9/6/17; 9/14/17; 10/29/17; 11/11/17; finished 12/11/17

SCIENTIFIC FORMULAS—

MATHEMATICS

PHYSICS

ASTRONOMY

ROCKET SCIENCE

Capital |
Low-case |
Greek Name |
English |

Alpha | a | ||

Beta | b | ||

Gamma | g | ||

Delta | d | ||

Epsilon | e | ||

Zeta | z | ||

Eta | h | ||

Theta | th | ||

Iota | i | ||

Kappa | k | ||

Lambda | l | ||

Mu | m |

Nu n Xi x Omicron o

Lambda l Mu m

Nu | n | ||

Xi | x | ||

Omicron | o | ||

Pi | p | ||

Rho | r | ||

Sigma | s | ||

Tau | t | ||

Upsilon | u | ||

Phi | ph | ||

Chi | ch | ||

Psi | ps | ||

Omega | o |

POWERS tera=10^12 giga=10^9 mega=10^6 myria=10^4 kilo=10^3 hecto=10^2

icosa=20 quindeca=15 hendeca=11 dec=10 non=9 octo=8 hepta=7 hexa=6 penta=5 tetra=4

tri=3

bi=2

uni=1

semi=.5

deci=10^-1

centi=10^-2

milli=10^-3

micro=10^-6

nano=10^-9

pico=10^-12

femto=10^-15

atto=10^-18

PRACTICAL MATHEMATICS FORMULAS PLATONIC SOLIDS—

1. Tetrahedron

Surface area=Sqrt3 x edge length^2 Volume=sqrt2/12 x edge length^3 2. Cube

Surface area=6 x edge length^2 volume=edge length^3

3. Octahedron

Surface area=2 x sqrt3 x edge length^2

volume=sqrt2/3 x edge length^3

4. Dodecahedron

Surface area=3 x sqrt(25+10 x sqrt5) x edge length^2 volume=(15+7 x sqrt5)/4 x edge length^3

5. Isocahedron

Surface area=5 x sqrt3 x edge length^2

volume=(5 x (3+sqrt5))/12 x edge

length^3

CIRCLE-

Diameter D = 2 x Radius

Circumference- C = 2 x Pi*Radius

area- A = Pi x Radius^2

SPHERE-

Surface area. A = 4 x Pi x Radius^2

volume V = 4/3 x Pi x Radius^3

Diameter of a sphere. d=cuberoot(3/4 x Pi x volume) x 2 SQUARE, RECTANGLE, PARALLELOGRAM

Area A=side 1 x side 2

VOLUME OF SQUARE, RECTANGLE, PARALLELOGRAM V=side 1 x side 2 x side

PYRAMID

Surface area=base area+.5 x slant length

Volume=base x depth x height/3

CYLINDER

Surface area=2 x pi x radius x (radius+height)

Volume=PI X radius^2 x length

CONE

Surface area=pi x radius x (radius+base to apex length) Volumes=Pi x radius^2 x height/3

TORUS

Surface area=4 x pi^2 x radius torus x radius of solid part volume=2 x pi^2 x radius torus x radius solid part^2 PYTHAGOREAN THEORM-

a^2+b^2=c^2

a=length of one right angle’s leg

b=length of other right angle’s leg

c=length of hypotenuse

LAW OF SINES-

a/sinA=b/sinB=c/sinC=2 x R=a x b x c/2 x area of triangle R=(a x b x c)/(squareroot((a+b+c) x (a+b-c) x (b+c-a)) Area of triangle=1/2 x a x b x sinC

LAWS OF COSINE-

c^2=a^2+b^2-2 x a x b x cosC

cosC=(a^2+b^2+c^2)/2 x a x b

AREA OF A TRIANGLE-

area=base x height x 1/2

AREA OF AN EQUILATERAL TRIANGLE- area=(length of a side )^2 x SQRT(3)/4 AREA OF A TRAPEZOID-

A=(top side+bottom side) x height/2

HERON’S FORMULA (area of any triangle)- area=SRQT(s x (s-side 1) x (s-side 2) x (s-side 3)) s=1/2 x (a + b + c)

SLOPE-

m=(y-y1)/(x-x1)

(Y1 and x1 are locations on coordinate plane) POINT SLOPE EQUATION OF A LINE-

Y -y1=slope(x-x1)

(Y1 and x1 are locations on coordinate plane) SLOPE INTERCEPT FORM FOR A LINE- y=slope(x)+(y intercept)

DISTANCE FORMULA-

distance=square root((x-x1)^2+(y-y1)^2+(z-z1)^2)) (z1, y1, and x1 are locations on coordinate system) ALGEBRA FORMULAS-

(a+b)^2=a^2+2 x a x b+b^2

(a-b)^2=a^2-2 x a x b+b^2

x^2-a^2=(x+a) x (x-a)

x^3-a^3=(x-a) x (x^2+a x x+a^2) x^3+a^3=(x+a) x (x^2-a x x+a^2) a/b+c/d=(a x d+b x c)/b x d) a/b-c/d=(a x d-b x c)/b x d

a/b x c/d=a x c/b x d

CONIC SECTIONS-

Circle- (x-g)^2+(y-h)^2=radius^2

(g=x coordinate, h=y coordinate) Parabola-

y^2+/-4ax

(a=x coordinate)

x^2=+/-4ay

(a=y coordinate)

ellipse-

x^2/a^2+y^2/b^2=1,

(a=x, b=y coordinates, or a=y, b=x coordinates) Hyperbola-

x^2/a^2-y^2/b^2=1,

(a=x, b=y coordinates, or a=y, b=x coordinates) QUADRATIC EQUATION- x=(-b+/-squareroot(b^2-4ac))/2a

LAWS OF EXPONENTS-

a^x x a^y=a^(x+y)

a^x/a^y=a^(x-y)

(a^x)^y=a^(X x Y) (a*b)^x=a^x x b^x a^0=1

a^1=a

LAWS OF LOGARITHMS-

log(base a)(M x N)=log(base a)(M)+log(base a(N) log(base a)(M/N)=log(base a)M-log(base a)(N) logM^r=r X x logM

log(base a)(M)=logM/loga

TRIGONOMETRY-

sine-o/h

cosine=a/h

tangent=o/a

cosecant=h/o

secant=h/a

cotangent=a/o

(a=adjacent side of right triangle)

(o=opposite side of right triangle)

(h=hypotenuse of right triangle)

Pythagorean identities-

sin^2(x)-cos^2(x)=1

sec^2(x)-tan^2(x)=1

csc^2-cos^2(x)=1

Product relations-

Sinx-tanx x cosx

cosx=cotx x sinx

tanx=sinx x secx

cotx=cosx x cscx

Secx-cscx x tanx

cscx=secx x cotx

Trigonometry functions-

sinx=x-x^3/3!+x^5/5!-x^7/7!

cosx=1-x^2/2!+x^4/4!-x^6/6!

Inverse trigonometry functions-

sin-1x=x+(1/2 x 3) x x^3+(1 x 3/2 x 4 x 5) x x^5+(1 x 3 x 5/2 x 4 x 6 x 7) x x^7+… cos-1x=pi/2-(x+(1/2 x 3) x x^3+(1 x 3/2 x 4 x 5) x x^5+(1 x 3 x 5/2 x 4 x 6 x 7) x x^7+… tan-1x=x-x^3/3+x^5/5-x^7/7+…

cot-1x=pi/2-x+x^3/3-x^5/5+x^7/7-…

Hyperbolic functions-

sinhx=x+x^3/3!+x^5/5!+x^7/7!+…

coshx=1+x^2/2!+x^4/4!+x^6/6!+…

Inverse hyperbolic functions-

sinh-1x=x-(1/2 x 3) x x^3+(1 x 3/2 x 4 x 5) x x^5-(1 x 3 x 5/2 x 4 x 6 x 7) x x^7+… tanh-1x=x+x^3/3+x^5/5+x^7/7+…

Nth TERM OF AN ARITHMETIC SEQUENCE- Nth term=a+(number of terms-1)*d

(a=1st term, d=common difference)

SUM OF n TERMS OF AN ARITHMETIC SERIES- Sum-n/2 x (a+nth term)

(a=1st term, d=common difference)

Nth TERMS OF A GEOMETRIC SEQUENCE-

a(n)=a x r^(n-1), (r cannot equal 0.)

(a=1st term, r=common ratio)

SUM OF THE n TERMS OF A GEOMETRIC SEQUENCE- s=a x ((1-r^n)/(1-r))

(r cannot equal 0, 1)

(n=number of terms, r=common ratio)

SUM OF AN INFINITE SERIES-

s=n/(1-r)

(If absolute value of r<1)

(n=number star with)

(r=how much keep multiplying x with forever) (s=sum of infinite series)

COMBINA TIONS-

C(n,r)=n!/r!(n-r)!

PERMUT A TIONS-

P(n,r)=n!(n-r)!

BINOMIAL FORMULA-

(a x x-b)^n

CALCULUS (DIFFERENTIATION)-

d/dx (x^n)=n x x^(n-1)

d/dx sinx= cost

d/dx cosx= -sinx

d/dx tanx=sec^2(x)

d/dx cotx=-csc^2(x)

d/dx sex-secs x tanx

d/dx cscx= -cscx x cotx

d/dx e^x=e^x

d/dx lnx=1/x

d/dx (u+v)=du/dx+dv/dx

d/dx(c x u)=c x du/dx

dy/dx=dy/dx x du/dx

(chain rule)

d/dx (u x v)=(v x (du/dx)-(u x (dv/dx)

(product rule)

d/dx(u/v)=(v x du/dx-u x dv/dx/)v^2

(quotient rule)

du=du/dx(dx)

CALCULUS (INTEGRATION)-

The definite integral of t from a to b for definite integral f(t)=F(b)-F(a)

Indefinite integral of x^r dx=x^(r+1)/(r+1)+c, (r cannot equal -1)

Indefinite integral of 1/x dx=ln(absolute value (x))+c

Indefinite integral of sinx dx=cosx+c

Indefinite integral of cosx dx=sinx+c

Indefinite integral of e^x dx=e^x+c

Indefinite integral of (f(x)+g(x))dx=indefinite integral f(x)+indefinite integral g(x) Indefinite integral of c x f(x) dx=c x (indefinite integral f(x))

indefinite integral of (u)dv=u x v-indefinite integral (v)du

(integration by parts)

CENTER OF MASS-

Center of mass (x)=((mass1) x (center of mass1)+(mass2) x (center of mass x 2))/ (mass-1+mass-2)

(A point representing the mean position of the matter in a body of system.) VECTOR ANALYSIS

Norm (magnitude of a vector)=sqrt(x^2+y^2+z^2)

Dot product u (dot) v=(u1) x (v1)+(u2) x (v2)+(u3) x (v3)=||u|| ||v|| cos(theta)

(theta is the angle between u and v, 0<=theta<=Pi)

Cross productÂ u x v=((u2) x (v3)-(u3) x (v2))i-((u1) x (v3)-(u3) x (v1))j+((u1) x (v2)-

(u2) x (v1))k

||u x v||=||u|| x ||v|| sin(theta)

(Theta is angle between u and v, 0<=theta<=Pi)

2 vectors orthogonal if their dot product v and u=0 or transpose vector v and vector u=0.

Exponential growth and decay—

y=A exp^k*t

A=starting number of for example bacteria, t=length of growth time, k=constant, y=number of bacteria after t time

OUTLINE OF PHYSICS FORMULAS

*** Straight line motion- ***

Velocity (meters/second)=distance (meters)/time (seconds) v=d/t (constant velocity)

v=2 x d/t (accelerating)

Distance (meters)=velocity (meters/second) x time (seconds) d=v x t

time (seconds)=distance (meters)/velocity (meters/second) t=d/v

t=sqrt(2 x distance/acceleration)

Acceleration (meters/second^2)=

((meters/second (end)-meters/second (start)/)time (seconds))/2 a=(d2/t-d1/t)/2

Acceleration (meters/second^2)=2 x distance (meters)/time (second)^2 a=2 x d/t^2

Final velocity (meters/second)=initial velocity (meters/second)+ acceleration (meters/second^2 x time (seconds)

v(f)=v1+a x t

velocity^2=initial velocity+2 x acceleration x distance

v^2=v1+2 x a x d

Average velocity (meters/second)=initial velocity+final velocity/time v(average)=(v1+v2)/t

Average velocity=initial velocity+1/2 x acceleration x time v(average)=v1+1/2 x a x t

Distance (meters)=initial velocity x time+1/2 x acceleration x time^2 d=v1+1/2 x t x a x t^2

Distance=acceleration x time^2/2

d=a x t^2/2

Newton’s 2nd law of motion

Force (newtons)=mass (kilograms) x acceleration (meters/second^2) F-m x a

Falling bodies

velocity=gravity (9.81 meters/second^2 for the earth) x time

v=g x t

How far fallen in meters=1/2 x gravity x time^2 d=1/2 x g x t^2

Time fallen=sqrt(2 x height/gravity)

t=sqrt(2 x h/g)

velocity=sqrt(2 x gravity x height) v=sqrt(2 x g x h)

*** Circular motion- *** Uniform circular motion

Moment of inertia=mass x distance from axis^2

m(inertial)=m x d^2

Angular velocity=angular displacement/change in timeÂ (radians/second) v(angular)=d/t

Angular momentum=moment of inertia x angular velocity m(angular)=m(inertial) x v(angular)

Centripedal acceleration

Centripedal acceleration=velocity^2/radius of path (radians/second^2) a(centipedal)=v^2/r

Torque (newtons-meter)

Centripetal force

Centripetal force=mass x velocity^2/radius of path

f(centripedal)=m x v^2/r

Gravitation

gravitation (newtons)=G x (mass(1) x mass(2))/radius^2

(G=6.67 x 10^-11)

f=G x (m1 x m2)/r^2

Fundamental forces in nature— strong

W eak

Electromagnetic

Gravity

*** Energy- ***

work

work (joules)=force (newtons) x distance (meters)

w=f x d

work=work output/work input x 100%

w=w(o)/w(i) x 100

Power

Power (watts)=work (joules)/time (seconds)

p=w/t

horsepower=746 watts

weight=mass x gravity

w=m x g

momentum=mass (kilograms) x velocity (meters/second) momentum=m x v

Energy

kinetic energy

KE (joules)=1/2 x mass (kilograms)x velocity (meters/second)^2 ke=1/2 x m x v^2

Potential energy

PE (joules)=mass x gravity (9.81 meters/second^2) x height (meters)

pe=m x g x h

Rest energy

Rest energy (joules)=mass x 300,000,000^2

Conservation of energy

Momentum-Â (kilograms-meters/second)

Linear momentum

L. momentum=mass (kilograms)x velocity (meters/second) m(momentum)=m x v

Conservation laws

Conservation of mass-energy

Conservation of linear momentum

Angular momentum

Conservation of angular momentum

Conservation of electric charge

Conservation of color charge

Conservation of weal isospin

Conservation of probability

Conservation of rest mass

Conservation of baryon number

Conservation of lepton number

Conservation of flavor

Conservation of parity

Invariance of charge conjugation

Invariance under time reversal

CP sysmetry

Inversion or reversal of space, time, and charge

(there is a one-to-one correspondence between each of the conservation laws and a differentiable symmetry in nature.)

Impulse

impulse=force (newtons) x time (seconds)

i=f x t

*** Relativity- ***

special relativity

Lorentz transformation

*** Fluids- ***

Density

Specific gravity

kilograms/meter^3

Pressure

pressure=force/area

pressure=newtons/meters^3

p=f/d^3

Pressure in a fluid

pressure=density (kilograms/meters^3) x depth (meters) x weight (kilograms) p=d(density) x d(depth) x w

p=kg/d^3 x d x m

**Archimede’s principle-the upward buoyant force that is exerted on a body immersed in a fluid, whether fully or partially submerged, is equal to the weight of the fluid that the body displaces.**

**Bernoulli’s principles-an increase in the speed of a fluid occurs **

**simultaneously with a decrease in pressure or a decrease in the fluid’s potential energy.****
**v^2/2+gz+p/Ï=constant

*v *is the fluid flow speed at a point on a streamline,

*g *is the gravitational acceleration

*z *is the elevation of the point above a reference plane,

with the positive *z*-direction pointing upward so in the direction

opposite to the gravitational acceleration,

*p *is the pressure at the chosen point, and

*Ï *is the density of the fluid at all points in the fluid.

