
Calculus for Beginners and Artists (2003) - lewisjoe
http://www-math.mit.edu/~djk/calculus_beginners/
======
jszymborski
I've discovered Calculus Made Easy recently and it's wonderful. The edition
edited by Martin Gardner is particularly good, with amazing preliminary
chapters [0].

I've spent many years "pretending to understand" calculus, but things I
remember gnawing at me, like limits & infinitesmals, are accompanied with
context and history such that you can finally put yourself into the
conversation and understand that my confusion is simply due to only getting a
fraction of the story.

You can read the full text for free here [1]

[0]
[https://openlibrary.org/books/OL351037M/Calculus_made_easy](https://openlibrary.org/books/OL351037M/Calculus_made_easy)

[1] [http://calculusmadeeasy.org/](http://calculusmadeeasy.org/)

~~~
hyperpallium
[a critique, not a request for help] I found it relied too much on faith: you
can ignore _this_ small quantity, but not _this_ one. What's the threshold?
Why?

Fractional and negative powers are assumed to work as a generalization of
positive integer powers, without proof.

But, TBF, all maths education requires a lot of faith. e.g. the unique prime
factorization theorem is assumed in high school, not proven.

~~~
jgwil2
If I recall correctly, the author explicitly states in the introduction that
mathematicians will hate the book precisely because it skips over proofs and
takes a pragmatic approach. If you're the kind of person who wants to
understand the proofs before using them, you will need to supplement this book
with other material.

~~~
hyperpallium
He spends much time on "minute" quantities at the start, but the explanation
doesn't really make sense. To me, it's a mental model I can't trust, like
rickety stairs.

The book is explicitly calculus-as-a-bag-of-tricks; monkey-see, monkey-do. As
he says:

    
    
      What one fool can do, another can.
      (Ancient Simian Proverb.)
    

Fair enough on proofs. BTW in the free gutenburg edition (maybe MG differs):
"prologue" doesn't mention _proofs_ , but that textbook writers make it
difficult (no "introduction" \- also no "preface", except to the 2nd ed):

> The fools who write the textbooks of advanced mathematics ... seem to desire
> to impress you with their tremendous cleverness by going about it in the
> most difficult way.

~~~
jgwil2
Yeah, this sounds right. I think MG contextualizes it a little bit more. I
actually like the book for the exercises and MG's footnotes especially, but as
I worked through it I had to consult other resources to make sure I wasn't
missing anything. The Khan Academy Youtube channel was very helpful for this.

------
enriquto
I sort of dislike this kind of "philosophical" introduction to calculus. Maybe
I don't have the spirit of an artist.

The best way to start with calculus is the one by Gilbert Strang, who explains
_everything_ on the first page of his book (and the rest of the book are
"just" examples).

The first half of the first page shows a drawing of the speedometer and
odometer of a car and it explains what they are, and that they are not
independent but related in a special way. On the second half of the first page
it says that differential calculus is the task of computing the speed from the
distance, and integral calculus is the task of computing the distance from the
speed. Then it says a lovely sentence "this is not an analogy, this is the
real deal and we have already started with the subject, and this is actually
all there is to it". Then in the rest of the page it explains in a couple of
sentences how can you compute speeds from distances and vice-versa, and why
you need a constant of integration, and so on. It also proves the fundamental
theorem of calculus. The rest of the book consists in concrete examples and a
few more constructions, up to Taylor series.

~~~
TheOtherHobbes
I can't imagine someone with an arts background getting anything at all out of
this introduction. The Gilbert Strang approach is much clearer - although it's
still not obvious why an artist would ever need calculus.

In fact there are some applications in generative design and 3D animation, but
it's still low on the list of essential art skills. And if you really need
those effects - and know enough math to understand how to code them - you can
copy code from a cookbook without having to derive it from first principles.

~~~
jacobolus
There are a bunch of mathematical artists who don’t do “generative design” or
“3D animation”. Textiles, sculpture, painting, architecture, ...

Many of them use calculus in one way or another.

