#### Vector operations

Then the following operations on vectors are true

${\displaystyle \nabla \cdot R(\theta )=x(\theta )=RCos(\theta )=Re^{\pm j(\theta +{\frac {\pi }{2}})}}$
${\displaystyle \nabla \times R(\theta )=y(\theta )=RSin(\theta )=Re^{\pm j\theta }}$

In real coordinate , with any point (x,y) correspond to (r,θ)

 Scalar value Vector ${\displaystyle r={\sqrt {x^{2}+y^{2}}}}$${\displaystyle \theta =Tan^{-1}{\frac {y}{x}}}$${\displaystyle x=rCos\theta }$${\displaystyle y=rSin\theta }$ ${\displaystyle r(\theta )=x(\theta )+y(\theta )=r(Cos\theta +Sin\theta )}$${\displaystyle \nabla \cdot r(\theta )=x(\theta )}$${\displaystyle \nabla \times r(\theta )=y(\theta )}$

In complex coordinate , with any point (x,y)

 Scalar value Vector ${\displaystyle z={\sqrt {x^{2}+jy^{2}}}}$${\displaystyle \theta =Tan^{-1}{\frac {jy}{x}}}$${\displaystyle x=zCos\theta }$${\displaystyle y=jzSin\theta }$ ${\displaystyle z(\theta )=x(\theta )+y(\theta )=z(Cos\theta +Sin\theta )}$${\displaystyle \nabla \cdot z(\theta )=x(\theta )}$${\displaystyle \nabla \times z(\theta )=jy(\theta )}$
 Scalar value Vector ${\displaystyle z^{*}={\sqrt {x^{2}-jy^{2}}}}$${\displaystyle \theta =Tan^{-1}{\frac {-jy}{x}}}$${\displaystyle x=zCos\theta }$${\displaystyle y=-jzSin\theta }$ ${\displaystyle z(\theta )=x(\theta )-j(\theta )=z(Cos\theta -jSin\theta )}$${\displaystyle \nabla \cdot z(\theta )=x(\theta )}$${\displaystyle \nabla \times z(\theta )=-jy(\theta )}$

From above, we have

${\displaystyle Cos\theta ={\frac {z+z^{*}}{2}}}$
${\displaystyle jSin\theta ={\frac {z-z^{*}}{2}}}$
${\displaystyle -jSin\theta ={\frac {z^{*}-z}{2}}}$

### Calculus

#### Change in variables

For any function f(x) . Over the interval of ${\displaystyle x}$ to ${\displaystyle x+\Delta x}$

Change in variable x

${\displaystyle \Delta x=(x+\Delta x)-x}$

Change in function f(x)

${\displaystyle \Delta f(x)=f(x+\Delta x)-f(x)}$

#### Rate of change

Rate of change

${\displaystyle {\frac {\Delta f(x)}{\Delta x}}={\frac {f(x+\Delta x)-f(x)}{\Delta x}}}$

#### Limit

${\displaystyle \lim _{x\to a}f(x)}$

Finite Limit We call ${\displaystyle L}$ the limit of ${\displaystyle f(x)}$ as ${\displaystyle x}$ approaches ${\displaystyle c}$ if ${\displaystyle f(x)}$ becomes arbitrarily close to ${\displaystyle L}$ whenever ${\displaystyle x}$ is sufficiently close (and not equal) to ${\displaystyle c}$ .

When this holds we write

${\displaystyle \lim _{x\to c}f(x)=L}$

or

${\displaystyle f(x)\to L\quad {\mbox{as}}\quad x\to c}$

Infinite Limit We call ${\displaystyle L}$ the limit of ${\displaystyle f(x)}$ as ${\displaystyle x}$ approaches infinity if ${\displaystyle f(x)}$ becomes arbitrarily close to ${\displaystyle L}$ whenever ${\displaystyle x}$ is sufficiently large.