*** Heat- ***

internal heat

Temperature

Heat

1 kilocalorie=3.97 british thermal units (BTU)

1 BTU=.252 kilocalories

Specific heat capacity

Heat transferred=mass (kilograms) x specific heat capacity x temperature change (kelvin)

h=m x h x t

change of state

Heat of fusion

Heat of vaporization

pressure and boiling point

*** Kinetic theory of matter- ***

Ideal gases

Boyle’s law

pressure(1) x volume(1)=pressure(2) x volume(2)

(temperature constant)

P1 x v1=p2 x v2

Absolute temperature scale

Temperature kelvin=temperature (celsius)+273.15

Charlie’s law

volume(1)/temperature(1)=volume(2)/temperature(2)

(pressure constant)

v1/t1=v2/t2

Ideal gas law

pressure(1) x volume(1)/temperature(1)=pressure(2) x volume(2)/temperature(2) P1 x v1/t1=p2 x v2/t2

Kinetic energy of gases

Molecular energy

KE (joules)=3/2 x K x temperature (kelvin)

(K=boltzmann’s constant=1.38 x 10^-23 joules/kelvin

ke=3/2 x k x t

solids and liquids

Atoms and molecules

*** Thermodynamics- ***

3 laws of thermodynamics

The four laws of thermodynamics are:

Zeroth law of thermodynamics: If two systems are in thermal equilibrium with a third system, they are in thermal equilibrium with each other.

This law helps define the notion of temperature.

1st law of thermodynamics: When energy passes, as work, as heat, or with matter, into or out from a system, the system’s inertial energy changes in accord with the law of conservation of energy. Equivalently, Perpetual motion machines of the 1st kind (machines that produce work without the input of energy) are impossible.

2nd law of thermodynamics: In a natural thermodynamic process, the sum of the entropies of the interacting thermodynamic systems increases. Equivalently, perpetual motion machines of the 2nd kind (machines that spontaneously convert thermal energy into mechanical work) are impossible.

3rd law of thermodynamics: The entropy of a system approaches a constant value as the temperature approaches absolute zero. With the exception of non-crystalline solids (glasses) the entropy of a system at absolute zero is typically close to zero, and is equal to the logarithm of the product of the quantum ground states.

entropy

The entropy of a system approaches a constant value as the temperature

absolute zero.

Mechanical equivalent of heat

Mech. Equiv. heat=4,185 x joules/kilocalories Mech. Equiv. heat=778 x foot-pounds/BTU

Heat engines

Engine efficiency

efficiency=1-heat temperature absorbed/heat temperature given off eff=1-h(temp. Absorbed)/h(temp. Given off)

Conduction

Convection

Radiation

*** Electricity- ***

Electric charge

Charge of proton=1.6 x 10^-19 coulombs

Charge of electron= -1.6 x 10^-19 coulombs

Electric charge=current (amperes) x time taken (seconds)

Coulomb’s law

Electric force (newtons)=K x charge-1 (coulombs) x charge-2 (coulombs)/ distance (meters)^2

(K=9 x 10^9 newton-meter^2/coulomb^2)

F=KÂ x c1 x c2/d^2

Atomic structure

Mass of proton=1.673 x 10^-27 kilograms

Mass of neutron=1.675 x 10^-27 kilograms Mass of electron=9.1 x 10-31 kilograms Ions

Electric field

Electric field (newton/coulomb)=force (newtons)/charge (coulombs) E=f/c

force=charge x electric field

Electric lines of force

Potential difference

volts=work/charge

(1 volt= 1 joule/coulomb)

volt=electric field (newtons/coulomb)x distance (meters)

v=E x d

Electric field (newtons/coulombs)=volts/distance

E=v/d

Potential Difference=current (amperes)x resistance (ohms)=

energy transferred/charge (coulombs)

pd=I x r=e/c

Electric current-

Electrical energy=voltage (volts)x current (amperes)x time taken (seconds) e=v x I x t

Electric current

Electric current (amperes)=charge (coulombs)/time interval (seconds)

(1 ampere=1 coulomb/second)

I=c/t

Electrolysis

Ohm’s law

Electric current (amperes)=volts/resistance (ohms)

(resistance (1 ohm))=1 volt/ampere)

I=v/r

resistance (ohms)=voltage (volts)/current (amperes)

r=v/I

voltage (volts)=current (amperes)x resistance (ohms)

v=I x r

Resisters in series

resistance=resistance(1)+resistance(2)+resistance(3)

R=r1+r2+r3

Resisters in parallel 1/resistance=1/resistance(1)+r1/resistance(2)+1/resistance(3) 1/R=1/r1+1/r2+1/r3

Kirchoff’’s law

current law=Summation (current)=0

Voltage law=summation (voltage)=0

Capacitance

1 farad=1 coulomb has 1 volt between plates

capacitance (farad)=charge (coulombs)/voltage (volts)

Work stored=work (charging)=1/2 x capacitance x voltage^2

W=w=1/2 x C x v

electric power

power (watts)=work done per unit time (joules)=voltage (volts)x charge

(coulombs)/time (seconds)

p=w=v x c/t

Power (watts)=current (amperes) x voltage (volts)=current (amperes)^2

x resistance (ohms)=voltage (volts)^2/resistance (ohms)

p=i x pd=i^2 x r=pd^2/r

Alternating current

power (watts)=1/2 x peak voltage x peak current x

cos (phase angle between current and voltage sine waves)

p=1/2 x v x I x cos(theta)

*** Magnetism- ***

Magnetic field

1 tesla=1 newton/ampere-meter

(tesla=weber/meter^2)

(1 gauss=10^-4 teslas)

B (magnetic field)=K x I (straight line current)/distance (meters)

(K=2 x 10^-7 newton/amperes^2

B=K x I/d

Magnetic field on a moving charge

Magnetic field on a current

B=Pi(3.14) x K x I/r

F=I x L x B

Magnetic field of solenoid

B=2 x Pi x K x N (number of turns)/L (length of solenoid) x I (current amperes) Forces between 2 currents

(K=2 x 10^-7 newton/amperes^2

F/L=K x I(1) x I(2)/d

Lorentz force

F=charge x electric field+charge x velocity (cross product) magnetic field F=qE+qv x B

*** electromagnetism ***

Maxwell’s equations

**Maxwell’s equations **are a set of partial differential **equations **that, together with the Lorentz force law, form the foundation of classical electromagnetism, classical optics, and electric circuits.

*** Electromagnetic induction- *** Generator

Motors

Alternating current

I=I (max)/sqrt(2)=.707 I (max)

V=I (max)sqrt(2)=.707 V (max)

Transformer

Primary voltage/secondary voltage=primary turns/secondary turns Primary current/secondary current=secondary turns/primary turns DC circuits

*** Waves- ***

Frequency

1 hertz=1 cycle/second

W avelength

wave velocity (meters/second)=frequency (hertz) x wavelength (meters)

v=f x w

Acoustics

Optics

Electromagnetic waves

Velocity of light=c=3 x 10^8 meters/second=186,282 miles/second

Doppler effect

Frequency found by observer with respect to sound=frequency produced by source x (velocity sound+velocity observer)/(velocity sound-velocity source) f(o)=f(s) x (v+v(o))/(v-v(s))

Frequency found by observer with respect to light=frequency produced by source x sqrt((1+relative velocity/c)/(1-relative velocity/c))

f(o)=f(s) x sqrt((1+v/c)/(1-v/c))

reflection and refraction of light

Index refraction=n=c/velocity in medium (meters/second)

r=c/v

Interference, diffraction, polarization

Particles and waves

*** Quantum physics- ***

**Uncertainty principle-**the velocity of an object and its position

cannot both be measured exactly, at the same time, even in

theory.

The **Schrodinger equation **is used to find the allowed energy levels of quantum mechanical systems (such as atoms, or transistors). The associated wave function gives the probability of finding the particle at a certain position. … The solution to this **equation **is a wave that describes the quantum aspects of

a system.

The **Pauli exclusion principle— **

Quantum theory of light

Quantum energy=E=h x f

Planck’s constant=h=6.63 x 10^-34 joules-second

X-rays

Electron volt=planck’s constant x frequency

eV=h x f

Electron KE=x-ray photon energy

f=eV/h

momentum=kinetic energy (joules)/speed of light^2 (meters/second) p=ke/c^2

Kinetic Energy=planck’s constant x frequency=

planck’s constant x speed light/wavelength

ke (joules)=6.63 x 10^-34 joules-second x frequency hertz(meter/second)=

The **Pauli exclusion principle **is the quantum

mechanical

principle which states that two or more identical fermions (particles with

half-integer spin)

cannot occupy the same quantum state within a quantum system

simultaneously.

6.63 x 1-^-34 x 300,000,000 meters/f (meters) joules=h x f=h x c/lambda

Electron volt

1 eV=1.6 x 10^-19 joules

1 KeV=10^3 eV

1 MeV=10^6 eV

1 GeV=10^9 eV

Kinetic energy=1/2 x mass x velocity^2=

planck’s constant x frequency-electon volts

ke (joules)=1/2 x m x v^2=6.63 x 10^-34 joules-second x f-eV Matter waves

De Broglie wavelength=lambda=h/m x v (momentum=m x v)

wavelength=planck’s constant/(mass x meters/second) lambda=h/m x v

Solid state physics

***Nuclear and atomic physics- ***

Nucleus

Mass (proton)=1.673 x 10^-27 kilograms=1.007277 u

Mass (neutron)=1.675 x 10^-27 kilograms=1.008665 u Nuclear structure

Binding energy

Mass defect= change m=((number protons x mass hydrogen)+ (number neutrons x mass neutrons))-m

(mass hydrogen=1.007825 u)

Fundamental forces

Gravity

Electromagnetic

Weak interaction

Strong force

Fission and fusion

Radioactivity

Alpha particle=helium nuclei

Beta particle=electron

Gamma rays=high energy photons with frequencies greater than x-rays neutron>proton+electron

proton>neutron+positron Radioactive decay and half-life Elementary particles and antimatter

ASTRONOMY FORMULAS

1. How to find the DISTANCE in parsecs to a star- distance=10^((apparent magnitude-absolute magnitude+5)/5) 2. APPARENT magnitude

apm=log d x 5-5+abm

apparent=log(distance) x 5 – 5 + absolute magnitude

3. How do you calculate the absolute magnitude of a star abm=((log L/log 2.516)-4.83)

Absolute magnitude=((log(number of sun’s luminosity of star)/ log2.516)-4.83

abm=-(5 x log d-5-apm)

Absolute magnitude= -(5 x log(distance parsecs)-5-apparent magnitude)

4. To find brightness of a star/number of suns-

L=10^((abm-4.83) x (log 2.516))

LUMINOSITY of star=10^-((absolute magnitude of star-4.83)*(log2.516))

luminosity increase=2.512^([magnitude increase]+4.83)

luminosity=mass^3.5Â (for main sequence stars)

Luminosity (watts)=4 x pi x radius(meters)^2 x temperature (kelvin)^4 x

5.67 x 10^-8 watts maters^-2 kelvin^-4

5. Mass binary system

Suppose in an example, we calculate the masses of 2 stars in a binary star system: if the period of star a is 27 years and its distance from the common center of mass is 19 AUs, the

Distance^3/period^2=19^3/27^2=6859/729=9.4 solar masses for the total mass of the 2 stars.

The velocity of star a is 30,000 km./second and star b is 10,000 km/second, so 30,000/10,000=3.

The mass of star b is 9.4/(1+3)=2.25 solar masses.

The mass of star a is 9.4-2.25=7.15 solar masses.

So star a is 7.15 solar masses, and star b is 2.25 solar masses, and both added up equals 9.4 solar masses, the combined mass of the 2 stars.

6. Radius of a star

radius=(temperature sun (kelvin)/temperature star (kelvin))^2 x (2.512^(absolute magnitude sun-absolute magnitude star)^1.2)

7. size of star/orbit/object-

size object miles=arcseconds size object x distance parsecs x 864,000

8. LT=10^10 x m(star)/m(sun)^-2.5

lifetime=(10^10) x (mass of star/mass of sun)^-2.5

9. To find ARC SECOND measurement of object size from parsec DISTANCE and visa versa and SIZE of an object–

SIZE OF STAR IN ARCSECONDS–arcseconds=1/d (parsecs) x number of suns size ARCSECONDS– parallax=1/distance parsecs

DISTANCE (parsecs)– d=1/arcseconds

10. Galaxy distance in millions of light years-

d=13,680 x rsh+8.338

distance (millions of light years)=13,680 x red shift+8.336

11. velocity of galaxy in kilometers/second-

v=300,000 x rsh

velocity (kilometers per second)=300,000 x redshift

11.5. approximate number of stars in a galaxy=luminosity in number of suns galaxy/.02954

12. redshift-

Rsh=mly-8.338/13,680

redshift=(light years (millions)-8.336)/13,680

13. Volume of a galaxy=4/3 x Pi x a x b^2

a=major axis, b=minor axis, (for elliptical galaxies)

14. Number of stars=volume/distance between stars^3

15. Average Distance between stars in light years=cube root(volume/number of stars) 16. Number of stars span across longest axis of galaxy=

(3/4 x volume)/(Pi x b^2 x n^3)

b=minor axis length light years, n=distance between stars light years

17. Escape velocity from a galaxy meters/second=

Mass galaxy kilograms x 1.989 x 10^30/(number of stars x

4,827,572.324)^2

18. Surface gravity

gravity(meters/second^2)=mass of star number of suns x 1.99 x

10^30 x 6.67 x 10^-11/(size of star number of suns x 864000000 x 1.62/2)^2 19. Titus-bode law-

Distance (astronomical units)=3*2^n+4/10

(n=-infinity, 0, 1, 2, 3, â€¦)

Mercury=-infinity

Venus=0

earth=1

mars=2

Asteroid belt=3

jupiter=4

Etc

20. kepler’s 3 laws of planetary motion-

1. Planets travel in elliptical orbits.

2. Equal areas are covered in in equal times in the elliptical orbit.

3. The distance in astronomical units to the 3rd power equals the time to travel one complete orbit in years to the 2nd power.

(time years)^2=(radius orbit astronomical units)^3

D=P^(2/3)

P=D^(3/2)

21. Velocity to achieve orbit=sqrt(G x M/distance from center of the earth) 22. Escape velocity=sqrt(G x M/r)

23. Four types of eccentric orbits

circle eccentricity=0

ellipse eccentricity= 0-1

parabola eccentricity=1

hyperbola eccentricity>1

24. Eccentricity=(greatest orbital distance-closest orbital distance)/(closest orbital distance+greatest orbital distance) e=(d(greatest)-d(closest))/(d(closest)+d(greatest))

25. Calculations: orbits, periods of orbits, perihelions, aphelions and eccentricities (for example a comet)

CIRCLE

Circular formula eccentricity=0

(x-h)^2+(y-k)^2=r^2. (#1)

(h=x coordinate and y=y coordinate ;r=radius of orbit)

period=2 x pi x sqrt(radius^3/(6.67 x 10^-11 x mass of body the body is orbiting)) Velocity in orbit=sqrt(6.67 x 10^-11 x

mass of body the body is orbiting/radius of orbit)

Centripetal acceleration=velocity^2/radius of orbit

ELLIPSE

Use ellipse formula x^2/a^2+y^2/b^2=1

Then calculate from 2 coordinates in AUs with formula

x^2 x b^2 + y^2 x a^2=a^2 x b^2

find a and b (the closest and furthest approaches)

Period years of orbit=distance (AU)^3/2

time=2 x pi x sqrt(a^3/G x M)

distance=period^2/3

velocity=sqrt(G x M x (2/r-1/a))

eccentricity=(0<e<1)

eccentricity=sqrt(1-b^2/a^2)

a (semimajor axis)=(perihelion + aphelion)/2

Semi minor axis (b)=sqrt(-(eccentricity^2-1) x a^2)

Simple way to calculate and orbit

1=X^2+Y^2. >Â 1=x^2+semimajor axis^2+y^2/semiminor axis^2

time in seconds to get to orbital position from 0 degrees going counterclockwise t=radians at current position of orbiting body/360 x sort(semi

major axis^3/6.67 x 10^-11 x 1.989 x 10^30)

to find the angle at which the orbiting body is at from 0 degrees going counterclockwise x^2=p^2+1

q=sqrt(p)

r=sqrt(x)

s=q/r

arcsin(s)=angle of the orbiting body with respect to the focal body

to find distance to orbiting body from foci

x^2=p^2+p^2

cosy=x^2/P^2 x x^2

m=arccosy

cosm=1/n

q=1/cosm

q=distance to orbiting body from foci

To find formula for the orbit, use ellipse formula

x^2/a^2+y^2/b^2=1

PARABOLA

eccentricity=1

y=a x X^2+b x X +c

Calculate from 2 coordinates of the body in its orbit (a and b coordinates in

both instances)

Solve for x, then y, and will have the formula for the parabolic orbit.