Check out the Bridges conference,
[http://bridgesmathart.org/](http://bridgesmathart.org/)

------
gibsonf1
I know of no better way to both learn, truly understand, and enjoy calculus
than the visual approach of 3Blue1Brown in the Calculus series of videos:
[https://www.youtube.com/watch?v=WUvTyaaNkzM&list=PLZHQObOWTQ...](https://www.youtube.com/watch?v=WUvTyaaNkzM&list=PLZHQObOWTQDMsr9K-rj53DwVRMYO3t5Yr)

~~~
hackermailman
I found the best way to truly understand and enjoy calculus was to learn it as
it was historically developed
[https://youtu.be/HRD9X-2Bmdw](https://youtu.be/HRD9X-2Bmdw) and then learning
about the problems they ran into and how we ended up with modern Calculus
after Euler and Lagrange tried to correct these problems
[https://youtu.be/fCZ8jJCVinU](https://youtu.be/fCZ8jJCVinU) (these lectures
cover Stillwell's book Mathematics and it's History)

Interesting in the second lecture is how Australia does 'photo radar' on one
stretch of highway, where it records you going through a gate at one point,
then many kilometers away records you again from another gate, then
establishes your average velocity between the two gates using Calculus and
sends you a ticket if your calculated average shows you must have exceeded the
speed limit during some point between the gates.

~~~
krick
> best way to truly understand and enjoy calculus was to learn it as it was
> historically developed

In fact, I'd argue it's the best way to understand almost anything, especially
in math. Many topics I found somewhat confusing at school or university or
whatever got _really_ simple once I learned about their history. Little by
little I come to feel that most of great inventions or discoveries made by
people we regard as geniuses are often brilliant at how clear, beautiful and
somewhat unexpected the solution was, but it's actually very rarely
_complicated_ and usually seems like the most natural thing in the world, when
told about how Fourier/Laplace/Leibniz/etc discovered it, and not hidden
behind standard school math curriculum.

That's a part of why I love 3Blue1Brown videos so much, and why I love Morris
Kline books. And it always makes me kind of sad feeling how much time I wasted
trying to come to terms with something that always was just unnatural
explanation.

~~~
gshubert17
Morris Kline's book "Calculus: An Intuitive and Practical Approach" is still
in print. I prefer the paper version because the Kindle version has many
formatting problems.

------
sunraa
Priceless: [http://www-
math.mit.edu/~djk/calculus_beginners/chapter00/se...](http://www-
math.mit.edu/~djk/calculus_beginners/chapter00/section01.html)

0.1 What You Should Know

To study calculus it is essential that you are able to breathe. Without that
ability you will soon die, and be unable to continue.

Beyond that, you will need some familiarity with two notions: the notion of a
number, and that of a function.

------
diggan
Seems like a handy resource but seems the navigation is a bit annoying to deal
with. Quick hack with `wget --mirror` + `pandoc` for joining the documents, I
got the following single-page layout: [https://cloudflare-
ipfs.com/ipfs/QmZetPzAP5bC3uXVmNUKU2DU22L...](https://cloudflare-
ipfs.com/ipfs/QmZetPzAP5bC3uXVmNUKU2DU22LPBmxfkSdGqYaGMqmgp7/)

(I take no responsability for any formatting errors. Math seems to render
correctly, but probably some links are broken)

------
ClearAndPresent
Speaking as an artist...

The tone from the beginning (at least in Chapter 0) struck me as what I might
class "fey wankery". The wry style gets into the way of communication. Is
there some misapprehension amongst STEM explainers that people who are less
familiar with maths need to be treated like skittish, possibly mentally-
challenged deer?

Or is it a nod to Socratic dialogue except one of two is a moron?

Then Chapter One leaps into rational numbers, set theory and fractions and
commits the usual errors: asymptotic curve into complexity, no history, no
vivid metaphorical visualisations and (a personal peeve) no historical or
causal explanations.

Chapter One literally starts with a question ("What are numbers?") and does
not ever answer it. You can count them, some of them are natural (what does
that mean?), you can perform operations on them. But what are they, Professor
Kleitman? An abstract concept that can be derived from our ability to discern
a first order similarity between both similar and dissimilar objects? Is it
too vast a question for a introduction to calculus? Then don't use the
question in your section header.

Operations? "There are addition, subtraction, multiplication and division."
Why? Why isn't there redition? What's redition? I don't know, it's an
operation I made up, but you seem to have plucked four relations between
natural numbers from the air and asked me to assume that's acceptable.

The text continues in similar, tedious terms, walking us through the basics on
stepping stones of assumption and unearned trust.

Educating beginners and "artists" doesn't require you to speak to us like five
year olds. Take a leaf from Feynman or Fuller's books and talk to us like
adults but do the work in creating vibrant metaphor.

Just my opinion and apologies to Professor Kleitman.

To end on a positive note, the opening pages of Gilbert Strang's book, linked
by another poster here, were much more effective in conceptualising the need
for and use of calculus by anchoring it in a strong metaphorical example.

In barely a page or two, I understand a relationship between velocity and
distance - and that time is involved - and how that relationship can be
geometrically envisioned.

Further, I'm already pondering how one might deal with a more realistic car
journey with a variable velocity, something that calculus will address later
on.

The difference? No infantilisation; instead a clear and applicable
metaphorical example.

