
Too Much Calculus – Gilbert Strang (2001) [pdf] - nkurz
http://www-math.mit.edu/~gs/papers/essay.pdf
======
ekidd
As a programmer, I've found far more uses for linear algebra than for
calculus. It shows up in robotics, in machine learning, in game development,
and in many other cool subfields.

I didn't get very much out of my university linear algebra course. (It was
pure theory with nearly zero applications.) I don't think I really developed a
good intuition for linear algebra—and why it's so useful—until I read Jim
Hefferon's free textbook:

[http://joshua.smcvt.edu/linearalgebra/](http://joshua.smcvt.edu/linearalgebra/)
(GFDL license, LaTeX source available)

(I get nothing out of recommending this open source textbook other than the
joy of sharing a book I loved.)

~~~
ams6110
As a programmer, I've used calculus and linear algebra exactly never in my
professional (20+ years) career. It just doesn't really come up doing web
apps, sysadmin or dba work.

~~~
taeric
A lot of this is whether or not you are looking for it, honestly. It is easy
to stay away from it, though it is also easy to find if you look for it.

------
xamuel
(Speaking from my experience as a calc teacher at a big U.S. state U.; mileage
may vary elsewhere) Most non-STEM students won't see the 'Calculus I, Calculus
II, Calculus III' that starts the essay so dramatically. Increasingly, liberal
arts students are able to get by with no more than a "math appreciation"
course and maybe what amounts to high school algebra. The "math appreciation"
course might actually touch on very neat topics for these students
(mathematics of voting, etc.) so I'd say math curriculum has already been
successfully reformed for them.

As for the average STEM student, in my experience, most freshmen wouldn't be
ready for linear algebra (unless by "linear algebra" you mean "matrix
arithmetic"). They have enough trouble understanding functions in Calculus I:
linear algebra's "functions are basically matrix multiplication" feature would
NOT help.

Another thing to keep in mind is calculus has had much more impact on
history/society than linear algebra. The fundamental theorem of calculus is a
legitimately earth-shattering breakthrough: none of the theorems in linear
algebra come anywhere near it in cultural importance.

Pragmatically, calculus is the most suitable mathematics for the type of
testing-oriented, uniformizable/transferable pedagogy that a university really
(in cold practical terms) must depend on for first-year students. It would be
extremely difficult to force the kind of uniformity onto linear algebra that
would be necessary if you seriously want to have millions of transfer students
moving around between thousands of universities every quarter, gossiping about
which linear algebra teacher gives the easiest grades, etc.

~~~
davidwihl
I'm amazed that you say that calculus has had a much greater impact on society
when virtually every database, graphics manipulation or economic analysis uses
principles of linear algebra.

Can you substantiate "millions of transfer students between thousands of
universities every quarter"? That seems like outlandish hyperbole.
Irrespective of the precise number, a system with momentum is not sufficient
justification for its continuation.

~~~
d_e_solomon
FWIW, I have a math degree and work in tech.

Both are incredibly important. I would say that linear algebra and discrete
methods are more important for computing, but calculus has had a larger impact
on science especially physics. Newtonian physics - a revolution in thinking
and unification of mathematics and science - is intricately tied to calculus.

Even in the example of economics that you cite, calculus is used heavily for
the theoretical underpinnings for finding equilibria and deriving min/max.

~~~
FeymanFan78
I agree that Calculus is a more readily applicable mathematics that easily
applied to many things. It really bothers me when people downplay the
usefulness of Calculus also the person that posted the fundamental concepts of
Linear Algebra are generally a little beyond the Grasp of people that haven't
even been through the Rigors of Calculus 1 - Calculus 3.

------
jeffreyrogers
I think Strang has some good points, and his advice is true for the majority
of students who won't go on to use substantial amounts of mathematics or need
to learn much math in the future. However, if you _do_ want to learn more math
beyond calculus and linear algebra, then a thorough understanding of calculus
is absolutely a prerequisite and you honestly probably haven't learned it well
enough taking the typical sequence of classes.

For example, I'm currently studying probability theory at a fairly rigorous
level (building up from measure theory) and my relative lack of skill with
calculus is the main stumbling block. And I was recently talking with one of
my professors who said that the biggest obstacle he sees students facing in
trying to understand advanced mathematics isn't the difficulty of the math
itself, but in their lack of mastery of the fundamental calculations that are
necessary to follow the ideas and to come up with them on your own. Fluency
with calculus is an enormous part of this. That said, I agree with the point,
which is that for most students linear algebra is more useful.

Actually, I might go a bit farther and say that most students don't need to
learn much math at all, at least in the way it is taught. Math is pretty
poorly taught for a subject that is actually quite interesting once you start
to understand it better (part of the problem is that in order to get to the
interesting parts you need to first master all the boring stuff), and most
people won't use any nontrivial math in their future lives.