Then will have the equation of the orbit like the quadratic equation above with

numerical figures for a and b.

Velocity of body in parabolic orbit

v=sqrt(2 x 6.67 x 10^-11 x mass of body being orbited/radius of body)

Period of orbit does not have an orbit so undefined.

eccentricity=sqrt(1+b^2/a^2)

time in seconds to get to orbital position from 0 degrees going counterclockwise– t=radians at current position of orbiting body/360 x sqrt(semi

major axis^3/6.67 x 10^-11 x 1.989 x 10^30)

to find the angle at which the orbiting body is at from 0 degrees going counterclockwise– g^2+(distance between foci/2 (‘p))^2=d^2

(d^2-p^2-g^2)/(-2(p^2)(g^2))=cosx

cos-1x=angle x

p^2-1^2=g^2

(1^2-p^2-g^2)/(-2(p^2)(g^2))=cosL

cos-1L=angle L

90-x-L=m

m-x=n

n is the angle the orbiting body is at from 0 degrees going counterclockwise HYPERBOLA

eccentricity>1

x/a-y/b=1. (#1)

xb-ya=ab.Â (#2)

Calculate from 2 coordinates of the body in its orbit (a and b coordinates in both instances) using (equation #2).

Solve for x, then y, and will have the formula (for #1 above) for the hyperbolic orbit.

Then will have the equation of the orbit like equation #1 with numerical figures for a and b.

Velocity of body at infinity in a hyperbolic orbit

velocity at an infinite distances away=sqrt(velocity^2-escape-velocity^2)

Period of orbit does not have na orbit so undefined.

26. EXAMPLES OF CALCULATING ORBITS

STEPS TO CALCULATE AN ELLIPTICAL ORBIT

Suppose we measure 2 coordinates of a comet, one at (4AU,1AU) and the other at (0AU,3AU).

1) We plug in each of the coordinates, the 1st equals the x and the 2nd equals the y, each of the 2 coordinates into b^2 x x^2+a^2 x y^2=a^2 x b^2.

Then we subtract the 2 equations from each other. Next, we solve for both a and b. The resulting equation is x^2/2.376^2+y^2/1.68^2=1

and this is the equation of the orbit for the 2 given coordinates. The ellipse equation is x^2/2.376^2+y^2/1.68^2=1.

Solving for x and y gives

x=sqrt((1-5.65 x y^2)/2.8) and y=sqrt((1-2.8 x x^2/5.65).

2) perehelion=1.68AU, aphelion=2.376AU. (figures for constants a and b) 3) semimajor axis=A=(perihelion+aphelion)/2=(1.68+2.376)/2=1.9958AU 4) Semi minor axis=smaller figure of a and b=1.68AU.

5) focii=amaller figure of a and b=1.68AU.

6) distance center of ellipse to the foci=A-perihelion=.3158AU

7) period=aphelion^3/2=2.82 years (1,029.83 days)

8) eccentricity=1-perihelion/A=.158

9) velocity at any instant=1.99 x 10^30 x 6.67 x 10^-11 x (1/radius=1/A)=

At perihelion=20.07 miles/second, at aphelion=10.74 miles/second

10. Suppose we want to find the time from perihelion to a distance of 2 AU and its velocity there. We use the formulas

Semimajor axis^2+semimajor axis^2=x^2

cosx=x^2/2 x semimajor axis^2 x x^2; angle=arccosx

time=angle/360 x sqrt(semimajor axis^3/6.67 x 10^-11 x 1.989 x 10^30)

The answer for time atÂ 2 AU=82.79/360 x sqrt(2.7186 x 10^34/1.32733 x 10^20)=

.22997 x 14311435.6=38.09 days at a velocity there at 12.995 miles/second.

HALLEY’S COMET

period=75.986 years.

focii=.6AU.

perihelion=.6AU.

aphelion=35.28AU.

Semimajor axis=17.94Au

eccentricity=.9855.

Velocity at perihelion=33.23 miles/second, aphelion=3.124 miles/second

STEPS TO CALCULATE A CIRCULAR ORBIT

Suppose 2 coordinates were recorded of a celestial body, one (3,2,646), and the other (1,3.873). It the celestial body has a circular orbit, the squares of each set of coordinates added together will equal the same defendant number. In this example,

3^2+2.646^2=15, and 1^2+3.873^2=16, so this is a circular orbit where the equation of the orbit is 4^2=x^2+y^2, where the 4 in the 4^2 is the radius of the circle, and the x and y are the

coordinates of the circular orbit. The eccentricity of of a circular orbit is equal to zero. STEPS TO CALCULATE A PARABOLIC ORBIT

Formulas-

velocity=sqrt(2 x G x Mass central body/radius of orbiting body from central body) trajectory=(4.5 x G x M x Time seconds^2)^1/3

To find out whether 2 coordinates measurements of an orbiting body if a parabolic orbit, say coordinates (3,27) and (2,12), we need to set up a parabolic equation

y=b x x^2, then put into it separately the 2 coordinates. 27=b x 3^2, solve for b to arrive at equals to 3.

12=b x 2^2, b also equals 3.

Since b in both equations are equal to each other, the orbiting object is in a parabolic orbit. The equation for the orbit is y=3 x x^2. The period of orbit is infinitely long since the orbiting object never returns. The vertex of the orbit is (0,0). The foci is equal to 3/4. We arrive at this by always using 4 x p, and setting it equal to 3, the number equal to b. When 4 x p=3 is solved, p=3/4. So the focus is at (0,3/4).the velocity of the orbiting body at its closest approach to the central body is equal to 30.05 miles/second.

STEPS TO CALCULATE A HYPERBOLIC ORBIT

Suppose 2 coordinates of an orbiting celestial body are recored as being at (28.28, 10)AU and (34.64, 14.14)AU positions. We try using the hyperbolic equation

1=x^2/a^2-y^2/b^2, solve for a and b, and the result equals 1, so this is a hyperbolic orbit. a=20 and b=10. The equation for the orbit is

1=x^2/20^2-y^2/10^2.

Solving for x and y yields

x=sqrt(400-4 x y^2), and y=sqrt(x^2/4-100(.

The orbit’s eccentricity is equal to sqrt(a^2+b^2)/a=sqrt(400+100)/400=1.118, which is greater than 1, so this is a hyperbolic orbit.

For focii=sqrt(a^2+b^2)=22.33AU from the sun’s position, which is equal to 22.33- a=22.33-20=20AU, which makes the focci=(20,0).

The period of this orbiting body is undefined since it will never return. To find the semimajor axis, we use the velocity formula

v=sqrt(6.67 x 10^-11 x 1.99 x 10^30 x (2/r-1/semimajor axis)).

r=2.33AU in meters and v=618,000 meters/second. Solving for the semimajor axis yields -1208.12.

Let us determine the velocity of the orbiting object at say 5AU from the sun.

v=sqrt(6.67 x 10^-11 x 1,99 x 10^30 x (2/5AU in meters-1/1208.12))=18.77 kilometers/second, or 11.64 miles/second.

If we wanted to calculate the velocity of the object when it gets as far away as the nearest star, 4.3 light years away,

it would be traveling 25.76 meter/second there.

27. Formulas to find masses, radius, and luminosities of WHITE DWARFS (mass<=.75 suns)

(radius<=.00436 suns)

(luminosity<=.00365 suns)

radius=mass^18.68 mass=radius^.052966

luminosity=mass^19.5198 mass=luminosity^.05123

luminosity=radius^1.04494 radius=luminosity^.95699

28. Formulas to find masses, radius, and luminosities of MAIN SEQUENCE stars luminosity=mass^3.5

mass=luminosity^(.2857)

radius=(temperature kelvin sun/temperature kelvin star)^2 x

(2.512^(absolute magnitude sun-absolute magnitude star)

note main sequence star’s masses can be found by knowing the star’s luminosity and its temperature.

Type star Mass radius. Temperature luminosity lifespan

O. +16. +6.6 +33,000 kelvin 55,000 to >200,000 >9.77 m/yrs

B 2.1-16. 1.8-6.6 10,000-33,000 kelvin. 42-24,000. 9.77 m/yrs-1.57

b/yrs

A 1.4-2.1. 1.4-1.8. 7,500-10,000 kelvin. 5.1-24. 1.57 b/yrs-4.3 b/yrs F. 1.04-1.4. 1.15-1.4. 6,000-7,500 kelvin. 2.4-5.1. 4.3 b/yrs-9.07 b/yrs

G. .8-1.04. .96-1.15. 5,200-6,000 kelvin .38-1.2. 9.07 b/yrs-17.47 b/yrs K. .45-.8. .7-.96. 3,700-5,200 kelvin. .08-.38. 17.47 b/yrs-73.62 b/yrs M. <=.45. <=.7. 2,000-3,700 kelvin <.002-.08. >73.62 b/yrs

29. Formulas to find SUBGIANT masses, radii, and luminosities MASS TO LUMINOSITY

O type subgiant stars luminosity=100,000-1 million luminosity^.295=mass

luminosity^.2=radius

mass^.675=radius

B type subgiant stars luminosity=350-40,000 luminosity^.245=mass luminosity^.205=radius

mass^.84=radius

A type subgiant stars luminosity=20-200 luminosity^.203=mass luminosity^.256=radius

mass^1.26=radius

F type subgiant stars luminosity=8-300 luminosity^.19=mass

luminosity^.37=radius

mass^1.94=radius

G type subgiant star luminosity=1.1-6 luminosity^.15=mass luminosity^.5=radius mass^3.57=radius

K type subgiant stars luminosity=3-45 luminosity^.1mass luminosity^.625=radius mass^4.15=radius

30. Masses, radius, and luminosities for GIANT stars- O type giant stars luminosity=30,000 to >2 million luminosity^.27=mass

luminosity^.206=radius

mass^.756=radius

B type giant stars luminosity=330-35,000 luminosity^.24=mass luminosity^.223=radius mass^.936=radius

A type giant star luminosity=10-8,000 luminosity^.227=mass luminosity^.244=radius mass^1.08=radius

F type giant stars luminosity=10-100 luminosity^.195=mass luminosity^.363=radius mass^1.86=radius

G type giant stars luminosity=15-75 luminosity^.19=mass luminosity^.57=radius mass^2.89=radius

K type giant stars luminosity=30-275 luminosity^.11=mass luminosity^.625=radius mass^5.7=radius

M type giant stars luminosity=800-3,700 luminosity^.098=mass luminosity^.65=radius

mass^6.7=radius

31. Masses, radius, and luminosities for BRIGHT GIANT stars- O type bright giant stars luminosity=65,000-320,000 luminosity^.27=mass

luminosity^.22=radius

mass^.795=radius

B type bright giant stars 25,000-40,000 luminosity^.23=mass

luminosity^.3=radius

mass^1.23=radius

A type bright giant stars luminosity=3,000-15,000 luminosity^.24=mass

luminosity^.37=radius

mass^1.665=radius

F type bright giant stars luminosity=500-2,500 luminosity^.23=mass

luminosity^.44=radius

mass^2.1=radius

G type bright giant stars luminosity=150-1,000 luminosity^.22=mass

luminosity^.4=radius

mass^2.3=radius

K type bright giant stars luminosity=200-3,500 luminosity^.21=mass

luminosity^.58=radius

mass^6=radius

M type bright giant stars luminosity=600-25,000 luminosity^.19=mass

luminosity^.65=radius

mass^8.1=radius

32. Finding masses, radius, and luminosities for SUPERGIANT stars Supergiant stars luminosity=120,000-2 million luminosity^.213=mass

luminosity^.56=radius

mass^2.63=radius

33. Close formula for mass to radius and radius to mass of STARS MASSES 40 to GREATER THAN 200 SOLAR MASSES

(crudely estimated formulas) luminosity^.325=mass

luminosity^.519=radius

mass^1.6=radius

34. MASSES OF STARS AND THEIR FATES Masses .07-10 suns white dwarf

Masses .5-8 suns planetary nebulas

Masses >8 suns supernovas

Masses 10-29 suns neutron stars

masses>29 suns black holes

35.

**Vega- Cl Effective relative **

**asstemperaturechromaticity ty (D65) **

[ **mass **

**sequence radius **

**main- seque nce stars **~0.00 003%

0.13%

0.6%

3%

7.6%

12.1%

76.45 %

**Chromatici sequence **

**on of Hydr all **

**Main- Main- **

**Main-**

**sequence luminosity ogen **

**Fracti **

**(solar (solar radii)(bolometric lines masses) ) **

**O **30,000 K blue

**B **10,000-30,00blue white

blue 16

6.6 >30,000

W eak

deep blue 2.1-16

0K white um

**A **7,500-10,000white blue white 1.4-2.1 1.4-1.8 5-25 Stron Kg

**. F **6,000-7,500 yellow white K

**. G **5,200-6,000 yellow K

**K **3,700-5,200 light orange K

**M **2,400-3,700 orange red K

white

yellowish

white

pale yellow 0.45-0.8 orange

light orange 0.08-0.45 red

Medi um

Weak

Very weak very weak

1.04-1.4

1.15-1.4 0.96-1.15 0.7-0.96 0.7 (<)

1.5-5 0.6-1.5 0.08-“0.6 <0.08

0.8-1.04

The Hertzsprung Russell diagram relates stellar classification with absolute magnitude, luminosity, and surface temperature.

**36. DISTRIBUTION OF TYPES OF STARS IN GALAXY- **

Giants and supergiants .946%

O star .0000256%

B stars .1105%

A stars .51085%

F stars 2.5545%

G stars 6.446%

K stars 10.313%

M stars 65.0295%

White dwarfs 8.515%

Brown dwarfs 1-10% (4.98% average estimate)

Neutron stars .8515%

Black holes .08515

DRAKE EQUATION ESTIMATE OF PERCENTAGE OF ADVANCED CIVILIZATIONS OF SYSTEMS STARS

Applies to 10% of all stars

37. SEVERAL ABSOLUTE AND APPARENT MAGNITUDES WITH LUMINOSITIES LIST ABSOLUTE MAGNITUDES LIST

Gamma ray burst. -39.1 374,000 trillion suns

Quasars -33. 1,360 trillion suns

supernovas. -19.3. 4.49 billion suns

Supernova 1978a. -15.66. 157 million suns

Pistol star. -10.75 1.7 million suns

deneb -8.38. 192,424 suns

Betelgeuse. -5.5. 13,558 suns

Sun 4.83. 1 sun

Proxima centauri. 11.13. 1/331 suns

Sun in Andromeda galaxy. 29.07. 1/4.98 billion suns

Venus 29.23. 1/5.8 billion suns

Hubble telescope viewing limit. 31. 1/29.4 billion suns

James webb telescope viewing limit. 34. 1/466 billion suns

APPARENT MAGNITUDES LIST

sun. -26.72. 23.74 trillion suns

Full moon. -12.6. 9.38 million suns

Venus. -4.4. 4,922 suns

Sirius. -1.6. 373 suns

Most energetic gamma ray burst 12.2 billion light years away- 3.77. (374,000 trillion suns)

Sun seen by us if it were in andromedas galaxy. 53.31 1/(2.48 x 10^19) suns

Type 2 supernova in Andromeda as seen from here— apparent magnitude— 4.94

Venus in Andromeda as seen from here— apparent magnitude— 53.47

Deneb in Andromeda as seen from here— apparent magnitude— 15.86

38. Formulas to find temperature kelvin from spectral class—

Temp.=1500 x spectral class number+10000

O0=20, O1=19,…, B9=1, A0=0

Temp.=187.27 x spectral class number+5880

A0=22, A1=21,…, G1=1, G2=0

Temp.=132.22 x spectral class number+3500

G2=18, G3=17,…, K9=1, M0=0, M1=-1, M2=-2,…, M9=-9

**Spaceflight formulas—**

**Meaning of variables in the formulas— v=velocity (meters/second)**

**vi=velocity initial (meters/second) vf=velocity final (meters/second) vexh=exhaust velocity (meters/second) isp=seconds**