~~~
mathgenius
This question "What are numbers?" I find way more fruitful to contemplate than
any answer I have so far found. And why should we be adults anyway? Adults
already know all the answers. Ask this question like a five year old would.

------
kstenerud
[http://www-math.mit.edu/~djk/calculus_beginners/chapter03/se...](http://www-
math.mit.edu/~djk/calculus_beginners/chapter03/section02.html)

We can determine the linear function which takes value f(a) at a and f(b) at b
by the following formula:

    
    
                    x-b        x-a
        f(x) = f(a) --- + f(b) ---
                    a-b        b-a
    

The first term is 0 when x is b and is f(a) when x is a, while the second term
is 0 when x is a and is f(b) when x is b. The sum of the two is therefore f(a)
when x is a and f(b) when x is b. And it is a linear function. Linear
functions have a term that is x multiplied by some constant, and may also have
a constant term as well.

Umm... whut?

This has me completely lost. I don't see how this can be geared towards
beginners and artists.

~~~
baby
1\. X-b befomes 0 if x=b right?

2\. This means anything multiplying this x-b becomes 0 as well when x=b right?

His definition of linear is also wrong in my book.

Linear: f(x) = a x

Affine: f(x) = a x + b

(For some a, b)

~~~
kstenerud
Is that comment for me or about the referenced page? If it's for me, I'm not
sure what you're saying.

~~~
baby
I'm trying to explain what he meant. Which step don't you understand
specifically in my comment?

~~~
kstenerud
I'm trying to understand why there's f(a) and f(b), why is b being subtracted
from x, why it's divided by a-b, why there's a similar thing for f(a), why the
two are being added, why the zero values are important, and what the whole
thing actually means.

~~~
llamaz
It's demonstrating how to find the line that passes through two points x=a,
x=b on the function y=f(x).

Without having read the actual page, I assume he's going to move a and b close
to eachother to approximate the tangent of the function at a point.

edit: In higher level math, you use circles/spheres or parabolas/paraboloids
to approximate functions, but in high school level calculus you stick to using
a straight line to approximate a function

~~~
kstenerud
Unfortunately, he doesn't do that. There's an even more complicated formula
under that which is supposed to be related to the slope of some line and a
ratio between b and a (I think?) and then I'm guessing he reduces it, but
doesn't explain how he did it.

I've tried learning calculus 3-4 times during my life, using materials for an
"absolute beginner", and this has always been my experience, as if one were
teaching programming by going from "This is a variable. It can store data."
directly to "A monad is a monoid in the category of endofunctors." To this day
I have no idea what calculus even is, or what it's for.

------
prvc
From reading a few excerpts only, I am reminded of a lot of popular science
writing, which too often adopts a condescending casual tone full of flimsy,
inaccurate metaphors and chatty asides, and which too often omits simple
statements of the details of the subject at hand (often less complex than the
verbiage used for the adulterated version).

The writing becomes so far removed from the subject, that while those already
familiar with it will be able to guess what the author was getting at, those
who are not will remain in the dark, leading to a zero increase in the
reader's knowledge in each case. Despite being in the former category with
respect to basic calculus, I can't help but feel a sense of umbrage on behalf
of the latter.