~~~
cbgb
This may be true for mathematics in the analytical tradition (things like
Topology, Measure Theory, Real/Complex Analysis), but Linear Algebra is far
more important in, unsurprisingly, algebraic disciplines. These include Number
Theory, Field Theory, and Mathematical Logic.

While studying mathematics in college, once I had finished my Real Analysis
requirement, I jumped headfirst into the algebraic side of things and never
found myself using any sort of calculus. Even in my Topology course, we
focused much more on using techniques from Real Analysis than specifically
calculus topics (the former just being a generalization of the latter).

Probability theory is somewhat deceptive in its classification, since much of
it "feels" a lot like a discrete mathematics course; however, much of the
concepts, like you say, are underpinned by measure-theoretic principles, which
is heavily analytic. It makes sense that calculus would come in handy in a
much deeper study of probability theory.

~~~
jeffreyrogers
That's a good point, my own studies have been biased towards the analytical
side of mathematics, but I can definitely see how calculus could be less
useful in other fields, e.g. I don't remember ever needing to do an integral
when learning algorithms. And the opinions of my professors are of course
biased as well, since they've spent their entire careers on analysis.

~~~
SpaceManNabs
When analyzing algorithms, I have used calculus as a shortcut to many
calculations. I believe that learning calculus is not just about its direct
applications. Calculus often comes up when just dealing with your set-up.

You might make a discrete model for a problem and end up using calculus for
approximations, equivalent calculations, or reasoning.

------
SixSigma
I watched Prof. Strang's Linear Algebra courseware, great stuff. He is a
wonderful teacher.

[http://ocw.mit.edu/courses/mathematics/18-06-linear-
algebra-...](http://ocw.mit.edu/courses/mathematics/18-06-linear-algebra-
spring-2010/)

As someone who hadn't studied it for 20 years, it was invaluable to revisit
the subject as an adult with the ability to reflect "ah, I can use that for...
" instead of "Ok but when am I ever going to use this"

~~~
tptacek
Those lectures are fantastic. Pro-tip: watch them at 2.5x speed (you'll keep
up just fine, and you can always pause to play with the concepts in Sage or
whatever).

Bonus: when you switch him back to normal speed, he sounds stoned out of his
mind.

~~~
YAYERKA
This is a great tip. I've been using it for a few years now to watch various
lectures on youtube--especially mathematics. Professors lecturing mathematics
tend to speak slowly since they don't want to say anything erroneous. Speeding
them up is usually amazing. For youtube; open the javascript console in your
browser and type '$('video').playbackRate = 2.5;'. I've found that each
lecturer usually has their own magic number regarding speed--after watching
someone for a few hours and varying the speed you can usually find it.

------
jkbyc
My linear algebra professor insisted that we, as computer science graduates,
know and can reproduce most if not all the proofs of the theorems presented in
the class. He didn't have such demands for math majors. His point was that we
as computer scientists / software engineers / programmers will always have to
understand and construct all the minutiae of the programs we write (otherwise
they won't work). In contrast, mathematicians could use a lot of it "just" as
tools.

I also had to go through a lot of heavy calculus courses which were very
proof-centric and built from first principles. Rarely we skipped a proof of a
theorem and we were expected to be able to reproduce and understand all of
them at exams. I hated it often back then. However, these days I can
appreciate the rigor it has taught me and I would wish I could send some of
the programmers I met to similar courses so that they would learn to think
clearly and precisely and be able to spot unmet or missing assumptions.

