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Partial Differential Equations Oliver Knill, Harvard University October 7, 2019
Partial Differential Equations
Oliver Knill,Harvard University
October 7, 2019
In Pursuit of the Unknown17 Equations That
Changed the World
Ian Stewart
“Stewart has a genius for explanation. . . . Mathematics doesn’t come more entertaining than this.”
—NEW SCIENTIST
“Combines a deep understanding of math with an engaging literary style.”
—THE WASHINGTON P OST
“Possibly mathematics’ most energetic evangelist.”
—THE SPECTATOR (LONDON)
“Stewart is able to write about mathematics for general readers. He can make tricky ideas
simple, and he can explain the maths of it with aplomb. . . . Stewart admirably captures
compelling and accessible mathematical ideas along with the pleasure of
thinking about them. He writes with clarity and precision.”
—LOS ANGELES TIMES
“A highly gifted communicator, able not only to explain the motivation of mathematicians
down the centuries but to elucidate the resulting mathematics with both clarity
and style. The whole is leavened by his inimitable understated wit.”
—THE TIMES EDUCATION SUPPLEMENT
$26.99 US / $30.00 CAN
ISBN 978-0-465-02973-0
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M ost people are familiar with history’s
great equations: Pythagoras’s theorem,
for instance, or Newton’s Law of Gravity,
or Einstein’s theory of relativity. But the way these
mathematical breakthroughs have contributed to human
progress is seldom appreciated. In his new work, In
Pursuit of the Unknown, celebrated mathematician Ian
Stewart untangles the roots of our most important
mathematical statements to show that equations have long
been a driving force behind nearly every aspect of our lives.
Using seventeen of our most crucial equations,
Stewart illustrates that many of the advances we now
take for granted—in science, philosophy, technology, and
beyond—were made possible by mathematical discoveries.
For example, the Wave Equation allowed engineers to
measure a building’s response to earthquakes, saving
countless lives; without the Wave Equation, moreover,
scientists would never have discovered electromagnetic
waves, which in turn led to the invention of radio and
television. The equation at the heart of information
theory, devised by Claude Shannon, forms the basis for
modern digital communication systems, which have
revolutionized everything from politics to business to
interpersonal relationships. And the Black-Scholes model,
used by bankers to track the prices of financial derivatives
over time, led to massive growth in the financial sector,
thereby contributing to the banking crisis of 2008—the
effects of which we are still feeling today.
(continued on back flap)
$26 .99 US / $30.00 CAN(continued from front flap) MATHEMATICS
Praise for Ian Stewart≥ Δ Φ ∞ Φ Δ Φ ∞ Φ Δ ≤ ≥ Δ Φ ∞ Φ Δ Φ ∞ Φ Δ ≤
6.25” x 9.5”S: 1-1/8”B: 7/8”
BASICHC
4/COLOR
FINISH:MATTE POLYSPOT GLOSS ON EQUATIONS ON FRONT COVER
Ian Stewart is Emeritus Professor of
Mathematics, active researcher at Warwick University in
England, and author of many books on mathematics. His
writing has also appeared in publications including New
Scientist, Discover, and Scientific American. He lives in
Coventry, England.
Jacket design by Jennifer Carrow03/12
An approachable, lively, and informative guide to the
mathematical building blocks that form the foundations of
modern life, In Pursuit of the Unknown is also a penetrating
exploration of how we have long used equations to make
sense of, and in turn influence, our world.
© A
vril
Stew
art
A Member of the Perseus Books Groupwww.basicbooks.com
In Pursuit of the Unknown17 Equations That
Changed the World
Ian StewartStewart
In Pursuit of the Unknown
17 Equations That Changed the W
orld
Ian Stewart
Transport Equation
f = f t x
Signal processing
Advection
Traveling waves
f = f t x
Heat Equation
f = f t xx
f = f t xx
Heat propagation
Diffusion
Smoothing
Evolution of Good
Nowak Automaton
Wave equation
f = f tt xx
Light
Sound
Particles
I i Good vibrationsWave Equation
displacements e c o n d p a r t i a l / \ ^ s e c o n d p a r t i a l
d e r i v a t i v e \ / > T d e r i v a t i v e
w i t h r e s p e c t / - * \ V ^ ^ - ^ w i t h r e s p e c tt o t i m e ' * ' t o s p a c e
speed squared
What does it say?The acceleration of a small segment of a violin string isproportional to the average displacement of neighbouringsegments.
