
The Art of Approximation in Science and Engineering [pdf] - drpossum
http://web.mit.edu/6.055/book/book-draft.pdf
======
532nm
Note: The draft linked in the title [1] has been superseded by the finished
book [2] (free pdf download here [3]), which is even better!

This is probably the most useful book I have read during my studies in physics
and I highly recommend it to everybody with some technical interest because it
focuses on teaching general and useful tools (using science and engineering
examples (mainly physics)).

During the first years of my studies I did reasonably well, but it always felt
like I was just manipulating symbols on paper, and the results didn't mean
much to me -- it was all just theory (and if the results would have come out
some other way I probably would have believed that too). This book taught me
how to be reckless and throw away unnecessary complexity in order to make said
theory simple enough to apply it easily to real-world problems, which suddenly
made my theory knowledge much, much more useful (previously it was mainly good
for passing exams). Now, during my PhD I still use the methods from the book
almost daily!

Compared to the earlier book draft [1] that has been floating on the web for
many years, the final book is much improved, mainly because it focuses on
general tools rather than specific physics topics. However, if you want to
have a quick demonstration of how powerful the methods from the book are, I'd
recommend to read chapter 9.3 about waves in [1], which unfortunately only
partially made it into the final book. Within a few pages you quickly derive
all the properties of waves in different regimes and draw practical
conclusions such as the speed limit for boats, the speed of tsunamis, why bugs
walking on water don't generate waves, etc!

[1] [http://web.mit.edu/6.055/book/book-
draft.pdf](http://web.mit.edu/6.055/book/book-draft.pdf)

[2] [https://mitpress.mit.edu/books/art-insight-science-and-
engin...](https://mitpress.mit.edu/books/art-insight-science-and-engineering)

[3]
[https://www.dropbox.com/s/bmqwzc8qqt5lv9p/9017.pdf?dl=1](https://www.dropbox.com/s/bmqwzc8qqt5lv9p/9017.pdf?dl=1)

~~~
denzil_correa
The author of the book Sanjoy Mahajan also has a very entertaining book and
online course titled "Street Fighting Math" [0]. Previous discussions on HN
about the book [1, 2, 3].

[0] [http://streetfightingmath.com](http://streetfightingmath.com)

[1]
[https://news.ycombinator.com/item?id=14343058](https://news.ycombinator.com/item?id=14343058)

[2]
[https://news.ycombinator.com/item?id=11523197](https://news.ycombinator.com/item?id=11523197)

[3]
[https://news.ycombinator.com/item?id=1548013](https://news.ycombinator.com/item?id=1548013)

------
mlthoughts2018
One of my favorite quotes, which I heard from an older grad student long ago
but I don’t know the real origins of it:

“There are exactly 3 things an applied mathematician can do to solve a
problem. The first is to Taylor expand something. I don’t know what the other
two are.”

~~~
CalChris
What I learned in _Linear Systems_ EE120 at Berkeley was that there are two
kinds of problems: linear problems and problems you can’t solve.

~~~
btrettel
I don't like that attitude. There are many non-linear problems which can
easily be solved accurately numerically or exactly.

So few people even try to solve problems exactly now that it seems like magic
when someone does it. I can recall once instance where a colleague seemed
awestruck that I solved Bernoulli's differential equation exactly when I
noticed that a problem we were working on could be expressed that way:
[https://en.wikipedia.org/wiki/Bernoulli_differential_equatio...](https://en.wikipedia.org/wiki/Bernoulli_differential_equation)

If you do take this approach, I'd advise against explaining how you solved the
problem. My colleague was not impressed that I looked up how to solve the
problem, and thought what I did was "only" a trick. But there are many tricks
for solving differential equations. Knowing which trick to pull out is part of
my job as far as I'm concerned.

On that note, I'd recommend this website and the related books for finding
exact solutions or useful changes of variables:
[http://eqworld.ipmnet.ru/](http://eqworld.ipmnet.ru/)

