The art of approximation in science and engineering

250 Pages·2010·1.58 MB·English

by  Sanjoy Mahajan

1 1
π pi 3
G Newton’sconstant 7·10−11 kg−1m3s−1
c speedoflight 3·108 ms−1
kB Boltzmann’sconstant 10−4 eVK−1
e electroncharge 1.6·10−19 C
σ Stefan–Boltzmannconstant 6·10−8 Wm−2K−4
msun Solarmass 2·1030 kg
R Earthradius 6·106 m
earth
θmoon/sun angulardiameter 10−2
ρair airdensity 1 kgm−3
ρ rockdensity 5 gcm−3
rock
(cid:126)c 200 eVnm
Lwater heatofvaporization 2 MJkg−1
vap
γwater surfacetensionofwater 10−1 Nm−1
a Bohrradius 0.5 Å
0
a typicalinteratomicspacing 3 Å
NA Avogadro’snumber 6·1023
E combustionenergydensity 9 kcalg−1
fat
E typicalbondenergy 4 eV
bond
e2/4π(cid:15)0 fine-structureconstantα 10−2
(cid:126)c
p airpressure 105 Pa
0
νair kinematicviscosityofair 1.5·10−5 m2s−1
νwater kinematicviscosityofwater 10−6 m2s−1
day 105 s
year π·107 s
F solarconstant 1.3 kWm−2
AU distancetosun 1.5·1011 m
P humanbasalmetabolicrate 100 W
basal
Kair thermalconductivityofair 2·10−2 Wm−1K−1
K ... ofnon-metallicsolids/liquids 1 Wm−1K−1
K ... ofmetals 102 Wm−1K−1
metal
cair specificheatofair 1 Jg−1K−1
p
cp ... ofsolids/liquids 25 Jmole−1K−1
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Contents
Preface v
Part 1 Organizing complexity 1
1 Divide and conquer 3
2 Abstraction 27
Part 2 Lossless compression 45
3 Symmetry and conservation 47
4 Proportional reasoning 67
5 Dimensions 85
Part 3 Lossy compression 113
6 Easy cases 115
7 Lumping 133
8 Probabilistic reasoning 151
9 Springs 175
Part Backmatter 231
Long-lasting learning 233
Bibliography 237
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4 4
iv
Index 241
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Preface
An approximate analysis is often more useful than an exact solution!
This counterintuitive thesis, the reason for this book, suggests two ques-
tions.
One question is: If science and engineering are about accuracy, how can
approximate models be useful? They are useful because our minds are a
small part of the world itself. When we represent a piece of the world in
our minds, we discard many aspects – we make a model – in order that
the model fit in our limited minds. An approximate model is all that we
can understand. Making useful models means discarding less important
information so that our minds may grasp the important features that
remain.
This perhaps disappointing conclusion leads to a second question: Since
every model is approximate, how do we choose useful approximations?
The American psychologist William James said [10, p. 390]: ‘The art of
being wise is the art of knowing what to overlook.’ This book therefore
developsintelligenceamplifiers: toolsfordiscardingunimportantaspects
of a problem and for selecting the important aspects.
These reasoning tools are of three types:
1. Organizing complexity
− Divide and conquer
− Abstraction
2. Lossless compression
− Symmetry and conservation
− Proportional reasoning
− Dimensions
3. Lossy compression
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5 5
6 6
vi Preface
− Easy cases
− Probabilistic reasoning
− Lumping
− Spring models
The first type of tool helps manage complexity. The second type helps
remove complexity that is merely apparent. The third type helps discard
complexity.
With these tools we explore the natural and manmade worlds, using ex-
amplesfromdiversefieldssuchasquantummechanics,generalrelativity,
mechanical engineering, biophysics, recreational mathematics, and cli-
mate change. This diversity has two purposes. First, the diversity shows
how a small toolbox can explain important features of the manmade and
engineered worlds. The diversity provides a library of models for your
own analyses.
Second, the diversity separates the tool from the
details of its use. A tool is difficult to appreciate Technique Example
abstractly, without an example. However, if you
see only one use of a tool, the tool is difficult to
distinguish from the example. An expert, familiar with the tool, knows
where the idea ends and the details begin. But when you first learn a
tool, you need to learn the boundary.
Ananswerisasecondexample. Totheextentthat
the second example is similar to the first, the tool
plus first use overlaps the tool plus second use. Penumbra
T
Theoverlapincludesapenumbraaroundthetool.
The penumbra is smaller than it is with only one
example: Two uses delimit the boundaries of the
tool more clearly than one example does.
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Preface vii
More clarity comes using an example from a dis-
tantfield. Thepenumbrashrinks,whichseparates
the tool from examples of its use. For example,
using dimensional analysis in a physics problem
and an economics analysis clarifies what part of
the illustration is specific to physics or economics
T
and what part is transferable to other problems.
Focus on the transferable ideas; they are useful in
any career!
This book is designed for self study. Therefore, please try the problems.
The problems are of two types. The first type are problems marked with
a wedge in the margin. They are breathers during an analysis: a place to
developyourunderstandingbyworkingoutthenextstepsinananalysis.
Thoseproblemsareansweredinthesubsequenttextwhereyoucancheck
your thinking and my analysis – please let me know of any errors! The
second type of problem, the numbered problems, give practice with the
tools, extend a derivation, or develop a useful or enjoyable model. Most
numbered problems have answers at the end of the book.
Ihopethatyoufindthetools,problems,andmodelsusefulinyourcareer.
And I hope that the diversity of examples connects with and aids your
curiosity about how the world is put together.
Bon voyage!
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1
Part 1
Organizing
complexity
1 Divide and conquer 3
2 Abstraction 27
The first solution to the messiness and complexity of the world, just as
with the mess on our desks and in our living spaces, is to organize the
complexity. Two techniques for organizing complexity are the subject of
Part 1.
The first technique is divide-and-conquer reasoning: dividing a large
problem into manageable subproblems. The second technique is abstrac-
tion: choosing compact representations that hide unimportant details in
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2
order to reveal important features. The next two chapters illustrate these
techniques with many examples.
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