Why is that important?It predicts that the string will move in waves, and it generalisesnaturally to other physical systems in which waves occur.
What did it lead to?Big advances in our understanding of water waves, soundwaves, light waves, elastic vibrations... Seismologists usemodified versions of it to deduce the structure of the interiorof the Earth from how it vibrates. Oil companies use similarmethods to find oil. In Chapter 11 we will see how it predictedthe existence of electromagnetic waves, leading to radio,television, radar, and modern communications.
Wave Tango
Laplace Equation
f + f = 0 xx yy
Chladni Patterns
For networks
For networks
Burgers Equation
f + f f = 0 t x
Schocks
Schrödinger equation
i f = f t xx
+ V(x) f
Quantum weirdnessSchrodinger's Equation
square root ofminus one
constantdividedby2rc
rate ofchange quantum
wave function
Hamilton ianoperator
withrespectto time
What does it say?The equation models matter not as a particle, but as a wave,and describes how such a wave propagates.
Why is that important?Schrodinger's equation is fundamental to quantummechanics, which together with general relativity constitutetoday's most effective theories of the physical universe.
What did it lead to?A radical revision of the physics of the world at very smallscales, in which every object has a 'wave function' thatdescribes a probability cloud of possible states. At this level theworld is inherently uncertain. Attempts to relate themicroscopic quantum world to our macroscopic classicalworld led to philosophical issues that still reverberate. Butexperimentally, quantum theory works beautifully, andtoday's computer chips and lasers wouldn't work without it.
Navier Stokes
f + f f t x
+ F(u,p)
= f xx
The ascent of humanityNavier-Stokes Equation
density velocity pressure s t ress body fo rces
^ ^ ^ i ^ ^ i i it i m e d e r i v a t i v e g r a d i e n t
dot productdivergence
What does it say?It's Newton's second law of motion in disguise. The left-handside is the acceleration of a small region of fluid. The right-hand side is the forces that act on it: pressure, stress, andinternal body forces.
Why is that important?It provides a really accurate way to calculate how fluids move.This is a key feature of innumerable scientific andtechnological problems.
What did it lead to?Modern passenger jets, fast and quiet submarines, Formula 1racing cars that stay on the track at high speeds, and medicaladvances on blood flow in veins and arteries. Computermethods for solving the equations, known as computationalfluid dynamics (CFD), are widely used by engineers to improvetechnology in such areas.
1 Mio Dollars
Gifted
Sofia Kowalevsky
Eiconal Equation
f x + f y2 2 = 1
Maxwell equations
div(B) = 0 div(E) = 4 πρ curl(E) = -B'/c
curl(B) = E'/c+4 πj/c
HI
Waves in the etherMaxwell's Equations
V-E = 0 VxE=electricfield
magneticfield
i c ^ pcurl speed of
light
VxH =
magneticfield
;rate of changewith respect to time
electricfield
What do they say?
Electricity and magnetism can't just leak away. A spinningregion of electric field creates a magnetic field at right angles tothe spin. A spinning region of magnetic field creates an electricfield at right angles to the spin, but in the opposite direction.
Why is that important?It was the first major unification of physical forces, showingthat electricity and magnetism are intimately interrelated.
What did it lead to?The prediction that electromagnetic waves exist, travelling atthe speed of light, so light itself is such a wave. This motivatedthe invention of radio, radar, television, wireless connectionsfor computer equipment, and most modern communications.
f + f f t x
= f-x fxx2
Black Scholes
The Midas formulaBlack-Scholes Equation
rate of changeof rate of change
volatility
iiliSlifciBliil
price of commodity
price of financial derivative—^ ra te o f change
risk-free interest rate
What does it say?It describes how the price of a financial derivative changes overtime, based on the principle that when the price is correct, thederivative carries no risk and no one can make a profit byselling it at a different price.
Why is that important?It makes it possible to trade a derivative before it matures byassigning an agreed 'rational' value to it, so that it can becomea virtual commodity in its own right.
What did it lead to?Massive growth of the financial sector, ever more complexfinancial instruments, surges in economic prosperitypunctuated by crashes, the turbulent stock markets of the1990s, the 2008-9 financial crisis, and the ongoing economicslump.
The End
O. Knill, October 7, 2019