~~~
CalChris
From your cite:

    
    
      Transformation to a linear differential equation
    

That was the lesson of the class, transform what seems to be a non-linear
problem into a solvable linear problem.

~~~
btrettel
That's a good strategy, but you should have other tools for non-linear
problems in your toolbox too.

I mostly want to push back against the idea that non-linear means unsolvable.
My example transformed the problem to a linear one, but that's not the only
possibility. Look at my other link for many other examples. Autonomous
ordinary differential equations are one possible class of ODEs which don't
need to be transformed to a linear system to be solved:
[https://en.wikipedia.org/wiki/Autonomous_differential_equati...](https://en.wikipedia.org/wiki/Autonomous_differential_equation)

You can often solve a non-linear autonomous ODE by direct integration.

------
honk
I've been fortunate enough to take a class with Sanjoy (The author) covering
this book. It was an incredible experience, and he's one of the best mentors
I've ever met.

One of the things I found most interesting from the class that you couldn't
get from the book was a particular method of assessment. He would pose a
question, and allow you to give a weighted answer across a few options, to
better understand your uncertainty. It was a very useful pedagogical
technique, and allowed me to quantify my understanding in a very tangible way.

I highly recommend the whole book, but especially the first section, on
breaking things down.

------
btrettel
Dimensional analysis, one of the topics in the book, is a favorite of mine.
Unfortunately the more useful parts of the subject are not that well known.
Fortunately, this book seems to have a fairly standard introduction to the
subject.

Most people seem to think dimensional analysis is limited to checking that the
dimensions are consistent ("dimensional homogeneity"). But dimensional
homogeneity is a constraint which can simplify problems, sometimes even
allowing them to be solved up to a constant. The latter often requires
additional reasoning to determine which variables are relevant and which are
not. Kolmogorov famously made use of this approach to obtain his "5/3" law in
turbulence:
[https://micromath.wordpress.com/2008/04/04/kolmogorovs-53-la...](https://micromath.wordpress.com/2008/04/04/kolmogorovs-53-law/)

The details of dimensional analysis seem only to be taught to engineers taking
fluid dynamics or heat transfer classes. I understand why: the Navier-Stokes
equations for fluid motion typically can't be solved in practice (for the most
part), but we usually know which quantities are involved because they appear
in the Navier-Stokes equations. Then it's useful to use that as a basis for
analysis. In other areas, you can usually solve the equations, so dimensional
analysis may not be necessary. Add on top of that the use of _physical_ models
(not mathematical or computational models), which may be scale models like a
small scale airplane in a wind tunnel. Dimensional analysis provides a basis
for scaling.

I'd recommend reading the relevant chapter of the book if you're interested.
Dimensional analysis is most useful for physical problems, but you sometimes
can generalize the idea of a "dimension" such that two things have different
dimensions when it does not make sense to add them.

~~~
nestorD
A very nice example is the use of dimensional analysis to cook a turkey :
[http://physicsblog.otterbein.edu/?p=34](http://physicsblog.otterbein.edu/?p=34)

------
msangi
Is this a follow up to Street Fighting Mathematics, from the same author?
[https://mitpress.mit.edu/books/street-fighting-
mathematics](https://mitpress.mit.edu/books/street-fighting-mathematics)

~~~
532nm
You can consider it as such, yes. The main difference is that whereas street
fighting mathematics was focused more on the mathematical side, this book
focuses more on physics. However, the spirit and bold worry-free approach is
exactly the same.

------
armenb
The textbook of the draft can be accessed at:

[https://ocw.mit.edu/resources/res-6-011-the-art-of-
insight-i...](https://ocw.mit.edu/resources/res-6-011-the-art-of-insight-in-
science-and-engineering-mastering-complexity-fall-2014/online-textbook/)

------
fizixer
It doesn't have any mention of the author(s).

The MIT course website lists Mahajan and Abeyaratne as the instructors, but
not clear if one, both, or neither are the authors.