**m=mass (kilograms)**

**mi=initial mass (kilograms)**

**mf=final mass (kilograms)**

**mr=mass ratio**

**a=acceleration (meters/second^2) f=force (newtons)**

**d=distance (meters) **

**t=time (seconds)**

**ke=kinetic energy (joules)**

**ed=energy density (joules/kilograms) p=power (watts)**

**spp=specific power (kilowatts/kilograms) mm=momentum (meters x kilograms) i=impulse (thrust x seconds)**

**fr=fuel rate (kilograms/second) mw=molecular weight texh=temperature exhaust (kelvin) eff=propulsive efficiency**

**r=radius (meters)**

**ecc=eccentricity**

**g=acceleration due to gravity**

**(9.81 meters/second^2) **

**Rocket equation**

**velocity**

**v=vexh x ln(mi/mf)**

**v=(d x 2)/t (when accellerating) v=d/t (constant velocity) v=sqrt(2 x a x d) **

**Velocity of exhaust vexh=v/ln*(mi/mf)Â vexh=.25 x sqrt(texh/mw) **

**Isp**

**isp=vexh/9.81**

**isp=f/(fr x 9.81)**

**isp=vf/(ln(mr) x 9.81)**

**Mass ratio**

**mr=mi/mf**

**mr=e^(v/vexh)**

**mr=e^(vf/(isp x 9.81))**

**Mass final**

**mf=mi/(e^(vf/vexh)**

**Mass initial**

**mi=mf x e^(vf/vexh)**

**mass**

**m=f/a**

**m=2 x ke/v^2**

**Force**

**f=m*a**

**f=ke/d Â**

**f=9.81 x isp x fr**

**acceleration**

**a=f/m**

**Energy**

**ke=1/2 x v^2 x m****
**ke=d x f

**Energy density (rest mass energy) **

**ed=ke/m **

**Fuel flow rate **

**fr=f/vexh **

**fr=m/t **

**Distance â **

**d=v x t/2 (with respect to accelerating) **

**d=v x t (constant velocity) **

**d=ke/f **

**d=v^2/2 x a **

**Timeâ **

**t=(d x 2)/v (constant acceleration) **

**t=d/v (constant velocity) **

**t=((m x vf)^2/2)-(m x mi)^2/2) x (1/f) x (2/vi+vf) **

**Power p=ke/t **

**p=f x d/t **

**Specific power **

**sp=(p/1000)/m **

**Momentum **

**M=v x mÂ **

**Impulse â **

**i=f x tÂ **

**Antimatter needed (kilograms) m=ke/1.8 x 10^16 **

**Propulsive efficiency eff=2/(1+(vexh/vf)) **

**eff=(vf/vexh)^2/(e^(vf/vexh)-1)**

**(Maximum efficiency for ratio- vf/vexh<1.6) eff=f x g x isp/2 x p **

BEAUTIFUL FORMULAS

8/29/17-8/30/17;9/15/17;11/21/17-11/22/17;12/25/17; 4/19/18

CONTENTS

- Dirac’s equation
- Einstein’s field equation
- Maxwell’s equations
- General relativity
- Special relativity
- Schrodinger’s equation
- Uncertainty principle
- Gibb’s statistical mechanics
- Stephan-Boltzmann law
- e=mc^2
- Laplace equation
- De broglie relation-matter wave
- Navier-stokes equations
- Riemann zeta function
- Noether theorem
- Euler-lagrange equation
- Hamilton quanternion
- Standard model
- Lagrange formula
- cantor inequality
- Riemann hypothesis
- Hawking-Bekenstein entropy formula
- Heat equation
- wave equation
- poisson equation
- Wave-particle duality
- fundamental theorem of calculus
- Pythagorean theorem
- Gauss-Bonnet theorem
- universal law of gravitation
- Newton’s 2nd law of motion
- kinetic energy
- Potential energy
- 2nd law of thermodynamics
- principle of least action
- Spherical harmonics
- Cauchy residue theorem
- Callen-Symanzik equation
- Minimal surface equation
- Euler 9 point center
- Mandelbrot set
- Yang-Baxter equation
- Divergence theorem
- Baye’s theorem
- logistic map
- Einstein’s law of velocity addition
- Photoelectric effect formula
- Faraday law
- Cauchy momentum equation
- De moivre’s theorem
- Fourier transform
- prime counting function
- Murphy’s law
- Summation formula
- Logarithmic spiral
- Heron’s formula
- Quadratic equation
- Euler line
- Pythagorean triple formula
- Euler’s formula
- Simplex method
- Proof of infinity of prime numbers
- Harmonic series
- Euler sums
- Cubic equation
- Quartic equation
- quintic equation
- Lorentz equation
- Euler-lagrange formula
- Euler product formula
- Euler-maclaurin formula
- Pi
- Exponent
- Natural logarithm
- Conic sections
- exponential growth or decay
- Calculation an orbit I.e. a comet
- interesting number idea 1
- interesting number idea 2
- interesting number idea 3

EQUATIONS

- Dirac equation

**Dirac equation***(original)*

The Dirac differential equation from quantum mechanics was formulated in 1928 which predicted the existence of antimatter, which are particle of the same mass and spin, but have an opposite charges than their counterparts of matter.

2. Einstein field equation

The einstein field equations, or the einstein-hilbert equations is used to describe gravity in a classical way. It uses geometry to model gravity’s effects.

3. Maxwell’s equations

James clerk maxwell formulated 4 differential equations to describe how charged particles produce an electric and magnetic force. They calculate the motion of particles in electric and magnetic fields. They describe how electric charges and electric currents create electric and magnetic fields, and vice versa.

The 1st equation is used to calculate the electric field produced by a charge.

The 2nd equation is used to calculate the magnetic field.Â The 3rd equation, ampere’s law, shows how the magnetic fields circulate around electric currents and time varying electric fields. The 4th equation. Faradayâ€™s law, shows how the electric fields circulate around time varying magnetic fields.

4. General relativity

**Albert einstein, in 1915, formed the general theory of relativity which deals with space and time, two aspects of spacetime. Spacetime curves when there is gravity, matter, energy, and momentum. Central the the general theory of relativity is the principle of equivalence. The theory shows that light curves in an accelerating frame of reference. It also asserts that light will bend and it will slow down in the presence of a massive amount matter.**

5. SPECIAL RELATIVITY

The Lorentz Transformations (the mathematical basis for the special theory of relativity-

An object at rest still has energy, called its rest mass, which is-

E0=mc^2.

The special theory of relativity asserts that the speed of light is the same no matter what speed the observer travels. It also explains what is relative and what is absolute about time, space, and motion. It further describes how mass increases, length shrinks, time slows down for objects moving close to the speed of light, and that a person traveling close to the speed of light would age less than would a stationary person.

6. Schrodinger equation

This is a differential equation that is the basis of quantum mechanics. It is one of the most precise theories of how subatomic particles behave as fully as possible. This equation defines a wave function of a particle or group of particles that have a certain value at every point in space for every given time. the wave function contains all information that can be known about a particle or system. The wave function gives real values relating to physical properties such as position, momentum, energy, etc.

7. Uncertainty principle

This principle says that trying to pin a thing down to one definite position will make its momentum less well pinned down, and vice-versa.

8. GIBB’S STATISTICAL MECHANICS

**Statistical mechanics** is a branch of theoretical physics which uses probability theory to study the average behavior of a

mechanical system, where the state of the system is uncertain.

Statistical mechanics is commonly used to explain the thermodynamic behavior of large systems.

9. Stefan Boltzmann law

R=Ïƒ*T*^{4}

where Ïƒ is the **Stefan-Boltzmann constant**, which is equal to 5.670 373(21) x 10^{-8} W m^{-2} K^{-4}, and where R is the energy radiated per unit surface area and per unit time. *T*is temperature, which is measured in kelvin scale. this law is only usable for the energy radiated by blackbodies but is still useful none the less. In quantum physics, the **Stefan-Boltzmann law** (sometimes called **Stefan’s Law**) states that the black body radiation energy emitted by an object is directly proportional to the temperature of the object raised to the fourth power.

10. Mass energy equivalence

E=mc^2

In physics, **mass energy equivalence** asserts that anything having mass has an equivalent amount of energy and vice versa. these fundamental quantities are directly related to one another.

11. Laplace’s equation

In mathematics, **Laplace’s equation** is a second-order partial differential equation. The solutions of Laplace’s equation are the harmonic functions, which are important in many fields of science, notably the fields of electromagnetism, astronomy, and fluid dynamics, because they can be used to accurately describe the behavior of electric, gravitational, and fluid potentials. In the study of heat conduction, the Laplace equation is the steady-state heat equation.

12. DE BROGLIE RELATION/Matter wave

Î»=h/mv

Where Î» is the wavelength of the object, *h* is Planck’s constant, *m* is the mass of the object, and *v* is the velocity of the object. An alternate but correct version of this formula is

Î»=h/p

Where *p* is the momentum. (Momentum is equal to mass times velocity). These equations merely say that matter exhibits a particle-like nature in some circumstances, and a wave-like characteristic at other times.

13. Navier Stokes equations

The **Navier Stokes equations** describe the motion of fluids. The equations result from applying newton’s 2nd law to fluid dynamics with the belief that the fluid stress is the sum of a diffusing vicious term (in relation to the gradient of velocity), plus a pressure term. They are very useful because they describe the physics of many things. They may be used to model weather, ocean currents, water flow in a pipe, the air’s flow around a wing, and the motion of stars inside a galaxy. The Navier Stokes equations in their full and simplified forms help with the design of aircraft and cars, the study of blood flow, the design of power stations, the analysis of pollution, and many other things. Together with Maxwell’s equations they can be used to model and study magnetohydrodynamics. The Navier Stokes equations are also of great interest in a purely mathematical sense. Somewhat surprisingly, given their wide range of practical uses, mathematicians have not yet proven that in three dimensions solutions always exist (existence), or that if they do exist, then they do not contain any singularities (or infinity or discontinuity) (smoothness). These are called the navier-stokes existence and smoothness problems. The Navier Stokes equations dictate not position but rather velocity. A solution of the Navier Stokes equations is called a velocity field or flow field, which is a description of the velocity of the fluid at a given point in space and time. Once the velocity field is solved for, other quantities of interest (such as flow rate or drag force) may be found. This is different from what one normally sees in classical mechanics, where solutions are typically trajectories of position of a particle or deflection of a continuum. Studying velocity instead of position makes more sense for a fluid, however for visualization purposes one can compute various trajectories.

14. Riemann zeta function

Î¶(s)=âˆ‘n=1 to âˆž 1ns, Re(s)>1.

Where

Re(s) is the real part of the complex numbers. For example, if s=a+ib, then Re(s)=a. (where i^2=â-1)

Riemann zeta function Î¶(*s*) in the complex plane. The color of a point *s* shows the value of Î¶(*s*): strong colors are for values close to zero and hue encodes the value’s argument. The white spot at *s*= 1 is the pole of the zeta function; the black spots on the negative real axis and on the critical line Re(*s*) = 1/2 are its zeros.

In mathematics, the **Riemann zeta function**, is a prominent function of great significance in number theory. It is so important because of its relation to the distribution of prime numbers. It also has applications in other areas such as physics, probability theory, and applied statistics.

The riemann hypothesis is a conjecture about the distribution of the zeros of the Riemann zeta function. Many mathematicians consider the Riemann hypothesis to be the most important unsolved problem in pure mathematics.

15. Noether’s theorem

dX/dt=0

Emmy noether was an influential mathematician known for her groundbreaking contributions to abstract algebra and theoretical physics.

Noether’s theorem can be stated informally

If a system has a continuous symmetry property, then there are corresponding quantities whose values are conserved in time.

A more sophisticated version of the theorem involving fields states that:

To every differentiable symmetry generated by local actions, there corresponds a conserved current.

16. Euler Lagrange equation

In the calculus of variations, the **Euler Lagrange equation**, **Euler’s equation**, or **Lagrange’s equation** (although the latter name is ambiguous), is a second-order partial differential equation whose solutions are the functions for which a given functional is stationary.

Because a differentiable functional is stationary at its local maxima and minima, the Euler Lagrange equation is useful for solving optimization problems in which, given some functional, one seeks the function minimizing or maximizing it. This is analogous to format’s theorem in calculus, stating that at any point where a differentiable function attains a local extremum its derivative is zero. In Lagrangian mechanics, because of Hamilton’s principle of stationary action, the evolution of a physical system is described by the solutions to the Euler Lagrange equation for the action of the system. In \classical mechanics, it is equivalent to newton’s law of motion, but it has the advantage that it takes the same form in any system of generalized coordinates, and it is better suited to generalizations. In classical field theory there is an analogous equation to calculate the dynamics of a field.

17. Quaternion

*a* + *b*** i** +

*c*

**+**

*j**d*

*k*where *a*, *b*, *c*, and *d* are real numbers, and ** i**,

**, and**

*j***are the fundamental**

*k**quaternion units*.

In mathematics, the **quaternions** are a number system that extends the complex numbers. they are applied to in 3-dimensional space. A feature of quaternions is that multiplication of two quaternions is non-commutative. Hamilton defined a quaternion as the quotient of two directed lines in a three-dimensional space^{]} or equivalently as the quotient of two vectors.

Quaternions find uses in both theoretical and applied mathematics, in particular for calculations involving 3-dimensional rotations such as in 3-dimensional computer graphics, computer vision and crystallographic texture analysis. In practical applications, they can be used alongside other methods, such as euler angles and rotation matrices, or as an alternative to them, depending on the application.

18. Standard Model (mathematical formulation) for particle physics

19. LAGRANGE FORMULA

**Lagrangian mechanics** is a reformulation of classical mechanics.

In Lagrangian mechanics, the trajectory of a system of particles is derived by solving the Lagrange equations in one of two forms, either the *Lagrange equations of the first kind*, which treat constraints explicitly as extra equations, often using Lagrange multipliers; or the *Lagrange equations of the second kind*, which incorporate the constraints directly by judicious choice of generalized coordinates. In each case, a mathematical function called the **Lagrangian** is a function of the generalized coordinates, their time derivatives, and time, and contains the information about the dynamics of the system.

20. CANTOR’S INEQUALITY/Cantor’s theorem

In elementary set theory, **Cantor’s theorem** is a fundamental result that states that, for any set *A*, the set of all subsets of *A* (the power sets of *A*, ð’«(*A*)) has a strictly greater cardinality than *A* itself. For finite sets, Cantor’s theorem can be seen to be true by simple enumeration of the number of subsets. Counting the empty subset, a set with *n* members has 2* ^{n}* subsets, so that if card(

*A*) =

*n*, then card(ð’«(

*A*)) = 2

*, and the theorem holds because 2*

^{n}

^{n}*> n*is true for all non-negative integers.

the theorem implies that there is no largest cardinal number (colloquially, “there’s no largest infinity”

21. Riemann hypothesis

The **Riemann hypothesis** is a mathematical conjecture. Many people think that finding a proof of the hypothesis is one of the hardest and most important unsolved problems of pure mathematics.

The hypothesis is named after Bernhard riemann. It is about a special function, the riemann zeta function. This function inputs and outputs complex numbers values. The inputs that give the output zero are called zeros of the zeta function. Many zeros have been found. The “obvious” ones to find are the negative even integers. This follows from Riemann’s functional equation. More have been computed and have real part 1/2. The hypothesis states all the undiscovered zeros must have real part 1/2.

The functional equation also says all zeros (except the “obvious” ones) must be in the critical strip: real part is between 0 and 1. The Riemann hypothesis says more: they are on the line given, in the image on the right (the white dots). If the hypothesis is false, this would mean that there are white dots which are not on the line given.

If proven correct, this would allow mathematicians to better describe how the prime numbers are placed among whole numbers. The Riemann hypothesis is so important, and so difficult to prove, that the Clay Mathematics Institute has offered $1,000,000 to the first person to prove it.

22. HAWKING-BEKENSTEIN ENTROPY FORMULA

**black-hole thermodynamics** is the area of study that seeks to reconcile the laws of thermodynamics with the existence of Black-hole event horizons. As the study of the statistical mechanics of black-body radiation led to the advent of the theory of quantum mechanics, the effort to understand the statistical mechanics of black holes has had a deep impact upon the understanding of quantum gravity, leading to the formulation of the holographic principle.

The 2nd law of thermodynamics requires that black holes have entropy. If black holes carried no entropy, it would be possible to violate the second law by throwing mass into the black hole. The increase of the entropy of the black hole more than compensates for the decrease of the entropy carried by the object that was swallowed.

23. HEAT EQUATION

The **heat equation** is a parabolic partial differential equation that describes the distribution of heat (or variation in temperature) in a given region over time. In the heat equation in two dimensions predicts that if one area of an otherwise cool metal plate has been heated, say with a torch, over time the temperature of that area will gradually decrease, starting at the edge and moving inward. Meanwhile the part of the plate outside that region will be getting warmer. Eventually the entire plate will reach a uniform intermediate temperature.