------
caymanjim
Poking around this, it looks like there's a well-written introduction to
calculus buried inside, but the navigation is terrible. They should come up
with a better way to present this. A single long scrollable page, as clunky as
that might be, would be far superior to this.

~~~
diggan
Great minds think alike! I linked to a single-page version here:
[https://news.ycombinator.com/item?id=20434118](https://news.ycombinator.com/item?id=20434118)

------
krick
A point as minor as they come, but I really am not a fan of that embedded
chapter structure. It's absolutely ok to have multiple sub-topics in a topic
(that's why TOC exists in the first place!), but it also makes it so much more
enjoyable to read, when you can simply go through text linearly, without
worrying about what link on what level you clicked just before that.

------
nemetroid
Despite the title, I struggle to see who exactly this was written for.

The Javascript widgets are quite neat. Fiddling around with parameters, and
receiving instant graphical feedback, is great for developing an intuitive
understanding. But I find that the writing has several issues that probably
make it less clear to beginners than your average calculus textbook:

The writing is very brief, with typical mathematical terseness. This is
usually a good thing, since it lets you be very exact in your definitions. But
to be accessible to a beginner, this writing needs to be backed up with
concrete examples, and preferably a picture or two. In most cases I find that
it isn't, so the reader needs to internalize a lot of things on their own
before being able to move on.

The text is also sparse on showing its work. A positive example is near the
bottom of 5.1, showing how to derive the quotient rule, but most of the time
it looks more like example 1 in 6.1, with a lot of information given inline
before actually applying the mentioned operations to the expression:

> Suppose we substitute the function g which has values given by g(x) = x² + 1
> into the function f which takes values f(x) = x³ − 3.

> The substituted function f(g) has values f(g(x)) = (x² + 1)³ − 3.

> Let us compute the derivative of this function. The derivative of f(s) with
> respect to s is 3s², while the derivative of g(x) with respect to x is 2x.

> If we set s = g(x) which is x² + 1, and take the product of these two we
> get:

> [expression]

This type of mental expression manipulation is fine for someone that's had
practice with it, but probably not for a beginner, who would gain a lot from
having these kinds of things written out in a more structured way.

In these aspects, I find the text less clear than the calculus textbook i used
at university (which was not directed toward beginners or artists).

I agree a lot with prvc's comment[0] about being able to fill in the gaps, in
that the terseness and handwaviness can make this _look_ like a beginner-
friendly version to someone that already is familiar with the subject, but I
don't think changing tone is enough to make something beginner-friendly.

0:
[https://news.ycombinator.com/item?id=20435238](https://news.ycombinator.com/item?id=20435238)

------
lewisjoe
Came across this just when I was looking to deep dive into calculus, typed
lambda calculus, differentiation, etc. I somehow have a hunch learning these,
will help me down my road on language design and implementing compilers.

Only midway, but hands down this is the best material I came across on the
subject. As light-hearted as much as detailed.

Update: Thank you, all. I guess the dots would never connect down my road. But
I'm midway and the topic's still interesting enough for me to keep going.

~~~
mcncm
Just out of curiosity, why do you need to know differential calculus to write
a compiler, and what does it have to do with lambda calculus? Learning it is a
great thing to do, and it will certainly help you in other ways. But that kind
of programming _usually_ has little in common with the "mathematical analysis"
kind of calculus (unless you're specifically interested in _differentiable
programming_ , which is actually kind of big these days). Anyway, happy
studies!

~~~
lewisjoe
Oh, I have this bad habit of going down the rabbit hole of fundamentals if I
feel I lack the basics of a concept. While I still haven't grasped the need
for calculus in language design, I've come across pieces on how Lisp is based
on lambda calculus vs Haskell based on typed lambda calculus. Also, that any
problem that can be solved with turing machine, can be solved by lambda
calculus as well. While not much about compilers, I guess it will solidify my
understanding of language designs?

I'm not sure. I'll rather learn and hope the dots will connect later than
ignore the subject altogether.

~~~
wool_gather
Let's clarify here: "Calculus"[0] and the lambda calculus have almost nothing
to do with one another. They just share a word in their name, like the
Department of Mathematics and the Fire Department. There are several different
things called "the X calculus".

[0]:Could be called "numerical calculus" or "calculus of integration and
differentiation", perhaps.

~~~
JoelMcCracken
A while back I was trying to under why some things are called algebras and
others calculus. The answer I got is that the term calculus is related to the
introduction/reduction of variables or symbols, whereas an algebra is
manipulation of existing symbols.