When this holds we write

${\displaystyle \lim _{x\to \infty }f(x)=L}$

or

${\displaystyle f(x)\to L\quad {\mbox{as}}\quad x\to \infty }$

Similarly, we call ${\displaystyle L}$ the limit of ${\displaystyle f(x)}$ as ${\displaystyle x}$ approaches negative infinity if ${\displaystyle f(x)}$ becomes arbitrarily close to ${\displaystyle L}$ whenever ${\displaystyle x}$ is sufficiently negative.

When this holds we write

${\displaystyle \lim _{x\to -\infty }f(x)=L}$

or

${\displaystyle f(x)\to L\quad {\mbox{as}}\quad x\to -\infty }$

#### Differentiation

Let ${\displaystyle f(x)}$ be a function. Then

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

In this case we say that ${\displaystyle f}$ is differentiable at ${\displaystyle x}$ and its derivative at ${\displaystyle x}$ is ${\displaystyle f'(x)}$ .

#### Integration

Mathematics operation on a continuous function to find its area under graph . There are 2 types of integration

${\displaystyle \int f(x)dx=F(x)+C}$

Where ${\displaystyle F}$ satisfies ${\displaystyle F'(x)=f(x)}$

Suppose ${\displaystyle f}$ is a continuous function on ${\displaystyle [a,b]}$ and ${\displaystyle \Delta x={\frac {b-a}{n}}}$ . Then the definite integral of ${\displaystyle f}$ between ${\displaystyle a}$ and ${\displaystyle b}$ is

${\displaystyle \int \limits _{a}^{b}f(x)dx=\lim _{n\to \infty }A_{n}=\lim _{n\to \infty }\sum _{i=1}^{n}f(x_{i}^{*})\Delta x}$

Where ${\displaystyle x_{i}^{*}}$ are any sample points in the interval ${\displaystyle [x_{i-1},x_{i}]}$ and ${\displaystyle x_{k}=a+k\cdot \Delta x}$ for ${\displaystyle k=0,\dots ,n}$ .}}

### Applied Maths

#### Transformation Integral

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

Ex.

 Function of time Laplace transform Fourier transform ${\displaystyle f(t)}$ ${\displaystyle F(s)=\int f(t)e^{-st}dt}$ ${\displaystyle F(j\omega )=\int f(t)e^{-j\omega t}dt}$ ${\displaystyle L{\frac {d}{dt}}i}$ ${\displaystyle sL}$ ${\displaystyle j\omega L}$ ${\displaystyle {\frac {1}{L}}\int dt}$ ${\displaystyle {\frac {1}{sL}}}$ ${\displaystyle {\frac {1}{j\omega L}}}$ ${\displaystyle C{\frac {d}{dt}}v}$ ${\displaystyle sC}$ ${\displaystyle j\omega C}$ ${\displaystyle {\frac {1}{C}}\int dt}$ ${\displaystyle {\frac {1}{sC}}}$ ${\displaystyle {\frac {1}{j\omega C}}}$

#### Diferential equation

##### Ordinal differential equation
${\displaystyle A{\frac {d}{dt}}f(t)+Bf(t)=0}$
${\displaystyle {\frac {d}{dt}}f(t)+{\frac {B}{A}}f(t)=0}$
${\displaystyle {\frac {d}{dt}}f(t)=-sf(t)}$
${\displaystyle s={\frac {B}{A}}}$
${\displaystyle \int {\frac {df(t)}{f(t)}}=-s\int dt}$
${\displaystyle Lnf(t)=-st+c}$
${\displaystyle f(t)=e^{-st+c}}$
${\displaystyle f(t)=Ae^{-st}}$
${\displaystyle A=e^{c}}$

1st ordered differential equation of the form

${\displaystyle A{\frac {d}{dt}}f(t)+Bf(t)=0}$

Has solution

${\displaystyle f(t)=Ae^{-st}}$ Where ${\displaystyle T={\frac {A}{B}}}$

Ex.