Our teachers and the creators of the curriculum were not stupid or naive. They
knew well how much of hands on calculus we would use in practice vs. algebra
or linear algebra. One of their main reasons for including calculus was to
teach us rigor. I know that once someone overheard two teachers talking - the
calculus teacher was asking someone from a CS department what to teach the CS
graduates and the reply was that he can teach us whatever he wants, he just
should make sure to teach us to think. I think they were quite successful at
it even if it wasn't a very pleasant experience for most.

That also reminds me of this very relevant and true comic:
[http://savethepenguins.net/image/237344573](http://savethepenguins.net/image/237344573)
("Wtf, man. I just wanted to learn how to program video games.")

------
ludicast
I agree. As far as calculus goes, I am more enamored with books like Spivak's
([http://www.amazon.com/Calculus-4th-Michael-
Spivak/dp/0914098...](http://www.amazon.com/Calculus-4th-Michael-
Spivak/dp/0914098918)) that take a proof-centric approach to teach calculus
from first principles.

Incidentally, for those who want to learn linear algebra for CS in a mooc
setting there are 3 classes running at this very moment:

[https://www.edx.org/course/linear-algebra-foundations-
fronti...](https://www.edx.org/course/linear-algebra-foundations-frontiers-
utaustinx-ut-5-02x) (from UT Austin)

[https://www.edx.org/course/applications-linear-algebra-
part-...](https://www.edx.org/course/applications-linear-algebra-
part-1-davidsonx-d003x-1) (from Davidson)

[http://coursera.org/course/matrix](http://coursera.org/course/matrix) (from
Brown)

The first 2 use matlab (and come with a free subscription to it for 6 months
or so), the last python. One interesting part of the UT Austin class is that
it teaches you an induction-tinged method for dealing with matrices that let
you auto-generate code for manipulating them: [http://edx-org-
utaustinx.s3.amazonaws.com/UT501x/Spark/index...](http://edx-org-
utaustinx.s3.amazonaws.com/UT501x/Spark/index.html) .

And of course there are Strang's lectures too, but those are sufficiently
linked to elsewhere.

~~~
pakled_engineer
My calc I course in university was applied calculus without a text. I had to
go back and redo single variable by reading Spivak (and Polya's How to Solve
It) to figure out the proofs in Concrete Math by D. Knuth, et. al.

------
tel
You can probably do worse than to cut mathematics up into "algebra" (not just
linear) and "analysis" (calculus). They're almost completely diametric points
of view by the bulk modern mathematical canon [0].

There's a fair point to say that one shouldn't learn so much calculus as to
lose your linear algebra, but the opposite is just as true. There are a lot of
comments in this thread along the lines of saying that linear algebra is far
more useful than calculus in CS. This statement feel not exactly unfair but
certainly like it's a reactionary measure.

Whenever you have a notion of change you have some kind of calculus underlying
it. Even if you end up having to view that notion of change algebraically. For
instance, there's a famous functional programming result that there is a well-
defined "derivative of an algebraic data type" which allows you to talk about
changes in data "over time".

I'd also feel very bad for anyone who tried to learn too much probability
without studying quite a bit of calculus first.

[0] Sure there's discrete calculus and algebra underlies the mechanism of most
calculus computation. Likewise there are continuous groups and you can't avoid
how topology mixes the two concepts.

------
cageface
I really can't agree with this strongly enough. I had enough Calculus and
Differential Equations courses to kill a horse in college (Chemistry major)
and I've never used that knowledge once in my entire programming career. But I
find uses for linear algebra and matrices in coding all the time. Thank god
Strang's excellent course is available online for free.

~~~
andrepd
If you were looking to become a programmer, maybe you shouldn't have majored
in chemstry, where strong Calculus foundations make perfect sense :)

~~~
samatman
Chemist here, who also programs. Inorganic chemists do calculus all the time.
Organic chemists do arithmetic (stoichiometry) and some discrete analysis
(instrumentation, the computers do the heavy lifting), orgo is primarily a
linguistic discipline, i.e. we are expected to parse sentences like
"benzophenone in THF was refluxed in the presence of Pd black for 2 hours" and
to know, without being told, that there's a workup before the next step in the
reaction.