The heat equation is of fundamental importance in diverse scientific fields. In mathematics, it is the prototypical parabolic partial differential equation. In probability theory, the heat equation is connected with the study of brownian motion via the Fokker-planck equation.In financial mathematics, it is used to solve the black-scholes partial differential equation. The diffusion equation, a more general version of the heat equation, arises in connection with the study of chemical diffusion and other related processes.

The heat equation is used in probability and describes random walks. It is also applied in financial mathematics for this reason.

It is also important in riemannian geometry and thus topology: it was adapted by richard s. Hamilton when he defined the Ricci flow that was later used by Grigori perelmanto solve the topological poincare conjecture.

24. Wave equation

iâ„âˆ‚/âˆ‚tÎ¨(x,t)=H^Î¨(x,t)

where *i* is the imaginary number, *Ïˆ (x,t)* is the wave function, *Ä§* is the reduced planck constant, *t* is time, *x* is position in space, *Ä¤* is a mathematical object known as the *Hamilton operator*. The reader will note that the symbol âˆ‚/âˆ‚t denotes that the partial derivative of the wave function is being taken.

Equations that describe waves as they occur in nature are called **wave equations**. Waves as they occur in rivers, lakes, and oceans are similar to those of sound and light. The problem of having to describe waves arises in fields like acoustics, electromagnetic, and fluid dynamics.

Historically, the problem of a vibrating string such as that of a musical instruments was studied.Â In 1746, d’Alambert discovered the one-dimensional wave equation, and within ten years Euler discovered the three-dimensional wave equation.

In quantum mechanics, the **Wave function**, usually represented by *Î¨*, or *Ïˆ*, describes the probability of finding an electron somewhere in its matter wave. To be more precise, the *square* of the wave function gives the probability of finding the location of the electron in the given area, since the normal answer for the wave function is usually a complex number. The wave function concept was first introduced in the legendary schrodinger equation.

25. Poisson’s equation

âˆ‡^2Ï†=f.

(âˆ‚^2/âˆ‚x^2+âˆ‚^2/âˆ‚y^2+âˆ‚^2/âˆ‚z^2)Ï†(x,y,z)=f(x,y,z).

When f=0 identically we obtain laplace’s equation.

In mathematics, **Poisson’s equation** is a partial differential equation of elliptic type with broad utility in mechanical engineering and theoretical physics. It arises, for instance, to describe the potential field caused by a given charge or mass density distribution; with the potential field known, one can then calculate gravitational or electrostatic field. It is a generalization of laplace’s equation, which is also frequently seen in physics. Poisson’s equation may be solved using a green’s function.

26.Wave particle duality

**Wave particle duality** is perhaps one of the most confusing concepts in physics, because it is so unlike anything we see in the ordinary world.

Physicists who studied light in the 1700s and 1800s were having a big argument about whether light was made of particles shooting around like tiny bullets, or waves washing around like water waves. Light seems to do both. At times, light seems to go only in a straight line, as if it were made of particles. But other experiments show that light has a frequency and wavelength, just like a sound wave or water wave. Until the 20th century, most physicists thought that light was either one or the other, and that the scientists on the other side of the argument were simply wrong.

Wave particle duality means that all particles show both wave and particle properties. This is a central concept of quantum mechanics. Classical concepts like “particle” and “wave” do not fully describe the behavior of quantum-scale objects.

27. Fundamental theorem of calculus

The **fundamental theorem of calculus** is central to the study of calculus. It is the theorem that shows the relationship between the derivative and the integral and between the definite integral and the indefinite integral. It is broken into two parts, the first fundamental theorem of calculus and the second fundamental theorem of calculus.

The first fundamental theorem of calculus states that if the function f is continuous, then

d/dxâˆ«axf(t)dt=f(x)

This means that the derivative of the integral of a function f with respect to the variable t over the interval [a,x] is equal to the function f with respect to x. This describes the derivative and integral as inverse processes.

28. Pythagorean theorem

In mathematics, the **Pythagorean theorem** or **Pythagoras’s theorem** is a statement about the sides of a right triangle.

One of the angles of a right triangle is always equal to 90 degrees. This angle is the right angle. The two sides next to the right angle are called the legs and the other side is called the hypotenuse. The hypotenuse is the side opposite to the right angle, and it is always the longest side.

The Pythagorean theorem says that the area of a square on the hypotenuse is equal to the sum of the areas of the squares on the legs. In this picture, the area of the blue square added to the area of the red square makes the area of the purple square. If the lengths of the legs are *a* and *b*, and the length of the hypotenuse is *c*, then,

a^2+b^2=c^2.

Pythagorean Triples

Pythagorean Triples or Triplets are three whole numbers which fit the equation

a^2+b^2=c^2.

The triangle with sides of 3, 4, and 5 is a well known example. If a=3 and b=4, then

3^2+4^2=5^2

because

9+16=25. This can also be shown as 3^2+4^2=5.

The three-four-five triangle works for all multiples of 3, 4, and 5. In other words, numbers such as 6, 8, 10 or 30, 40 and 50 are also Pythagorean triples. Another example of a triple is the 12-5-13 triangle, because

12^2+5^2=13

29. Gauss Bonnet theorem

The **Gauss Bonnet theorem** or **Gauss Bonnet formula** in differential geometry is an important statement about surfaces which connects their geometry (in the sense of curvature) to their topology (in the sense of the euler characteristic).

30. Newton’s law of universal gravitation

Fg=Gm1m2/r2,

**Newton’s universal law of gravitation** is a physical law that describes the attraction between two objects with mass.

In this equation:

* F** _{g}* is the total gravitational force between the two objects.

*G* is the gravitational constant.

*m*_{1} is the mass of the first object.

*m*_{2} is the mass of the second object.

*r* is the distance between the centres of the objects.

In SI units, *F** _{g}* is measured in newtons (N),

*m*

_{1}and

*m*

_{2}in kilograms (kg),

*r*in meters (m), and the constant

*G*is approximately equal to 6.674Ã—10

^{-11}N m

^{2}kg

31. Newton’s 2nd law of motion

F=ma.

For a particle of mass *m*, the net force **F** on the particle is equal to the mass *m* times the particle’s acceleration **a.**

32. Kinetic energy

**Kinetic energy** is the energy that an object has because of its motion. This energy can be converted into other kinds, such as gravitational or electric potential energy, which is the energy that an object has because of its position in a gravitational or electric field.

Translational kinetic energy

The **translational kinetic energy** of an object is:

E translational=1/2mv^2

where m is the mass (resistance to linear acceleration or deceleration); v

is the linear velocity.

Rotational kinetic energy

The **rotational kinetic energy** of an object is:

E rotational=1/2I^2

where I is the moment of inertia (resistance to angular acceleration or deceleration, equal to the product of the mass and the square of its perpendicular distance from the axis of rotation);

Ï is the angular velocity.

33. Potential energy

**Potential energy** is the energy that an object has because of its position on a gradient of potential energy called a potential field.

Actual energy (*E* = hf) is nonzero frequency angular momentum.

Potential energy (rest mass) is zero frequency angular momentum.

The potential fields are irrotationally radial (“electric”) fluxes of the vacuum and divide into two classes:

The gravitoelectric fields;

The electric fields.

The potential energy is negative. It is not a mere convention but a consequence of conservation of energy in the zero-energy universe as an object descends into a potential field, its potential energy becomes more negative, while its actual energy becomes more positive, and, in accordance with the 2nd law of thermodynamics, tends to be radiated away, so that the object acquires a net negative potential energy, also known as the object’s binding energy.

In accordance with the minimal total potential energy principle, the universe’s matter flows towards ever more negative total potential energy. This cosmic flow is time.

Gravitational potential energy

**Self gravitating sphere**

The gravitational potential energy of a massive spherical cloud is proportional to its radius and causes the sphere to fall towards its own centre.

**Earth**

If an object is lifted a certain distance from the surface from the earth, the force experienced is caused by weight and height. Work is defined as force over a distance, and work is another word for energy.

Electric potential energy

**Electric potential energy** is experienced by charges both different and alike, as they repel or attract each other. Charges can either be positive (+) or negative (-), where opposite charges attract and similar charges repel.

Elastic potential energy

**Elastic potential energy** is experienced when a rubbery material is pulled away or pushed together. The amount of potential energy the material has depends on the distance pulled or pushed. The longer the distance pushed, the greater the elastic potential energy the material has.

34. Second law of thermodynamics

S (prime)-S>=0

The **second law of thermodynamics** says that when energy changes from one form to another form, or matter moves freely, entropy (disorder) increases, in a closed system.

Differences in temperature, pressure, and density tend to even out horizontally after a while. Due to the force of gravity, density and pressure do not even out vertically. Density and pressure on the bottom will be more than at the top.

Entropy is a measure of spread of matter and energy to everywhere they have access.

The most common wording for the second law of thermodynamics is essentially due to *Rudolf Clausius*:

It is impossible to construct a device which produces no other effect than transfer of heat from lower temperature body to higher temperature body

In other words, everything tries to maintain the same temperature over time.

There are many statements of the second law which use different terms, but are all equal. Another statement by Clausius is:

Heat cannot of itself pass from a colder to a hotter body.

An equivalent statement by Lord Kelvin is:

A transformation whose only final result is to convert heat, extracted from a source at constant temperature, into work, is impossible.

The second law only applies to large systems. The second law is about the likely behavior of a system where no energy or matter gets in or out. The bigger the system is, the more likely the second law will be true.

In a general sense, the **second law** says that temperature differences between systems in contact with each other tend to even out and that work can be obtained from these non-equilibrium differences, but that loss of thermal energy occurs, when work is done and entropy increases. Pressure, density and temperature differences in an isolated system, all tend to equalize if given the opportunity; density and pressure, but not temperature, are affected by gravity. A heat engine is a mechanical device that provides useful work from the difference in temperature of two bodies.

Quotes

The law that entropy always increases, holds, I think, the supreme position among the laws of nature. If someone points out to you that your pet theory of the universe is in disagreement with Maxwell’s equations a then so much the worse for Maxwell’s equations. If it is found to be contradicted by observation a well, these experimentalists do bungle things sometimes. But if your theory is found to be against the second law of thermodynamics I can give you no hope; there is nothing for it but to collapse in deepest humiliation.

–Sir Arthur Stanley Eddington, *The Nature of the Physical World* (1927)

The tendency for entropy to increase in isolated systems is expressed in the second law of thermodynamics — perhaps the most pessimistic and amoral formulation in all human thought.

—Greg Hill and Kerry Thornley. principia discordia(1965)

There are almost as many formulations of the second law as there have been discussions of it.

–Philosopher / Physicist P.W. Bridgman, (1941)

35. Principle of least action

The **principle of least action** a or, more accurately, the **principle of stationary action** â’s a variational principle that, when applied to the action of a mechanical system, can be used to obtain the equations of motion for that system. In relativity, a different action must be minimized or maximized. The principle can be used to derive newtonian, lagrangian and hamiltonian equations of motion, and even general relativity. The principle remains central in modern physics and mathematics, being applied in thermodynamics, fluid mechanics, the theory of relativity, mechanics, particle physics, and string theory and is a focus of modern mathematical investigation in morse theory. maupertuis principle and Hamilton’s principle exemplify the principle of stationary action.

36. SPHERICAL HARMONICS

In mathematics and physical science, **spherical harmonics** are special functions defined on the surface of a sphere. They are often employed in solving partial differential equations that commonly occur in science. The spherical harmonics are a complete set of orthogonal functions on the sphere, and thus may be used to represent functions defined on the surface of a sphere, just as circular functions (sines and cosines) are used to represent functions on a circle via fourier series. Like the sines and cosines in Fourier series, the spherical harmonics may be organized by (spatial) angular frequency. Further, spherical harmonics are basis functions for SO(3), the group of rotations in three dimensions, and thus play a central role in the group theoretic discussion of SO(3).

37. Cauchy Residue theorem

In complex analysis, a discipline within mathematics, the **residue theorem**, sometimes called **Cauchy’s residue theorem**, is a powerful tool to evaluate line integrals of analytic functions over closed curves; it can often be used to compute real integrals as well. It generalizes the cauchy integral theorem and cauchy integral formula. From a geometrical perspective, it is a special case of the generalized stoke’s theorem.

38. Callan Symanzik equation

In physics, the **Callan Symanzik equation** is a differential equation describing the evolution of the n-point correlation functions under variation of the energy scale at which the theory is defined and involves the beta-function of the theory and the anomalous dimensions.

39. MINIMAL SURFACE EQUATION

In mathematics, a **minimal surface** is a surface that locally minimizes its area. This is equivalent to having zero mean curvature. The term “minimal surface” is used because these surfaces originally arose as surfaces that minimized total surface area subject to some constraint. Physical models of area-minimizing minimal surfaces can be made by dipping a wire frame into a soap solution, forming a soap film, which is a minimal surface whose boundary is the wire frame. However the term is used for more general surfaces that may self-intersect or do not have constraints. For a given constraint there may also exist several minimal surfaces with different areas.

40. EULER’S 9 POINT CENTER/Nine-point center

In geometry, the **nine-point center** is a triangle center, a point defined from a given triangle in a way that does not depend on the placement or scale of the triangle. It is so-called because it is the center of the 9-point circle, a circle that passes through nine significant points of the triangle: the midpoints of the three edges, the feet of the three altitudes, and the points halfway between the orthocenter and each of the three vertices.

41 MANDELBROT SETS

The **Mandelbrot set** is a famous example of a fractals in mathematics.The Mandelbrot set is important for the chaos theory. The edging of the set shows a self-similarity, which is not perfect because it has deformations.

42. Yang Baxter equation

In physics, the **Yang Baxter equation** (or **star-triangle relation**) is a consistency equation which was first introduced in the field of statistical mechanics. It depends on the idea that in some scattering situations, particles may preserve their momentum while changing their quantum internal states. It states that a matrix

R, acting on two out of three objects, satisfies

(RâŠ—1)(1âŠ—R)(RâŠ—1)=(1âŠ—R)(RâŠ—1)(1âŠ—R)

In one dimensional quantum systems,

R is the scattering matrix and if it satisfies the Yang Baxter equation then the system is integrable. The Yang Baxter equation also shows up when discussing knot theory and the braid groups where

R corresponds to swapping two strands. Since one can swap three strands two different ways, the Yang Baxter equation enforces that both paths are the same.

43. DIVERGENCE THEOREM

In vector calculus, the **divergence theorem**, also known as **Gauss’s theorem** or **Ostrogradsky’s theorem**, is a result that relates the flow (that is, flux) of a vector field through a surface to the behavior of the vector field inside the surface. More precisely, the divergence theorem states that the outward flux of a vector field through a closed surface is equal to the volume integral of the divergence over the region inside the surface. Intuitively, it states that *the sum of all sources (with sinks regarded as negative sources) gives the net flux out of a region*. The divergence theorem is an important result for the mathematics of physics and engineering, in particular in electrostatics and fluid dynamics.

In physics and engineering, the divergence theorem is usually applied in three dimensions. However, it generalizes to any number of dimensions. In one dimension, it is equivalent to the fundamental theorem of calculus. In two dimensions, it is equivalent to green’s theorem. The theorem is a special case of the more general stoke’s theorem.

44. Bayes’ theorem

P(A|B)=P(B|A)P(A)P(B).

In probability theory and applications, **Bayes’ theorem** shows the relation between a conditional probability and its reverse form. For example, the probability of a hypothesis given some observed pieces of evidence and the probability of that evidence given the hypothesis.

45. Logistic map

xn+1=rxn(1-xn)

where

xn

is a number between zero and one that represents the ratio of existing population to the maximum possible population.

46. EINSTEIN’S LAW OF VELOCITY ADDITION/Velocity-addition formula

In relativistic physics, a **velocity-addition formula** is a three-dimensional equation that relates the velocities of objects in different reference frames. Such formulas apply to successive lorentz transformations, so they also relate different frames. Accompanying velocity addition is a kinematic effect known as thomas procession, whereby successive non-collinear Lorentz boosts become equivalent to the composition of a rotation of the coordinate system and a boost.

Standard applications of velocity-addition formulas include the doppler shift, doppler navigation, the aberration of light, and the dragging of light in moving water. It was observed by galilei that a person on a uniformly moving ship has the impression of being at rest and sees a heavy body falling vertically downward. This observation is now regarded as the first clear statement of the principle of mechanical relativity. The cosmos of Galileo consists of absolute space and time and the addition of velocities corresponds to composition of galilean transformations. The relativity principle is called galilean relativity. It is obeyed by newtonian mechanics.

According to the theory of special relativity, the frame of the ship has a different clock rate and distance measure, and the notion of simultaneity in the direction of motion is altered, so the addition law for velocities is changed. The cosmos of special relativity consists of Minkowski spacetime and the addition of velocities corresponds to composition of lorentz transformations. In the special theory of relativity Newtonian mechanics is modified into relativistic mechanics.

