eyes on - mvputten.org · boris v balakin, alex c hoffmann, pawel kosinski & lee d rhyne (2010)...
TRANSCRIPT
http://www.bbc.com/earth/story/20160120-the-most-beautiful-equation-is-eulers-identity2
Outline
I. Natural number e = 2.7183… (Euler)
II. Complex numbers and Euler’s identity
III. Ubiquity of e and pi
IV. Conclusions
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Boris V Balakin, Alex C Hoffmann, Pawel Kosinski & Lee D Rhyne (2010)
Terminal velocity sphere in oil and water
Terminal velocitysmooth transition in time to a constant velocity
Engineering Applications of Computational Fluid Mechanics Vol. 4, No. 1, pp. 116–126 (2010)
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V (t) =V0 1− e− t /τ( )
e = 2.718281828...
τ : time scale, set by viscosity. Relatively small for sphere dropped in oil; larger for sphere dropped in water
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Leonhard Euler (1727)
“Meditatio in Experimenta explosione tormentorum nuper instituta”
(Meditation upon experiments made recently on the firing of Canon) Swiss, 1707-1783, student of Johann
Bernouilli
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Compound interest problem in a savings account
Initial deposit
100% interest in a one time payout
Jacob Bernoulli (1654-1705)
Final balance
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1000
2031
1000 2016
Account Balance over time
once twice yearly
2046 2000
1000
15002250
2017 10332018 1067
2674
1065.33
Payout scheme:
10
1000
2031
1000 2016
Account Balance over time
once twice yearly
2046 2000
1000
15002250
2017 10332018 1067
2674
1065.33
Payout scheme:
2718
1000
1033.89
1068.94
1648.68
daily
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1+ gain = final deposit
initial deposit=
27181000
= 2.718daily, n=10950 in 30 yr
1+ gain = 1+ 1
n⎛⎝⎜
⎞⎠⎟
n
= 1+ 1n
⎛⎝⎜
⎞⎠⎟
1+ 1n
⎛⎝⎜
⎞⎠⎟K 1+ 1
n⎛⎝⎜
⎞⎠⎟→ e = 2.718281828...
large n limit
Payout over n periods (“installments”)
(n times)
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0
700
1400
2100
2800
2016 2026 2036 2046
Account balance by daily compound interest
2.71828... x 1000
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ex = 1+ x + 1
2x2 +
16x3 +L = 1+ x
1!+x2
2!+x3
3!+L
1!= 12!= 23!= 64!= 24n!= 1*2 * 3*L (n −1)*n
n! grows much faster than xn, so that the series converges for all x
n!≅ 2πn nne−n (Stirling formula)
ex = limn→∞
1+ xn
⎛⎝⎜
⎞⎠⎟
n
1+ 11!+12!+13!+14!+15!= 2.7167 e1 = 2.71828( )
1+ 21!+42!+83!+164!+325!
= 7.2667 e2 = 7.3891( )Example
Define
Can show
Definition of exponential function
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ddx
f (x) = f (x)
f (0) = 1
⎧
⎨⎪
⎩⎪
Unique solution: f (x) = ex
Alternate definition: slope(x)=exp(x)
x-axis
y-axis
tangent at x
ΔxΔy
ex
x
ex =ΔyΔx
graph
Exercise: show key property fn(x)=f(nx)
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Origin
positive real numbersNegative real numbers
“left” “right”
“up?”
“down?”
Leonhard Euler’s complex numbers (student of Johann Bernouilli)
(1707-1783)
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Square of a complex number
Reα
Im
2α
1-1
w = z2 described by
w = z 2
β = 2α
← z = 2, α= 13πw = z2 has length 4 and angle 2
3π
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Pythagoras on the unit circle
Im
Re axisα
x = cosα
y = sinα
z = x + iy = cosα + isinαx2 + y2 = 1cos2α + sin2α ≡ 1
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Powers on the unit circle S1
z = 1→ w = zn
w = 1, β = nα
cosα + isinα( )n = cosnα + isinnα
Im
Re axisα De Moivre
nα
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ex = 1+ x + 1
2x2 + 1
6x3 +L
Powers of the exponential function
ex( )n = enx
eix( )n = enix
eix = 1+ ix + 12(ix)2 + 1
6(ix)3 +L = 1+ ix − 1
2x2 − 1
6ix3 +L
eix = 1− 12x2 +L⎛
⎝⎜⎞⎠⎟+ i x − 1
6x3 +L⎛
⎝⎜⎞⎠⎟
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cosα + isinα( )n = cosnα + isinnα
eix( )n = enix
x =α :eiα = cosα + isinα
eiα( )n = einα = cosnα + isinnα
Powers on S1
Powers of eix
eiα = cosα + isinα :cosα =
eiα + e− iα
2= Reeiα
sinα =eiα − e− iα
2i= Im eiα
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Polar representation of complex numbers
z = r cosα + isinα( ) = reiα
zn = rneinα
cosα + isinα = cos α + 2πn( ) + isin α + 2πn( ) n∈¢( )
1= ei2πn n∈¢( )log z = log r + iα + i2πn n∈¢( )
eiα = ei α+2πn( ) n∈¢( )
N
N
N
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What have we learned
e and pi are associated by Euler’s identity
πππeee
ei0 = 1, e12π i= i, eiπ = −1, e
32π i= −i, e2π i = 1, etc