~~~
cageface
Exactly. I was an organic chemist. Never used a lick of Calculus. I didn't get
into programming until I'd already been working as a chemist for about five
years.

------
oleks
Perhaps amusingly, the Computer Science education at the University of
Copenhagen includes no mandatory Calculus course, but it does include a Linear
Algebra course. Even so, students have a hard time seeing what they'll use
Linear Algebra for, before later on in their education.

~~~
ajuc
Make them code simple wolfenstain 3d clone with opengl, directx or even
unity3d.

It was a revelation for me when I've tried to get into game programming, and
half of the math taught in primary and secondary school became immediately
useful, also linear algebra (which was only started at the final year of high
school). Made linear algebra courses on the university a lot more interesting.
There's a difference if you look at a class "what I need to pass an exam", and
"what I could use in my next game".

Also tinkering with 3d graphics makes thinking in vectors, planes, matrices,
dot and cross products intuitive after a few weeks.

BTW I wonder hwo many kids learnt basic linear algebra from Denthor's Asphyxia
tutorials :)

~~~
jimmaswell
I can't see myself really needing to apply any Linear Algebra concepts writing
a Wolfenstein clone in Unity3D. It does everything for you like that (shader
graphics, collision detection, vector math, etc). All you'd have to do is code
some simple AI for the enemy gameobjects and make the player gameobject in the
standard FPS shooting format, which there are pre-built defaults for in the
example projects. The majority of this project would be copy and paste with
minor modifications.

~~~
ajuc
Even "just" ai can teach you a lot. For example trying to calculate if enemy
can see you, or which path he should take is interesting.

Of course you can go far with "turn towards player and shoot" ai, but where's
the fun in that.

Actually it would be fun to give student's assigment like:

here's the AI interface, make a game using it, and write your bot. Then we
will run all the bots in free for all mode on all the games (except your bot
on your game), and we will grade by sum of the points.

~~~
jimmaswell
>trying to calculate if enemy can see you

Send a raytrace from the bot to the player and see if it hits something other
than the player, which is just one line

Pathtaking beyond simply moving towards the player is a lot harder, yeah.

~~~
ajuc
If you just send 1 ray from eye towards the center of the enemy (or head, or
whatever) - and this particular part of the body is occluded by a wall or a
chair, but the rest isn't - the result will be wrong.

You can accept that, use many rays, or do proper view frustum/body
model(simplified probably) intersection, after cutting the view frustum with
the terrain.

You can also consider lighting (so enemy can't see you hiding in a dark
corner), or even dynamic lighting (and give enemies light sources).

Everything becomes hard after you think about it enough :)

But I agree simple solutions + some tinkering with parameters works most of
the time (and gamers probably won't notice).

------
cpitman
Statistics are wildly important for both business and understanding the world
around you, and without solid calculus skills you cannot truly work with
statistics.

Its not something that comes up every day at work, but it is still pretty
often. For example, I was on a large team developing a complicated distributed
system. The performance requirements were uncertain (like 500000 clients +-
200000, sending 10 +- 30 messages per second, etc). The system itself was in
early development, hadn't been tuned, and we had only rough performance
numbers for it.

Given only that, I needed to purchase the correct amount of hardware for the
test lab, keeping in mind that the lead time for hardware was several months.
I needed both statistics and some basic calculus to build a model based on
what we knew, calculating variances for each intermediate and output, and then
getting 50%, 90%, 99% confidence level estimates.

In the end, we got pretty close to a bullseye. On the other hand, I learned
the important lesson to hide all the math from the client, lest they get
awfully confused.

------
kowdermeister
The Khan Academy lectures are a pretty way to start:
[https://www.youtube.com/playlist?list=PL835D2252574274C5](https://www.youtube.com/playlist?list=PL835D2252574274C5)

------
wwweston
Possibly understated... not only is there the three course sequence + diff eq,
there's usually also a two-sequence real analysis course (more or less
"careful calculus") and probably either PDEs or complex analysis to boot.

I took the discrete emphasis when I studied and I think I actually ended up
with fewer discrete classes (5, IIRC) than calculus classes (6).