47. PHOTOELECTRIC EFFECT FORMULA

The **photoelectric equation** involves; h = the Planck constant 6.63 x 10^{-34} J s. f = the frequency of the incident light in hertz (Hz) … E_{k} = the maximum kinetic energy of the emitted electrons in joules (J)

The **photoelectric effect** is the emission of electrons or other free carriers when light is shone onto a material. Electrons emitted in this manner can be called *photo electrons*. The phenomenon is commonly studied in electronic physics, as well as in fields of chemistry, such as quatuum chemistry or electrochemistry.

48. Faraday’s law of induction

**Faraday’s law of induction** is one of the basic laws of electromagnetism. The law explains the operation principles of generators, transformers and electric motors.

49. Cauchy momentum equation

The **Cauchy momentum equation** is a vector partial differential equation put forth by cauchy that describes the non-relativistic momentum transport in any continuum.

50. De Moivre’s formula

The process of mathematical induction can be used to prove a very important **theorem** in mathematics known as **De Moivre’s theorem**. If the complex number z = r(cos α + i sin α), then. The preceding pattern can be extended, using mathematical induction, to **De Moivre’s theorem**.

51. Fourier transform

The **Fourier transform** is a mathematical function that can be used to show the different parts of a continuous signal. It is most used to convert from time domain to frequency domain. Fourier transforms are often used to calculate the frequency spectrum of a signal that changes over time. This kind of signal processing has many uses such as cryptography, oceanography, speech recognition, or handwriting recognition. Fourier transforms can also be used to solve differential equations.

Calculating a Fourier transform requires understanding of integration and imaginary numbers. Computers are usually used to calculate Fourier transforms of anything but the simplest signals. The Fast Fourier Transform is a method computers use to quickly calculate a Fourier transform.

52. Prime-counting function

In mathematics, the **prime-counting function** is the function counting the number of prime numbers less than or equal to some real number *x*. It is denoted by Ï€(*x*) (unrelated to the number Ï€).

Number of primes in up to the number x=x/lnx

53. MURPHY’S LAW FORMULA

Here, PM is the Murphy’s probability that something will go wrong. KM is Murphy’s constant (equal to one) and FM is Murphy’s factor, a very small number.

**Murphy’s law** is an adage or epigram that is typically stated as: “Anything that can go wrong will go wrong”.

54. SUMMATION FORMULA

In mathematics, **summation** (capital Greek sigma symbol: **âˆ‘**) is the addition of a sequence of numbers; the result is their *sum* or *total*. If numbers are added sequentially from left to right, any intermediate result is a partial sum, prefix sum, or running total of the summation.

The numbers to be summed (called *addends*, or sometimes *summands*) may be integers, rational numbers, real numbers, or complex numbers. Besides numbers, other types of values can be added as well: vectors, matrices, polynomials and, in general, elements of any additive group (or even monoid).

55. Logarithmic spiral A **logarithmic spiral**, **equiangular spiral** or **growth spiral** is a special kind of spiral spiral curve which often appears in nature. The logarithmic spiral was first described by descarte and later extensively investigated by Jakob bernoulli, who called it *Spira mirabilis*, “the marvelous spiral”.

Logarithmic spirals in nature

In several natural phenomena one may find curves that are close to being logarithmic spirals. Here follows some examples and reasons:

The approach of a hawk to its prey. Their sharpest view is at an angle to their direction of flight; this angle is the same as the spiral’s pitch.

The approach of an insect to a light source. They are used to having the light source at a constant angle to their flight path. Usually the sun is the only light source and flying that way will result in a practically straight line.

The arms of spiral galaxies. Our own galaxy, the milky way. is believed to have four major spiral arms, each of which is roughly a logarithmic spiral with pitch of about 12 degrees, an unusually small pitch angle for a galaxy such as the Milky Way. In general, arms in spiral galaxies have pitch angles ranging from about 10 to 40 degrees.

The arms of tropical cyclones, such as hurricanes.

Many biological structures including spider webs and the shells of mullosks. In these cases, the reason is the following: Start with any irregularly shaped two-dimensional figure *F*_{0}. Expand *F*_{0} by a certain factor to get *F*_{1}, and place *F*_{1} next to *F*_{0}, so that two sides touch. Now expand *F*_{1} by the same factor to get *F*_{2}, and place it next to *F*_{1} as before. Repeating this will produce an approximate logarithmic spiral whose pitch is determined by the expansion factor and the angle with which the figures were placed next to each other. This is shown for polygonal figures in the accompanying graphic.

56. Heron’s formula

Heron’s formula states that the area of a triangle whose sides have lengths *a*, *b*, and *c*is

A=s(s-a)(s-b)(s-c),

where *s* is the semiperimeter of the triangle; that is,

s=(a+b+c)2.

In geometry, **Heron’s formula **gives the area of a triangle by requiring no arbitrary choice of side as base or vertex as origin, contrary to other formulas for the area of a triangle, such as half the base times the height or half the norm of a cross product of two sides.

57. Quadratic equation

x=-b+/- sqrt(b^2-4ac)/2a

A **quadratic equation** is an equation in the form of *ax*^{2} + *bx* + *c*, where *a* is not equal to 0. It makes a parabola (a “U” shape) when graphed on a coordinate plane.

Where the letters are the corresponding numbers of the original equation, *ax*^{2} + *bx* + *c* = 0. Also, *a* cannot be 0 for the formula to work properly.

The factored form of this equation is *y* = *a*(*x* âˆ’ *s*)(*x* âˆ’ *t*), where *s* and *t* are the zeros, *a* is a constant, and *y* and the two *x*s are ordered pairs which satisfy the equation.

58. Euler line

In geometry, the **Euler line**, is a line determined from any triangle that is not equilateral. It is a central line of the triangle, and it passes through several important points determined from the triangle, including the orthocenter, the circumcenter, the centroid, the Exeter point and the center of the 9-point circle of the triangle.

59. PYTHAGOREAN TRIPLES FORMULA

**Euclid’s formula** is a fundamental formula for generating Pythagorean triples given an arbitrary pair of integers *m* and *n* with *m* > *n* > 0. The formula states that the integers

a=m^2âˆ’n^2,

b=2mn,

c=m^2+n^2

form a Pythagorean triple

A **Pythagorean triple** consists of three positive integers *a*, *b*, and *c*, such that *a*^{2} + *b*^{2}= *c*^{2}. Such a triple is commonly written (*a*, *b*, *c*), and a well-known example is (3, 4, 5). If (*a*, *b*, *c*) is a Pythagorean triple, then so is (*ka*, *kb*, *kc*) for any positive integer *k*.

60. Euler’s formula

e^ix=cosx+isinx

**Euler’s formula **is a mathematical formula in complex analysis that establishes the fundamental relationship between the trigonometric functions and the complex exponential functions. Euler’s formula is ubiquitous in mathematics, physics, and engineering. The physicist richard feynmann called the equation “our jewel” and “the most remarkable formula in mathematics”.

When

x=Ï

Euler’s formula evaluates to

e^i+1=0

which is known as ruler’s identity.

61.

**Simplex method**, Standard technique in linear programming for solving an optimization problem, typically one involving a function and several constraints expressed as inequalities. The inequalities define a polygonal region (*see*polygon), and the solution is typically at one of the vertices. The simplex method is a systematic procedure for testing the vertices as possible solutions.

62. PROOF INFINITE NUMBER OF PRIME NUMBERS

Theorem.There are infinitely many primes.
Suppose that P = p_{1}p_{2}…p+1 and let _{r}p be a prime dividing P; then p can not be any of p_{1}, p_{2}, …, p, otherwise _{r}p would divide the difference P–p_{1}p_{2}…p=1, which is impossible. So this prime _{r}p is still another prime, and p_{1}, p_{2}, …, p would not be all of the primes._{r} |

63. Harmonic series (mathematics)

In mathematics, the **harmonic series** is the divergent infinite series:

Summation n=1 to infinity of 1/n=1+1/2+1/3+1/4+1/5+…

Its name derives from the concept of overtones, or harmonics in music: the wavelengths of the overtones of a vibrating string are

1/2,1/3,1/4, etc., of the string’s fundamental wavelengths. Every term of the series after the first is the harmonic mean of the neighboring terms; the phrase *harmonic mean* likewise derives from music.

The harmonic series can be counterintuitive to students first encountering it, because it is a divergent series even though the limit of the *n*th term as *n* goes to infinity is zero. The divergence of the harmonic series is also the source of some apparent paradoxes. One example of these is the “worm on a rubberband”. Suppose that a worm crawls along an infinitely-elastic one-meter rubber band at the same time as the rubber band is uniformly stretched. If the worm travels 1 centimeter per minute and the band stretches 1 meter per minute, will the worm ever reach the end of the rubber band? The answer, counterintuitively, is “yes”, for after *n* minutes, the ratio of the distance travelled by the worm to the total length of the rubber band is

1/100âˆ‘k=1 to n1^k

(In fact the actual ratio is a little less than this sum as the band expands continuously.) The reason is that the band expands behind the worm also; eventually, the worm gets past the midway mark and the band behind expands increasingly more rapidly than the band in front.

Because the series gets arbitrarily large as *n* becomes larger, eventually this ratio must exceed 1, which implies that the worm reaches the end of the rubber band. However, the value of *n* at which this occurs must be extremely large: approximately *e*^{100}, a number exceeding 10^{43} minutes (10^{37} years). Although the harmonic series does diverge, it does so very slowly.

Another problem involving the harmonic series is the jeep problem.

Another example is the block-stacking problem: given a collection of identical dominoes, it is clearly possible to stack them at the edge of a table so that they hang over the edge of the table without falling. The counterintuitive result is that one can stack them in such a way as to make the overhang arbitrarily large, provided there are enough dominoes.

A simpler example, on the other hand, is the swimmer that keeps adding more speed when touching the walls of the pool. The swimmer starts crossing a 10-meter pool at a speed of 2 m/s, and with every cross, another 2 m/s is added to the speed. In theory, the swimmer’s speed is unlimited, but the number of pool crosses needed to get to that speed becomes very large; for instance, to get to the speed of light (ignoring special relativity), the swimmer needs to cross the pool 150 million times. Contrary to this large number, the time required to reach a given speed depends on the sum of the series at any given number of pool crosses (iterations):

10/2âˆ‘k=1 to n1^k.

Calculating the sum (iteratively) shows that to get to the speed of light the time required is only 94 seconds. By continuing beyond this point (exceeding the speed of light, again ignoring special relativity), the time taken to cross the pool will in fact approach zero as the number of iterations becomes very large, and although the time required to cross the pool appears to tend to zero (at an infinite number of iterations), the sum of iterations (time taken for total pool crosses) will still diverge at a very slow rate.

64. EULER SUMS

precise sum of the infinite series:

∑n=1 to ∞1/n^2=1/1^2+1/2^2+1/3^2+⋯=**1.644934** or π^{2}/6

65. FORMULA FOR SOLUTION OF CUBIC EQUATION

In algebra, a **cubic function** is a function of the form f(x)=ax^3+bx^2+cx+d

in which *a* is nonzero.

Setting *f*(*x*) = 0 produces a **cubic equation** of the form

ax^3+bx^2+cx+d=0.

The solutions of this equation are called roots of the polynomial *f*(*x*). If all of the coefficients *a*, *b*, *c*, and *d* of the cubic equation are real numbers then there will be at least one real root (this is true for all odd degree polynomials). All of the roots of the cubic equation can be found algebraically. (This is also true of a quadratic or quartic (fourth degree) equation, but no higher-degree equation, by the abel-ruffini theorem). The roots can also be found trigonometrically. Alternatively, numeric approximates of the roots can be found using root-finding theorem like newton’s method.

The coefficients do not need to be complex numbers. Much of what is covered below is valid for coefficients of any field with characteristic 0 or greater than 3. The solutions of the cubic equation do not necessarily belong to the same field as the coefficients. For example, some cubic equations with rational coefficients have roots that are non-rational (and even non-real) complex numbers.

66. SOLUTION TO QUARTIC EQUATION

In algebra, a **quartic function** is a function of the form

f(x)=ax^4+bx^3+cx^2+dx+e,

where *a* is nonzero, which is defined by a polynomial of degree four, called a **quartic polynomial**.

Sometimes the term **biquadratic** is used instead of *quartic*, but, usually, **biquadratic function** refers to a quadratic function of a square (or, equivalently, to the function defined by a quartic polynomial without terms of odd degree), having the form

f(x)=ax^4+cx^2+e.

A **quartic equation**, or equation of the fourth degree, is an equation that equates a quartic polynomial to zero, of the form

ax^4+bx^3+cx^2+dx+e=0,

where *a* â‰ 0.

The derivative of a quartic function is a cubic function.

Since a quartic function is defined by a polynomial of even degree, it has the same infinite limit when the argument goes to positive or negative infinity. If *a* is positive, then the function increases to positive infinity at both ends; and thus the function has a global minimum. Likewise, if *a* is negative, it decreases to negative infinity and has a global maximum. In both cases it may or may not have another local maximum and another local minimum.

The degree four (*quartic* case) is the highest degree such that every polynomial equation can be solved by radicals.

**67. Quintic function**

**In algebra, a quintic function is a function of the form**

g(x)=ax^5+bx^4+cx^3+dx^2+ex+f,

where *a*, *b*, *c*, *d*, *e* and *f* are members of a field, typically the rational numbers, the real numbers or the complex numbers, and *a* is nonzero. In other words, a quintic function is defined by a polynomials of degree five.

If *a* is zero but one of the coefficients *b*, *c*, *d*, or *e* is non-zero, the function is classified as either a quartic function, cubic function, quadratic function or linear function.

Because they have an odd degree, normal quintic functions appear similar to normal cubic function when graphed, except they may possess an additional local maximum and local minimum each. The derivative of a quintic function is a quartic function.

Setting *g*(*x*) = 0 and assuming *a* â‰ 0 produces a **quintic equation** of the form:

ax^5+bx^4+cx^3+dx^2+ex+f=0.

Solving quintic equations in terms of radicals was a major problem in algebra, from the 16th century, when cubic and quartic equations were solved, until the first half of the 19th century, when the impossibility of such a general solution was proved, with the abel-ruffini theorem.

Finding the roots of a given polynomial has been a prominent mathematical problem. Solving linear, quadratic, cubic and quartic equations by factorization into radicals can always be done, no matter whether the roots are rational or irrational, real or complex; there are formulae that yield the required solutions. However, there is no algebraic expression for general quintic equations over the rationals in terms of radicals. This also holds for equations of higher degrees. Some quintics may be solved in terms of radicals. However, the solution is generally too complex to be used in practice. Instead, numerical approximations are calculated using Root-finding algorithms for polynomials. Some quintic equations can be solved in terms of radicals. These include the quintic equations defined by a polynomial that is reducible, such as *x*^{5} âˆ’ *x*^{4} âˆ’ *x* + 1 = (*x*^{2} + 1)(*x* + 1)(*x* âˆ’ 1)^{2}. For characterizing solvable quintics, and more generally solvable polynomials of higher degree, evariste galois developed techniques which gave rise to group theory and galois theory. Applying these techniques, arthur caylay found a general criterion for determining whether any given quintic is solvable.

68. Lorentz force

In physics (particularly in electromagnetism) the **Lorentz force** is the combination of electric and magnetic force on a point charge due to electromagnetic fields. A particle of charge *q* moving with velocity **v** in the presence of an electric field **E** and a magnetic field **B** experiences a force

F=qE+qv —B

(in SI units).

69. Euler-lagrange formula-

Lsubx(tsuby,q(t),qdot(t))-d/dtLsubx(t,q(t),qdot(t))=0

In the calculus of variations, the **Euler-Lagrange equation**, **Euler’s equation**, or **Lagrange’s equation** (although the latter name is ambiguous see disambiguation page), is a second-order partial differential equation whose solutions are the functions for which a given functional is stationary. It was developed by Swiss mathematician Leonhard Euler and Italian-French mathematician Joseph-Louis Lagrange in the 1750s.

70. Euler product formula In number theory, an **Euler product** is an expansion of a Dirichlet series into an infinite product indexed by prime numbers. The original such product was given for the sum of all positive integers raised to a certain power as proven by Leonhard Euler. This series and its continuation to the entire complex plane would later become known as the Riemann zeta function.