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e=2.71828..., pi=3.14159
eiπ +1= 0
Euler’s identity
Example:
“The most remarkable formula in mathematics” (Richard Feynman)
πππeee
Re
Im
α = π
i
1-1
1+ iπ
1!−π 2
2!−iπ 3
3!+L +
iπ 9
9!= −0.9760...+ 0.0069...i
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What have we learned
Complex numbers:
a natural extension of the real numbers
exponential map w=ez in the complex plane
πππeee
Im(z)
Re(z)
w is oscillatory
w grows exponentially
w decays exponentially
w is oscillatory
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Fourier series
=
+
+
+
low frequency,
large amplitude
high frequency,
small amplitude
f(x):
C1eiω0x
C2e2iω0x
C0
C3e3iω0x
(1768-1830)
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Fourier analysis of heat flow
u(t, x) :ut = uxx
u(0, x) = u0 (x)
⎧⎨⎪
⎩⎪
u = a(t)eikx :da(t)dt
= −k2a(t)
a(t) = a(0)e−k2t
https://commons.wikimedia.org/wiki/File:Heatequation_exampleB.gif
Higher modes quickly die out, leaving only k=0 at large times
exponential decay in time for k>0
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Q2. Heat flow
Consider a pair of metal bars of the same size and material, initially at T1=100 and T2=200. Upon bringing them into contact, what is their equilibrium temperature at late times?
[Hint: Heat is thermal energy, measured by temperature. Upon contact, heat flow sets in preserving total energy. The total energy at late time is the same as the total initial energy.]
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Lockyer, J.N., 1878, “Water-waves and Sound-waves”, Pop. Sc. Monthly, Vol. XIII (Appleton and Co., New York) p.166
u = Acosωt
Time harmonic motion
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u = Acos(ωt − kx) :
utt − c2uxx = (−ω
2 + c2k2 )u = 0
Dispersion relation with velocity c
ω
k
Linear dispersion relation (dispersionless medium)
ω = ck
Wave equation
Wave propagation
time, space
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Fourier analysis of wave motion
u(t, x) :utt = uxx u(0, x) = u0 (x)ut (0, x) = u1(x)
⎧
⎨⎪
⎩⎪
u = a(t)eikx : d 2a(t)dt 2
= −k2a(t), a(t) = a(0)e± ikt (u1 = 0)
Each Fourier mode propagates without decay: persistent wave motion
(shape may change, total energy and momentum are conserved)
Jean-Baptiste le Rond d’Alembert (1717-1983)
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Q3. Sound waves
The linear dispersion relation for sound waves shows that angular frequency and wave number have a constant ratio.
In a duet, A sings at A4 (440 Hz), B sings at E5 (660 Hz). What are the ratios the wave lengths of their sounds?
Hendrik ter Brugghen
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de Broglie matter wave
Erwin Schrodinger (1887-1991)
Louis de Broglie (1892-1987)
E = hω = ih ∂
∂tϕ
ψ (t, x) = a(x)e− iωtWave function
Probability ψ (x) 2 = a2 (x)
is eigenvalue in ih ∂∂tψ = Eψ
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Schrodinger wave function
Erwin Schrodinger (1887-1991)
ih ∂∂tψ = Hψ
H = Ek =
p2
2m→ H =
p2
2m, p = ih∂x
−ihΨ t =
h22m
Ψ xx
p = mv : Ek =12mv2 = p2
2m
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Exponential function in relativity
light conet = x(c = 1)
x
tMinkowski spacetime
1864-1909 1879-1955
world-line
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Hyperbolic functions
eiα = cosα + isinα
cosα =eiα + e− iα
2= Reeiα
sinα =eiα − e− iα
2i= Im eiα
cos2α + sin2α = 1
eλ = coshλ + sinhλ
coshλ = eλ + e−λ
2= cos iλ
sinhλ = eλ − e−λ
2= −isin iλ
cosh2 λ − sinh2 λ = 1
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Hyperbolic representation of a world-line
xa = (t, x), ua = dtds, dxds
⎛⎝⎜
⎞⎠⎟, ucuc = u
tut − uxux = 1
ut = coshλ, ux = sinλ
V =dxdt
=dx / dsdt / ds
=ut
ux= tanhλ
1
-1
lambda
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Rest mass energy
E = mut = mcoshλ = m 1+ sinh2 λ
pa = mua
coshλ = 1+ sinh2 λ ≅ 1+ 12sinh2 λ ≅ 1+ 1
2tanh2 λ λ <<1( )
m + 12mV 2 = E0 + Ek
E0 = mc2Restore velocity of light c:
relativistic four-momentum
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E = E02 + P2 ≅
P = mux = m dxdτ
Rest mass energy
P
E0 = mc2
relativistic x-momentum
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dispersion relation of
light
dispersion relation of m (books, your laptop, …)
E
Q4. Powering a laptop
Consider an new battery converting up to 1 g of mass into electrical power.
A laptop uses up to 100 W (Joules per second). For how many years can this battery power our laptop?
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