~~~
titanomachy
Although it's not explicitly stated in the article, I think Strang's remarks
are actually directed at non-math programs in STEM (I assume you are at least
a minor in math because of 3 analysis courses). Math majors will get plenty of
algebra either way, but it's the engineers and scientists who miss out from
overemphasising calc over algebra.

------
tdfx
I've never understood why Calculus is taught as a separate course. It seems so
detached from practicality. I've always thought students should take perhaps
4-6 semesters of Physics -- and introduce derivatives, integrals, etc. as
needed when they can be demonstrated to more easily solve the physics
problems. If the physics content was more spread out, there would be ample
time to introduce the necessary math concepts, instead of stuffing them into a
separate course.

~~~
AnimalMuppet
I remember, in my high school calculus class, the teacher said that there were
six or seven applications of derivatives, and more like 40 applications of
integrals. It's _way_ more than just physics.

------
clebio
What is the context of this short essay? Why did he feel the need to write it,
at that time?

------
EdwardCoffin
I don't disagree that linear algebra isn't given enough emphasis. However, the
first thing I thought of when I read this was something attributed to Richard
Hamming, who is known for the transcript of his talk that pops up on here
frequently, You and your Research [1]. I got this from a review of his book on
Amazon [2]:

> Hamming, in discussion, concluded his life thinking "the best tool to teach
> thinking was to teach the calculus."

I'm just thinking here that Strang is saying Linear Algebra is more useful for
_applying_, where Hamming would say Calculus is more useful for reasons that
go beyond applications (not that it doesn't also have applications).

[1]
[http://www.cs.virginia.edu/~robins/YouAndYourResearch.html](http://www.cs.virginia.edu/~robins/YouAndYourResearch.html)

[2]
[http://www.amazon.com/review/RWPF0EL3E7GJZ/ref=cm_cr_pr_perm...](http://www.amazon.com/review/RWPF0EL3E7GJZ/ref=cm_cr_pr_perm?ie=UTF8&ASIN=9056995014)

------
hexagonc
I had the fortune of taking 18.06 (Linear Algebra) from Gilbert Strang around
the time of this essay. And like other commenters, I will agree that he was a
very good teacher. I could tell that he held a great emotional attachment to
matrices; he'd often anthropomorphize them with genders and refer to them as
"little guys".

------
xlm1717
I think taking less calculus and more linear algebra is a mistake.

There is a really good book in two volumes that I feel shows how methods of
linear algebra can be extended using calculus to solve a wide range of
problems: A Course in Mathematics for Students of Physics:

[http://www.goodreads.com/book/show/2685577-a-course-in-
mathe...](http://www.goodreads.com/book/show/2685577-a-course-in-mathematics-
for-students-of-physics)

Volume 2 is more important. It starts demonstrating how methods of Linear
Algebra can be used to solve circuit problem, then generalizes this to higher
dimensions and from discrete to continuous spaces using the exterior
derivative. Seeing this demonstrated gave me a much clearer, more intuitive
understanding for both calculus and linear algebra.

------
hzhou321
I wish Gilbert Strang would teach calculus and write a book as he did for
linear algebra.

~~~
dfan
You are in luck!

Strang, Calculus, 2nd edition: [http://www.amazon.com/Calculus-Second-Edition-
Gilbert-Strang...](http://www.amazon.com/Calculus-Second-Edition-Gilbert-
Strang/dp/0980232740)

~~~
bewuethr
And the first edition is freely available at Open Courseware:
[http://ocw.mit.edu/resources/res-18-001-calculus-online-
text...](http://ocw.mit.edu/resources/res-18-001-calculus-online-textbook-
spring-2005/textbook/)

~~~
hzhou321
I wish every day is like this!

------
xxcode
I took Gilbert Strang's Mathematical Methods for Engineers course at MIT.
Can't seem to find the link for it, but I would recommend the online version
for anyone who wants to learn linear algebra.