${\displaystyle L{\frac {d}{dt}}i+iR=0}$
${\displaystyle i=Ae^{-{\frac {1}{T}}}t}$
${\displaystyle T={\frac {L}{R}}}$
${\displaystyle A{\frac {d^{2}}{dt^{2}}}f(t)+B{\frac {d}{dt}}f(t)+Cf(t)=0}$
${\displaystyle {\frac {d^{2}}{dt^{2}}}f(t)+{\frac {B}{A}}{\frac {d}{dt}}f(t)+{\frac {C}{A}}f(t)=0}$
${\displaystyle {\frac {d^{2}}{dt^{2}}}f(t)=-{\frac {B}{A}}{\frac {d}{dt}}f(t)-{\frac {C}{A}}f(t)}$
${\displaystyle {\frac {d^{2}}{dt^{2}}}f(t)=-2\alpha {\frac {d}{dt}}f(t)-\beta f(t)}$

With

${\displaystyle \beta ={\frac {C}{A}}}$
${\displaystyle \alpha ={\frac {B}{2A}}}$

Solution

• One real root . ${\displaystyle \alpha =\beta }$ . ${\displaystyle i(t)=Ae^{-\alpha t}}$
• Two real roots . ${\displaystyle \alpha >\beta }$ . ${\displaystyle i(t)=Ae^{(-\alpha \pm {\sqrt {\beta -\alpha }})t}}$
• Two complex roots . ${\displaystyle \alpha <\beta }$ . ${\displaystyle i(t)=Ae^{(-\alpha \pm {\sqrt {\alpha -\beta }})t}=A(\alpha )Sin\omega t}$

With

${\displaystyle \omega ={\sqrt {\beta -\alpha }}}$
${\displaystyle A(\alpha )=Ae^{-\alpha t}}$

Ex.

${\displaystyle L{\frac {d}{dt}}i+{\frac {1}{C}}\int idt+iR=0}$
${\displaystyle {\frac {d^{2}}{dt^{2}}}i+{\frac {R}{L}}{\frac {d}{dt}}i+{\frac {1}{LC}}i=0}$
${\displaystyle {\frac {d^{2}}{dt^{2}}}i+2\alpha {\frac {d}{dt}}i+\beta i=0}$

With

${\displaystyle \beta ={\frac {C}{A}}}$
${\displaystyle \alpha ={\frac {B}{2A}}}$

Solution

• One real root . ${\displaystyle \alpha =\beta }$ . ${\displaystyle i(t)=Ae^{-\alpha t}}$
• Two real roots . ${\displaystyle \alpha >\beta }$ . ${\displaystyle i(t)=Ae^{(-\alpha \pm {\sqrt {\beta -\alpha }})t}}$
• Two complex roots . ${\displaystyle \alpha <\beta }$ . ${\displaystyle i(t)=Ae^{(-\alpha \pm {\sqrt {\alpha -\beta }})t}=A(\alpha )Sin\omega t}$

With

${\displaystyle \omega ={\sqrt {\beta -\alpha }}}$
${\displaystyle A(\alpha )=Ae^{-\alpha t}}$
##### Special differential equation
${\displaystyle {\frac {d}{dt}}f(t)=f(t)}$
${\displaystyle \int {\frac {df(t)}{f(t)}}=\int dt}$
${\displaystyle Lnf(t)=t+c}$
${\displaystyle f(t)=e^{t+c}}$
${\displaystyle f(t)=Ae^{t}}$ with A = ${\displaystyle e^{c}}$

Equation of general form

${\displaystyle {\frac {d}{dt}}f(t)=f(t)}$

Has solution

${\displaystyle f(t)=Ae^{t}}$ with A = ${\displaystyle e^{c}}$

It follows that

${\displaystyle {\frac {d}{dt}}f(t)=sf(t)}$
${\displaystyle f(t)=Ae^{st}}$ with A = ${\displaystyle e^{c}}$

In general,

 ${\displaystyle {\frac {d}{dt}}f(t)=f(t)}$ ${\displaystyle f(t)=Ae^{t}}$ with A = ${\displaystyle e^{c}}$ ${\displaystyle {\frac {d}{dt}}f(t)=sf(t)}$ ${\displaystyle f(t)=Ae^{st}}$ with A = ${\displaystyle e^{c}}$