âˆpP(p,s)

71. Euler-maclaurin formula In mathematics, the **Euler-Maclaurin formula** provides a powerful connection between integrals (see calculus) and sums. It can be used to approximate integrals by finite sums, or conversely to evaluate finite sums and infinite series using integrals and the machinery of calculus. For example, many asymptotic expansions are derived from the formula, and Faulhaber’s formula for the sum of powers is an immediate consequence.

72. Pi

pi=C/d

(**pi** is equal to the circumference divided by the diameter).

Pi is an endless string of numbers

**Pi** is a mathematical constant. It is the ratio of the distance around a circle to the circle’s diameter. This produces a number, and that number is always the same. However, the number is rather strange. The number starts 3.141592……. and continues without end. Numbers like this are called irrational numbers.

The diameter is the longest straight line which can be fitted inside a circle. It passes through the center of the circle. The distance around a circle is known as the circumference. Even though the diameter and circumference are different for different circles, the number pi remains constant: its value never changes. This is because the relationship between the circumference and diameter is always the same.

A mathematician named Lambert also showed in 1761 that the number pi was irrational; that is, it cannot be written as a fraction by normal standards. Another mathematician named Lindeman was also able to show in 1882 that pi was part of the group of numbers known as transcendental, which are numbers that cannot be the solution to a polynomial equation.

Pi can also be used for figuring out many other things beside circles. The properties of pi have allowed it to be used in many other areas of math besides geometry, which studies shapes. Some of these areas are complex analysis, trigonometry, and series.

Today, there are different ways to calculate many digits of. This is of limited use though. Pi can sometimes be used to work out the area or the circumference of any circle. To find the circumference of a circle, use the formula C (circumference) = Ï times diameter. To find the area of a circle, use the formula (radius²). This formula is sometimes written as

A=r^2, where *r* is the variable for the radius of any circle and *A* is the variable for the area of that circle.

To calculate the circumference of a circle with an error of 1 mm:

4 digits are needed for a radius of 30 meters

10 digits for a radius equal to that of the earth

15 digits for a radius equal to the distance from the earth to the sun.

People generally celebrate March 14 as pi day because March 14 is also written as 3/14, which represents the first three numbers 3.14 in the approximation of pi.

73. Exponential function

In mathematics, an **exponential function** is a function that quickly grows. More precisely, it is the function

exp(x)=e^x, where e is ruler’s constant, an irrational number that is approximately 2.71828.

74. Natural logarithm

The **natural logarithm** of a number is its logarithm to the base of the mathematical constant e, where *e* is an irrational and transcendental number approximately equal to 2.718281828459. The natural logarithm of *x* is generally written as ln *x*, log_{e}*x*, or sometimes, if the base *e* is implicit, simply log *x*. The natural logarithm of *x* is the power to which *e* would have to be raised to equal *x*.

75. Conic sections

CONIC SECTIONS-

Circle-

(x-g)^2+(y-h)^2=radius^2

(g=x coordinate, h=y coordinate)

Parabola-

y^2+/-4ax

(a=x coordinate)

x^2=+/-4ay

(a=y coordinate)

ellipse-

x^2/a^2+y^2/b^2=1,

(a=x, b=y coordinates, or a=y, b=x coordinates)

Hyperbola-

x^2/a^2-y^2/b^2=1,

(a=x, b=y coordinates, or a=y, b=x coordinates)

76. Exponential growth and decay

y=A*exp^k*t

A=starting number of for example bacteria, t=length of growth time, k=constant, y=number of bacteria after t time

77. Calculating an orbit I.e. of a comet

Calculations: orbit, period of orbit, perihelion, aphelion and eccentricity

(for example a comet)â€”

Use ellipse formula x^2/a^2+y^2/b^2=1

Then calculate from 2 coordinates in AUs with formula

x^2 x b^2 + y^2 x a^2=a^2 x b^2

find a and bÂ (the closest and furthest approaches)

Period years of orbit^3=distance (a from above)^2

Period=cuberoot(distance AUs of â above)^2

Perihelion=d=a-c

c=(a^2-b^2)^1/2 (c=distance from focus to center of ellipse)

aphelion=d=c-b

perihelion=A x (1-eccentricity)

aphelion=A x (1+ eccenticity)

A (semimajor axis)=(perihelion + aphelion)/2

eccentricity=1-perihelion/A

eccentricity=aphelion/A-1

To find formula for the orbit, use ellipse formula

x^2/a^2+y^2/b^2=1, then use formula

x^2 x b^2+y^2 x a^2=a^2 x b^2,

Use 2 location coordinates from the orbit, plug in one of the coordinates

Into the 2nd formula, then plug in the 2nd coordinates into the same

formula. Subtract one of the resulting formulas from the other resulting

formula, then solve for a or b with the formula that results from the

subtraction. Plug in the solution to a or b that was solved into one of the

Pre-subtraction formulas to find the a or b that has not been found yet.

Now, we have the a and b constants, so we plug them into the ellipse

Formula, and thus have the equation for the orbit of the stellar body,

I.e. a comet.

78. Interesting math example #1

1×1=1

11×11=121

111×111=12321

1111×1111=1234321

11111×11111=123454321

111111×111111=12345654321

Etc

79. Interesting math example #2

1×8+1=9

12×8+2=98

123×8+3=987

1234×8+4=9876

12345×8+5=98765

Etc

80. Interesting math example #3

1=.9999999

0.999

In mathematics, **0.999…** (also written **0.9**, among other ways), denotes the repeating decimal consisting of infinitely many 9 after the decimal point (and one 0 before it). This repeating decimal represents the smallest number no less than all decimal number 0.9, 0.99, 0.999, etc.^{[1]} This number can be shown to equal 1. In other words, “0.999…” and “1” represent the same number. There are many ways of showing this equality, from intuitive arguments to mathematically rigorous proof. The technique used depends on target audience, background assumptions, historical context, and preferred development of the real numbers, the system within which 0.999… is commonly defined. (In other systems, 0.999… can have the same meaning, a different definition, or be undefined.) More generally, every nonzero terminating decimals has two equal representations (for example, 8.32 and 8.31999…), a property true of all base representations. The utilitarian preference for the terminating decimal representation contributes to the misconception that it is the only representation. For this and other reasons such as rigorous proofs relying on non-elementary techniques, properties, and/or disciplines math students can find the equality sufficiently counterintuitive that they question or reject it. This has been the subject of several studies in mathematics education.

UNIVERSAL PHYSICAL CONSTANTS

ATOMIC MASS UNITS-

1.6605402 x 10^-27 kilograms

(1/12 0f the mass of an atom of carbon-12)

AVOGADRO NUMBER-

6.0221367 x 10^23/moles

(mole=number of elementary entities that are in

carbon-12 atoms in exactly 12 grams of carbon-12)

BOHR’S MAGNETON-

9.2740154 x 10^-24 joules/tesla

(The magnetic moment of an electron caused by either its orbital

or spin angular momentum. Magnetic moment is a quantity that

determines the torque it will experience in an external magnetic

field. Torque is rotational force. A joule is equal to the work done on

an object when a force of 1 newton acts on the object in the direction

of motion through a distance of 1 meter: kilograms x meters^2/

seconds^2. A joule is also equal to 10 million ergs. A Tesla is a

derived unit of the strength of a magnetic field: kilograms/(seconds^2

x amperes.)

BOHR RADIUS-

5.29177249 x 10^-11 meters

(The mean radius of an electron around the nucleus of a hydrogen atom

at its ground state.)

BOLTZMANN CONSTANT-

1.3806513 x 10^-23 joules/kelvin

(A physical constant relating the average kinetic energy of

particles in a gas with the temperature of the gas. Kelvin

temperature scale is the primary unit of temperature measurement

in the physical sciences, but is often used in conjunction with the

celsius degree, which is of the same magnitude absolute zero in

kelvin is equal to -273.15 degrees celsius.)

ELECTRON CHARGE-

1.60217733 x 10^-19 coulombs

(Charge carried by a single electron. The coulomb is the quantity of

charge that has passed through the cross section of an electrical

conductor carrying one ampere within one second.)

ELECTRON CHARGE/MASS RATIO-

1.75881962 x 10^11 coulombs/kilograms

(The importance of the charge-to-mass ratio, according to classical

electrodynamics, is that 2 particles with the same charge-to-mass ratio

move in the same path in a vacuum when subjected to the same electric

and magnetic fields.)

ELECTRON COMPTON WAVELENGTH-

2.42631058 X 10^-12 meters

(A compton wavelength of a particle is equal to the wavelength of a

photon whose energy is the same as the mass of the particle. The

compton wavelength of an electron is the characteristic length scale of

quantum electrodynamics. It is the length scale at which relativistic

quantum field theory becomes crucial for its accurate description.)

ELECTRON MAGNETIC MOMENT-

9.2847701 x 10^-24 joules/tesla

(The electron is a charged particle of -1e, where e is the unit of

elementary charge. Its angular momentum comes from 2 types of

rotation: spin and orbital motion.)

ELECTRON MAGNETIC MOMENT IN BOHR MAGNETONS-

1.00159652193

(Bohr magneton is a physical constant and natural unit for expressing the

magnetic moment of an electron caused by either its orbital or spin

angular momentum. The electron magnetic moment, which is the

electron’s intrinsic spin magnetic moment, is approximately one Bohr

magneton.)

ELECTRON MAGNETIC MOMENT/PROTON MAGNETIC MOMENT-

658.21068801

ELECTRON REST MASS-

9.1093897 x 10^-31 kilograms

ELECTRON REST MASS/PROTON REST MASS-

5.44617013 x1 0^-4

This is how much less mass the electron is as compared to the proton.

(1,836.21 times lighter than proton)

FARADAY CONSTANT-

9.6458309 x 10^4 coulombs/mole

(The magnitude of electric charge per mole of electrons.)

FINE STRUCTURE CONSTANT-

.00729735308

(The strength of the electromagnetic interaction between elementary

particles.)

GAS CONSTANT-

8.3144710 x 10^joules/(mole x kelvin)

(A physical constant which is featured in many fundamental equations

in the physical sciences, such as the ideal gas law and the Nernst

equation.)

GRAVITATIONAL CONSTANT-

6.67206 x 10^-11 newtons x meters^3/(kilograms*second^2)

Denoted by letter G, it is an empirical physical constant involved

in the calculation of gravitational effects.

IMPEDENCE IN VACUUM-

3.767303134 x 10^2 ohms

(The wave-impedence of a plane wave in free space. Electric field

strength divided by the magnetic field strength.)

SPEED OF LIGHT-

299,792,458 meters/second

SPEED OF LIGHT IN A VACUUM SQUARED-

89,875,517,873,681,764 meters^2/seconds^2

MAGNETIC FLUX QUANTUM-

2.06783383 x 10^-15 webers

(The measure of the strength of a magnetic field over a given area

taken perpendicular to the direction of the magnetic field.)

MOLAR IDEAL GAS VOLUME-

22.41410×10^-3 meters^3/moles

(As all gases that are behaving ideally have the same number density,

they will all have the same molar volume. It is useful if you want to

envision the distance between molecules in different samples.)

MUON REST MASS-

1.8835327×10^-28 kilograms

(A muon is an elementary particle similar to an electron, with an electric

charge of -1 and a spin of 1/2, but with a much greater mass.)

NEUTRON COMPTON WAVELENGTH-

1.31959110 x 10^-15 meters

(Explains the scattering of photons by electrons. The compton

wavelength of a particle is equal to the wavelength of a photon

whose energy is the same as the mass of the particle.)

NEUTRINO REST MASS-

3.036463233*10^-35 kilograms

NUCLEAR MAGNETON-

5.0507866 X 10^-27 Henry/meters

(A physical constant of magnetic moment. Using the mass of a proton,

rather than the electron, used to calculate the Bohr magneton. unit of

magnetic moment, used to measure proton spin and approximately

equal to 1.1,836 Bohr magneton.)

PERMEABILITY CONSTANT-

12.5663706144 x 10^-7 Henry/meters

(Magnetic constant, or the permeability of free space, is a measure of

the amount of resistance encountered when forming a magnetic field

in a classical vacuum.)

PERMITTIVITY CONSTANT-

8.854187817 x 10^-12 farad/meters

(A constant of proportionality that exists between electric displacement

and electric field intensity in a given medium.)

PLANCK’S CONSTANT-

6.6260755×10^-34 joules/hertz

6.62607004×10^-34 meters^2 x kilograms/seconds

(This constant links the about of energy a photon carries with the

frequency of its electromagnetic wave.)

PROTON COMPTON WAVELENGTH-

2.4263102367 x 10^-12 meters

(The compton wavelength is a quantum mechanical property of a

particle. A convenient unit of length that is characteristic of an elementary

particle, equal to Planck’s constant divided by the product of the particles

mass and the speed of light.)

PROTON MAGNETIC MOMENT-

1.41060761 x 10^-26 joules/tesla

(The dipole of the proton. Protons and neutrons, both nucleons,

comprise the nucleus of an atom, and both nucleons act as small

magnets whose strength is measured by their magnetic moments.)

PROTON MAGNETIC MOMENT IN BOHR MAGNETONS-

1.521032202 x 10^-3

(A physical constant and the natural unit for expressing the magnetic

moment of an electron caused by either its orbit or spin angular

momentum.

PROTON MASS/ELECTRON MASS-

1,836.152701

PROTON REST MASS-

1.6726231 x 10^-27 kilograms

RYDBERG CONSTANT-

1.0973731534 x 10^7/meters

(A physical constant relating to atomic spectra, in the science of

spectroscopy. Appears in the Balmer formula for spectral lines of the

hydrogen atom.)

RYDBERG ENERGY-

13.6056981 electron-volts

(It corresponds to the energy of the photon whose wavenumber is the

Rydberg constant, I.e. the ionization of the hydrogen atom. It describe

the wavelengths of spectral lines of many elements.)

STEFAN-BOLTZMANN CONSTANT-

5.67051 x 10^-8 weber/(meters^2 x kelvin^4)

(The power per unit area is directly proportional to the 4th power of the

thermodynamic temperature. It is the total intensity radiated over all

wavelengths as the temperature increases, of a black body which is

proportional to to 4th power of the thermodynamic temperature.)

Table of physical constants

**Universal constants **

**Value **

**Quantity Symbol **

299 792 458 m⋅s−1defined Newtonian constant of gravitation G

**Relative standard uncertainty **

speed of light in vacuum c

6.67408(31)×10−11 m3⋅kg−1⋅s−2. Planck constant h

6.626 070 040(81) × 10−34 J⋅s. reduced Planck constantħ=h/2π

1.054 571 800(13) × 10−34 J⋅s.

4.7 × 10−5 1.2 × 10−8

1.2 × 10−8

**Electromagnetic constants ****Quantity **

**Symbol **

**Value** **(****SI ****units)
Relative standard uncertainty
**magnetic constant (vacuum permeability)μ0

4π × 10−7 N⋅A−2 = 1.256 637 061… × 10−6 N⋅A−2defined electric constant (vacuum permittivity)ε0=1/μ0c2 8.854 187 817… × 10−12 F⋅m−1defined

characteristic impedance of vacuumZ0=μ0c

376.730 313 461… Ωdefined

Coulomb’s constant ke=1/4πε0

8.987 551 787 368 176 4 × 109 kg⋅m3⋅s−4⋅A−2defined elementary charge e

1.602 176 6208(98) × 10−19 C.