------
jules
I don't think you can cleanly separate the topics. Most applications of linear
algebra come from calculus since after all you use calculus to linearize
something. In case it was already linear it's probably a linear differential
equation -> calculus again. Complex analysis is also closely related to linear
algebra via geometric algebra. Rather than teaching them separately and
emphasizing linear algebra more, it would make more sense to approach calculus
+ linear algebra as an integrated subject.

------
ivan_ah
Anyone interested in getting started with linear algebra can check out this
printable tutorial: Linear algebra in four pages:
[http://minireference.com/blog/linear-algebra-
tutorial/](http://minireference.com/blog/linear-algebra-tutorial/)

The page also links to another short tutorial which explains more of the
geometric intuition behind linear transformations.

Neither tutorial can serve as a full course, but reading them will give you a
good overview of what linear algebra is about.

------
laurentoget
14 years later, the popularity of machine learning and "big data", which
requires quite a bit of linear algebra, makes this only truer.

~~~
tel
It also requires quite a bit of calculus though!

~~~
laurentoget
Calculus is indeed a required tool if you want to design new machine learning
algorithm. But without linear algebra, good luck implementing an algorithm or
even using a library to apply it on a concrete case.

~~~
tel
I don't see why this is an either/or decision?

Without a firm understanding of probability then good luck interpreting or
fitting a model. Without a firm understanding of linear algebra good luck
implementing or using an algorithm.

And hell, if you have a streaming algorithm you probably no longer need linear
algebra but to grasp either the probabilistic or geometric interpretations
you'll probably need calculus.

------
jmilloy
> We need to present the mathematics that is most useful to the most students.

I wish it weren't this way! Mathematics is a wide and fascinating field, and I
think more students could find interest if our schooling reflected that. But
moving some of the detail and drudgery that is the calculus sequence to
subject matter that requires it, in order to make way for linear algebra, is a
great start.

~~~
jeffreyrogers
I agree completely, but calculus touches so much of advanced math that at some
point you have to learn it in depth (assuming you actually want to learn
advanced mathematics). I'd argue that linear algebra is a more interesting
subject for most people, but calculus contains more important ideas that are
crucial if you intend to continue studying mathematics.

~~~
jmilloy
I have a major in Mathematics, and I truly didn't need calculus after the
multivariable/vector calculus course I had to take my Freshman fall. Why did I
_have_ to do so much if it wasn't going to be necessary until graduate school
at the earliest, or perhaps never depending on specialization? And why was it
a prerequisite for my other math courses (in algebra, topology, real analysis,
number theory, and combinatorics)?

~~~
peterfirefly
How was topology motivated without talking about generalizations of limits?

(Honest question, no snark.)

~~~
jmilloy
It's a good question.

I think my first topology course, like in many intro analysis courses, shared
concepts from calculus without requiring any previous expertise from our
calculus courses. A rigorous approach to real analysis or point-set topology
will be very set-theoretic, and you build the general definitions of
continuity, compactness, etc from the ground up. So, I would have used
concepts from my analysis and topology courses _in_ calculus, if only it had
been taught in the other order.

I guess it feels like limits are a rigorous foundation for calculus, but they
are not calculus. In any case, I didn't need anything from differential or
integral calculus (or that entire _required_ course developing the theoretical
basis for Stokes...), which is what I mean to say.

------
lolwhat
I remember taking Calculus 1, 2, 3 but only having 1 linear algebra course and
some engineering math. Although I did well in them, as a developer, I wish I
payed more attention to linear algebra or at least wished that my uni offered
more courses related to it, like discrete math and maybe even statistics. Good
thing I still have my old books!

------
j2kun
To reinforce this point: here's just a fraction of what you can do with linear
algebra [1]

[1]: [https://medium.com/@jeremyjkun/here-s-just-a-fraction-of-
wha...](https://medium.com/@jeremyjkun/here-s-just-a-fraction-of-what-you-can-
do-with-linear-algebra-633383d4153f)

~~~
aet
To be fair, almost all that stuff uses calculus too.

------
evrim
heh, try teaching galois for a year, you'll understand why calculus is so
popular.

~~~
banachtarski
it's a beautiful branch of mathematics though.

~~~
evrim
everybody thinks they know algebra if they study linear algebra. this is worse
than not knowing about calculus i guess.

can't image teaching point-set topology, teaching uryshons proof to
undergrads, gee.

------
forkandwait
Another perspective: I am trying to teach myself physics from Morin's books,
and for the first time I see how well structured was my lower division calc
sequence d for this application.

------
nicholasdrake
i actually have taught some linear algebra concepts on youtube and have gotten
several hundred thousand views and some positive feedback. i think the key is
trying to figure out a framework of intuition to help students through all the
math but i liked the idea of four key equations we are trying to solve as a
framework as well

------
mjh2539
The way this was typeset...it's making my head hurt.

------
graycat
> Too Much Calculus

Not for me! Heck, I never took freshman calculus or _pre-calculus_ and,
instead read a book and started on sophomore calculus then continued with,
right, ordinary differential equations, advanced calculus for applications,
and advanced calculus, topology, modern analysis, measure theory, functional
analysis, exterior algebra, etc. Not enough calculus for me!

Strang, of course, is emphasizing linear algebra. Great! It's terrific stuff.
I had an abstract algebra course that started on vector spaces and matrix
theory. Then I got famous book from Princeton on multi-variable calculus and
read the first chapters on linear algebra, e.g., learned Gram-Schmidt. Then I
got a linear algebra book and read a few chapters and applied them to some
problems in classical mechanics. Then I wrote my math honors paper in group
representation theory, right, more linear algebra. The out of school, I wanted
to know linear algebra better and read a good book cover to cover, carefully.
Then I got Halmos, _Finite Dimensional Vector Spaces_ and read it cover to
cover. Wrote Halmos about his proof of the Hamelton-Cayley theorem and got
back a nice answwer!

Then I got another famous text on multi-variable calculus and read it cover to
cover, including the parts on exterior algebra. Then I did that again from
Spivak's _Calculus on Manifolds_ \-- right, a lot of both calculus and linear
algebra. Then I got Forsythe and Moler, _Computer Solution of Linear Algebraic
Systems_ , read it cover to cover, directed a project in interval arithmetic
for linear algebra, took a course in numerical analysis with a lot of linear
algebra for solution of partial differential equations, e.g., iterative
solutions (Gauss-Seidel) and wrote corresponding code.

I also used linear algebra in work with the fast Fourier transform, multi-
variate statistics, and more.

For more on linear algebra, I read the first part, linear algebra, of von
Neumann, _Quantum Mechanics_.

From a world class guy in linear algebra, I took an _advanced_ course in
linear algebra -- by then the course was beneath me and I led the class by a
wide margin. Favorite theorem -- the polar decomposition.

Read Coddington's book on differential equations -- gorgeous book.

The combination of linear algebra and calculus, with other math, continued.

At one time a little calculus did a lot: The initial value problem for the
little first order, linear ordinary differential equation

y'(t) = k y(t) ( b - y(t) )

saved FedEx, that is, kept it from going out of business. So, _useful
calculus_!

So, sure, I like linear algebra and like the combination!

I definitely do not see that I had too much calculus!

Here's what I saw, and still fume about: All I could get in chemistry was one
too simple course in high school and one too simple course in college. All I
could get in physics was one too simple course. I college, by the time I got
to Maxwell's equations, I didn't know enough calculus to do really well. In
college I never got math enough to do well with quantum mechanics.

But in grades 9-12 and then in the first two years of college, I was force fed
like a goose with six years of English literature, with each year yet another
play by Shakespeare. I agree that there was a good writer in England in the
1600s -- Newton! So it was also Chaucer, Milton, Wordsworth, ..., Dickens,
_the great natural order_ (all corresponding theorems and proofs omitted!),
etc. With good information about people? Nope.

Then there was history: It never got to the 20th century. It never touched on
technology, economics, or any other _causes_. There was plenty of time in
history courses; it's just that the courses didn't have much content.

If my startup works, then I'll get back to more in calculus, functional
analysis, stochastic processes, and mathematical physics. "More calculus,
Ma!".