Bohr magneton μB=eħ/2me 9.274 009 994(57) × 10−24 J⋅T−1.

conductance quantum

7.748 091 7310(18) × 10−5 S. inverse conductance quantum

G0−1=h/2e2

12 906.403 7278(29) Ω 2.3 × 10−10

Josephson constant

KJ=2e/h

4.835 978 525(30) × 1014 Hz⋅V−1 6.1 × 10−9

magnetic flux quantum

φ0=h/2e

2.067 833 831(13) × 10−15 Wb 6.1 × 10−9

nuclear magneton

μN=eħ/2mp

5.050 783 699(31) × 10−27 J⋅T−1 6.2 × 10−9

von Klitzing constant

RK=h/e2

25 812.807 4555(59) Ω 2.3 × 10−10

6.1 × 10−9

6.2 × 10−9 2.3 × 10−10

**Atomic and nuclear constants ****Quantity **

**Symbol **

**Value**[23][24] **(****SI ****units)
Relative standard uncertainty **Bohr radius

a0=α/4πR∞

5.291 772 1067(12) × 10−11 m 2.3 × 10−9

classical electron radius

re=e2/4πε0mec2m_

2.817 940 3227(19) × 10−15 m 6.8 × 10−10

electron mass

me

9.109 383 56(11) × 10−31 kg 1.2 × 10−8

Fermi coupling constant

GF/(ħc)3

1.166 3787(6) × 10−5 GeV−2 5.1 × 10−7

fine-structure constant

α=μ0e2c/2h=e2/4πε0ħc

7.297 352 5664(17) × 10−3 2.3 × 10−10

Hartree energy

Eh=2R∞hc

4.359 744 650(54) × 10−18 J 1.2 × 10−8

proton mass

mp

1.672 621 898(21) × 10−27 kg 1.2 × 10−8

quantum of circulation

h/2me

3.636 947 5486(17) × 10−4 m2 s−1 4.5 × 10−10

Rydberg constant

R∞=α2mec/2h

10 973 731.568 508(65) m−1 5.9 × 10−12

Thomson cross section

(8π/3)re2

6.652 458 7158(91) × 10−29 m2 1.4 × 10−9

weak mixing angle

sin2θW=1−(mW/mZ)2 0.2223(21)

9.5 × 10−3

Efimov factor

22.7

**Physico-chemical constants ****Quantity **

**Symbol Relative standard uncertainty **

**Value**[23][24] **(****SI ****units) **Atomic mass constant

mu=1u

1.660539040(20)×10−27 kg 1.2×10−8

Avogadro constant

NA,L

6.022140857(74)×1023 mol−1 1.2×10−8

Boltzmann constant

k=kB=R/NA

1.38064852(79)×10−23 J⋅K−1 5.7×10−7

Faraday constant

F=NAe

96485.33289(59) C⋅mol−1 6.2×10−9

first radiation constant

c1=2πhc2

3.741 771 790(46) × 10−16 W⋅m2 1.2 × 10−8

for spectral radiance

c1L=c1/π

1.191 042 953(15) × 10−16 W⋅m2⋅sr−1 1.2 × 10−8

Loschmidt constant

atT = 273.15 K and p = 101.325 kPa

n0=NA/Vm

2.686 7811(15) × 1025 m−3 5.7 × 10−7

gas constant

R

8.3144598(48) J⋅mol−1⋅K−1 5.7×10−7

molar Planck constant

NAh

3.990 312 7110(18) × 10−10 J⋅s⋅mol−1 4.5 × 10−10

molar volume of an ideal gas

atT = 273.15 K and p = 100 kPa

Vm=RT/p

2.271 0947(13) × 10−2 m3⋅mol−1

5.7 × 10−7

at T= 273.15 K and p= 101.325 kPa2.241 3962(13) × 10−2 m3⋅mol−1 5.7 × 10−7

Sackur–Tetrode constant

at

T= 1 K and p= 100 kPa S0/R=52/R=+ln[(2πmukT/h2)3/2kT/p]

−1.151 7084(14)1.2 × 10−6at T= 1 K and p = 101.325 kPa −1.164 8714(14)1.2 × 10−6

second radiation constant

c2=hc/k

1.438 777 36(83) × 10−2 m⋅K 5.7 × 10−7

Stefan–Boltzmann constant

σ=π2k4/60ħ3c2

5.670367(13)×10−8 W⋅m−2⋅K−4 2.3×10−6

Wien displacement law constant

b energy=hck−1/=hck^ 4.965 114 231…

2.8977729(17)×10−3 m⋅K

5.7×10−7

Wien’s entropy displacement law constant[25]

b entropy=hck−1/=hck^ 4.791 267 357…

3.002 9152(05) × 10−3 m⋅K 5.7 × 10−7

**Adopted values ****Quantity **

**Symbol
Value (**

**SI**

**units)**

Relative standard uncertaintyconventional value of Josephson constant[26]

Relative standard uncertainty

KJ−90

4.835 979 × 1014 Hz⋅V−1

0 (defined)

conventional value of von Klitzing constant[27]

RK−90

25 812.807 Ω

0 (defined)

constant

Mu=M(12C)/12

1 × 10−3 kg⋅mol−1 0 (defined)

of carbon-12

M(12C)=NAm(12C)

1.2 × 10−2 kg⋅mol−1 0 (defined)

molar mass

standard acceleration of gravity (gee, free-fall on Earth) gn

9.806 65 m⋅s−20 (defined) standard atmosphere

atm

101 325 Pa 0 (defined)

ATOMIC MASS UNITS-

1.6605402 x 10^-27 kilograms

(1/12 0f the mass of an atom of carbon-12)

AVOGADRO NUMBER-

6.0221367 x 10^23/moles

(mole=number of elementary entities that are in carbon-12 atoms in exactly 12 grams of carbon-12)

BOHR’S MAGNETON-

9.2740154 x 10^-24 joules/tesla

(The magnetic moment of an electron caused by either its orbital

or spin angular momentum. Magnetic moment is a quantity that determines the torque it will experience in an external magnetic

field. Torque is rotational force. A joule is equal to the work done on an object when a force of 1 newton acts on the object in the direction of motion through a distance of 1 meter: kilograms x meters^2/ seconds^2. A joule is also equal to 10 million ergs. A Tesla is a

derived unit of the strength of a magnetic field: kilograms/(seconds^2 x amperes.)

BOHR RADIUS-

5.29177249 x 10^-11 meters

(The mean radius of an electron around the nucleus of a hydrogen atom at its ground state.)

BOLTZMANN CONSTANT-

1.3806513 x 10^-23 joules/kelvin

(A physical constant relating the average kinetic energy of particles in a gas with the temperature of the gas. Kelvin temperature scale is the primary unit of temperature measurement in the physical sciences, but is often used in conjunction with the celsius degree, which is of the same magnitude absolute zero in kelvin is equal to -273.15 degrees celsius.)

COSMOLOGICAL CONSTANT-

*R=1/2Rg=8 x pi x 6.67 x 10^-11 x T *

(T=energy-momentum tensor)

The constant is a homogeneous energy density that causes the expansion of the universe to accelerate.

ELECTRON CHARGE-

1.60217733 x 10^-19 coulombs

(Charge carried by a single electron. The coulomb is the quantity of charge that has passed through the cross section of an electrical conductor carrying one ampere within one second.)

ELECTRON CHARGE/MASS RATIO-

1.75881962 x 10^11 coulombs/kilograms

(The importance of the charge-to-mass ratio, according to classical electrodynamics, is that 2 particles with the same charge-to-mass ratio move in the same path in a vacuum when subjected to the same electric and magnetic fields.)

ELECTRON COMPTON WAVELENGTH- 2.42631058 X 10^-12 meters

(A compton wavelength of a particle is equal to the wavelength of a photon whose energy is the same as the mass of the particle. The compton wavelength of an electron is the characteristic length scale of quantum electrodynamics. It is the length scale at which relativistic quantum field theory becomes crucial for its accurate description.)

ELECTRON MAGNETIC MOMENT- 9.2847701 x 10^-24 joules/tesla

(The electron is a charged particle of -1e, where e is the unit of elementary charge. Its angular momentum comes from 2 types of rotation: spin and orbital motion.)

ELECTRON MAGNETIC MOMENT IN BOHR MAGNETONS- 1.00159652193

(Bohr magneton is a physical constant and natural unit for expressing the magnetic moment of an electron caused by either its orbital or spin angular momentum. The electron magnetic moment, which is the electron’s intrinsic spin magnetic moment, is approximately one Bohr magneton.)

ELECTRON MAGNETIC MOMENT/PROTON MAGNETIC MOMENT- 658.21068801

ELECTRON REST MASS-

9.1093897 x 10^-31 kilograms

ELECTRON REST MASS/PROTON REST MASS- 5.44617013 x1 0^-4

This is how much less mass the electron is as compared to the proton. (1,836.21 times lighter than proton)

FARADAY CONSTANT-

9.6458309 x 10^4 coulombs/mole

(The magnitude of electric charge per mole of electrons.)

FINE STRUCTURE CONSTANT- .00729735308

(The strength of the electromagnetic interaction between elementary particles.)

GAS CONSTANT-

8.3144710 x 10^joules/(mole x kelvin)

(A physical constant which is featured in many fundamental equations in the physical sciences, such as the ideal gas law and the Nernst equation.)

GRAVITATIONAL CONSTANT-

6.67206 x 10^-11 newtons x meters^3/(kilograms*second^2)

Denoted by letter G, it is an empirical physical constant involved in the calculation of gravitational effects.

IMPEDENCE IN VACUUM-

3.767303134 x 10^2 ohms

(The wave-impedence of a plane wave in free space. Electric field strength divided by the magnetic field strength.)

SPEED OF LIGHT-

299,792,458 meters/second

SPEED OF LIGHT IN A VACUUM SQUARED- 89,875,517,873,681,764 meters^2/seconds^2

MAGNETIC FLUX QUANTUM-

2.06783383 x 10^-15 webers

(The measure of the strength of a magnetic field over a given area taken perpendicular to the direction of the magnetic field.)

MOLAR IDEAL GAS VOLUME- 22.41410×10^-3 meters^3/moles

(As all gases that are behaving ideally have the same number density, they will all have the same molar volume. It is useful if you want to envision the distance between molecules in different samples.)

MOLAR MASS CONSTANT- 1×10^-3 kilograms/moles (relates relative atomic mass and molar mass)

MOLAR MASS OF CARBON-12— 1.2×10^-2 kilograms/moles (relates atomic mass of carbon-12 and molar mass)

MUON REST MASS-

1.8835327×10^-28 kilograms

(A muon is an elementary particle similar to an electron, with an electric charge of -1 and a spin of 1/2, but with a much greater mass.)

NEUTRON COMPTON WAVELENGTH- 1.31959110 x 10^-15 meters

(Explains the scattering of photons by electrons. The compton wavelength of a particle is equal to the wavelength of a photon whose energy is the same as the mass of the particle.)

NEUTRINO REST MASS- 3.036463233*10^-35 kilograms

NUCLEAR MAGNETON-

5.0507866 X 10^-27 Henry/meters

(A physical constant of magnetic moment. Using the mass of a proton, rather than the electron, used to calculate the Bohr magneton. unit of magnetic moment, used to measure proton spin and approximately equal to 1.1,836 Bohr magneton.)

PERMEABILITY CONSTANT-

12.5663706144 x 10^-7 Henry/meters

(Magnetic constant, or the permeability of free space, is a measure of the amount of resistance encountered when forming a magnetic field in a classical vacuum.)

PLANCK CHARGE- 1.875545956×10^-18 coulombs (a quantity of electric charge)

PERMITTIVITY CONSTANT-

8.854187817 x 10^-12 farad/meters

(A constant of proportionality that exists between electric displacement and electric field intensity in a given medium.)

PLANCK’S CONSTANT-

6.6260755×10^-34 joules/hertz

6.62607004×10^-34 meters^2 x kilograms/seconds

(This constant links the about of energy a photon carries with the frequency of its electromagnetic wave.)

PLANCK CONSTANT (REDUCED)- 6.582119514×10^-16 eV-seconds

(h-bar, in which h equals h divided by 2pi, is the quantization of angular momentum.)

PLANCK’S LENGTH- 1.616229X10&-35 meters

(the scale at which classical ideas about gravity and space-time cease to be valid, and quantum effects dominate.)

PLANCK MASS- 2.17647X10^-8 kilograms

(derived approximately by setting it as the mass whose compton wavelength and schwarzschild radius are equal.)

PLANCK TIME- 5.3916×10^-44 seconds

(time needed for light to travel 1 planck length in a vacuum.)

PLANCK TEMPERATURE 1.416808×10^32 degrees kelvin

(if an object were to reach this temperature, the radiation it would emit would have a wavelength of 1.616×10^-35 meters, Planck’s length, at which point quantum gravitational effects become relevant.)

PROTON COMPTON WAVELENGTH- 2.4263102367 x 10^-12 meters

(The compton wavelength is a quantum mechanical property of a particle. A convenient unit of length that is characteristic of an elementary particle, equal to Planck’s constant divided by the product of the particles mass and the speed of light.)

PROTON MAGNETIC MOMENT-

1.41060761 x 10^-26 joules/tesla

(The dipole of the proton. Protons and neutrons, both nucleons, comprise the nucleus of an atom, and both nucleons act as small magnets whose strength is measured by their magnetic moments.)

PROTON MAGNETIC MOMENT IN BOHR MAGNETONS- 1.521032202 x 10^-3

(A physical constant and the natural unit for expressing the magnetic moment of an electron caused by either its orbit or spin angular momentum.

RYDBERG CONSTANT-

1.0973731534 x 10^7/meters

(A physical constant relating to atomic spectra, in the science of spectroscopy. Appears in the Balmer formula for spectral lines of the hydrogen atom.)

RYDBERG ENERGY-

13.6056981 electron-volts

(It corresponds to the energy of the photon whose wavenumber is the Rydberg constant, I.e. the ionization of the hydrogen atom. It describe the wavelengths of spectral lines of many elements.)

STANDARD ACCELERATION ON EARTH BY GRAVITY- 9.80665 meters/seconds^2

STANDARD ATMOSPHERE- 101.325 pascals

(pressure, temperature, density, and viscosity of the earth’s atmosphere.)

STEFAN-BOLTZMANN CONSTANT-

5.67051 x 10^-8 weber/(meters^2 x kelvin^4)

(The power per unit area is directly proportional to the 4th power of the thermodynamic temperature. It is the total intensity radiated over all wavelengths as the temperature increases, of a black body which is proportional to to 4th power of the thermodynamic temperature. This constant is used to link a star’s temperature to the amount of light it emits.)

MAGNETIC CONSTANT (VACUUM PERMIABILITY)- 1.256637061X10^-6 NEWTONS/AMPERES^2

ELECTRIC CONSTANT (VACUUM PERMITTIVITY)- 8.854187817X10^-12 FARAD/METER

CHARACTERISTIC IMPEDANCE OF VACUUM- 376.730313461 OHMS

COULOMB’S CONSTANT-

8.9875517873881764X10^8 KILOGRAMS METERS^3/

SECONDS^4XAMPERES^2

ELEMENTARY CHARGE-

1.6021766208X10^-19 COULOMBS

CONDUCTIVE QUANTUM-

7.748091731X10^-8 SECONDS

INVERSE CONDUCTIVE QUANTUUM- 12.9064037278 OHMS

JOSEPHSON CONSTANT-

4.835978525X10^14 HERTZ/VOLTS

MAGNETIC FLUX QUANTUM- 2.067831X10^-15 WEBERS

NUCLEAR MAGNETON-

5.050783699X10^-27 JOULES/TESLAS

VON KILTZING CONSTANT- 258.12807557 ohms

CLASSICAL ELECTRON RADIUS- 2.8179403227X10^-15 METERS

ELECTRON MASS-

9.10938356×10^-31 kilograms

FERMI COUPLING CONSTANT- 1.1663787X10^-5 GeV^-2

FINE-STRUCTURE CONSTANT- 7.2972525664X10^-3

HARTREE ENERGY-

4.35974465X10^-18 JOULES

PROTON MASS-

1.6726219×10^-27 kilograms

QUANTUM OF CIRCULATION-

3.6369475486X10^-4 METERS^2/SECONDS

THOMSON CROSS SECTION- 6.6524587158X10^-29 METERS^2

WEAK MIXING ANGLE- .2223

EFIMOV FACTOR- 22.7

FIRST RADIATION CONSTANT-

3.74177179X10^-16 WEBER-METERS^2

FIRST RADIATION CONSTAnt (for spectral radiance)- 1.191042953×10^-16 webers-meters^2-seconds/radius

Loschmidt constant-

2.6867811×10^25 /meters^3

Molar planck constant-

3.990312711×10^-10 joules-seconds/moles

Molar volume of an ideal gas (at t=273.15 k and p=100 kpa)- 2.2710947×10^-2meters^3/moles

Molar volume of an ideal gas (at t=273.15 k and p=101.325 kpa)- 2.2413962×10^-2 meters^3/moles

Sacker-tetrode constant (at t=273.15 k and p=100 kpa)- -1.1517084

Sacker-tetrode constant (at t=273.15 k and p=101.325 kpa)- -1.1648714

Second radiation constant-

1.438777×10^-2 meters kelvin

Wien displacement law constant- 2.8977729×10^-3 meters kelvin

Wien entropy displacement constant- 3.0029152×10^-3 meters kelvin

Conventional value of Josephson constant- 4.835979×10^14 hertz/volts

Conventional value of von Klitzing constant- 25812.807 ohms

2 PARAMETERS OF THE HIGGS FIELD POTENTIAL- V(H)=lambda*(R^2-v^2)=lambda*H^4-2*v^2*H^2*lambda*v^4

(H is the higgs field)

the higgs field is an energy field that is thought to exist everywhere in the universe. the field is accompanied by a fundamental particle called the higgs boson, which the field uses to continuously interact with other particles. the process of giving a particle mass is known as the higgs effect.