~~~
mruts
I always thought it related to continuous variables and the highly useful
concept of the infinitesimal. But maybe that definition works as well.

~~~
User23
I thought it was just yet another historical and linguistic oddity in English.

~~~
mruts
Calculus means "stone" in Latin. I suspect the relationship between stones and
counting is why we use it so much (though I'm not an expert on how/why Romans
used stones).

------
undershirt
> “If calculus is the language of the universe, then Steven Strogatz is its
> Homer.”

I saw someone reading this at a coffeeshop, it’s a new one:

[http://www.stevenstrogatz.com/books/infinite-
powers](http://www.stevenstrogatz.com/books/infinite-powers)

~~~
gshubert17
This is an excellent book, which is full with historical insights, references
to people I didn't know about (Sophie Germain, Mary Cartwright, Sofia
Kovalevskaya), and emphasizes intuition and applications rather than proofs.

------
WheelsAtLarge
I took 2 semesters of Calculus in high school. It was a "requirement" to get
college AP. I was convinced that I would have failed at college if I didn't
take it. I have yet to use it in my career. A waste of time. A 2 semester
class of drawing or Art would have been way more useful.

Why is calculus held in such high esteem in college prep?

Most students can safely skipt calculus. A 1-day general introduction would be
enough for 90%+ of college students.

~~~
mlevental
90% of college students aren't stem majors and so what you're saying is almost
tautology. for stem there is literally no branch that doesn't use calculus in
at least some way.

------
RichardCA
I never really felt like I understood Calculus until I watched the Mechanical
Universe on PBS.

Here's an example, demonstrating how Kepler's Second Law is an expression of
the conservation of angular momentum.

[https://www.youtube.com/watch?v=0tSmY4fn5xY#t=7m50s](https://www.youtube.com/watch?v=0tSmY4fn5xY#t=7m50s)

------
neom
As someone with extreme dyslexia and dyscalculia, this is fascinating. Because
numbers and words are images for me, my recall and then contextual association
of them is very slow, building different frameworks to understand them is
super important and I wish there was a greater emphasis on different memory
styles in learning outside of rote.

------
mrcactu5
Robert Ghrist has written an colorful and innovative calculus text.

[https://www.math.upenn.edu/~ghrist/calculus.html](https://www.math.upenn.edu/~ghrist/calculus.html)

[https://www.math.upenn.edu/~ghrist/notes.html](https://www.math.upenn.edu/~ghrist/notes.html)

[https://sites.math.washington.edu/~morrow/334_13/FLCT.pdf](https://sites.math.washington.edu/~morrow/334_13/FLCT.pdf)

------
halbgnarf
[http://www-math.mit.edu/~djk/calculus_beginners/chapter03/se...](http://www-
math.mit.edu/~djk/calculus_beginners/chapter03/section03.html)

> if y = z + a * (x - z) then 1.) f(y) = f(x) + a * (f(x) - f(z))

This is incorrect, isn't it?

It should be 2.) f(y) = f(z) + a * (f(x) - f(z)) if I'm not mistaken.

For example, consider f(n) = n + 1

\- z=1, x=2

\- f(z)=2, f(x)=3

\- f(z + a(x - z)) <=> f(1 + a); for a = -1, we should get f(0) = 1,

Substituting into 1.) yields 3 + -1 * (3 - 2) = 2 (incorrect) Substituting
into 2.) yields 2 + -1 * (3 - 2) = 1 (correct)

------
adamnemecek
I'll leave this here [https://github.com/MikeInnes/diff-
zoo](https://github.com/MikeInnes/diff-zoo)

------
putzdown
This site undermines its own intent of providing a friendly and accessible
introduction to calculus through poor and erroneous grammar. An editor could
make it very useful. Some examples:

“OK, but how does calculus models change?” error: subject-verb number
agreement. Try “model”.

“The fundamental idea of calculus is to study change by studying
‘instantaneous’ change, by which we mean changes over tiny intervals of time.”

So, a change is a changes? Stay with the singular: “by which we mean a change
over a tiny interval of time.”

“It turns out that such [tiny] changes tend to be lots simpler than changes
over finite intervals of time.”

Now we have a logical problem. Isn’t the set of tiny intervals of time a
subset of finite intervals of time? In ordinary usage, “tiny” is finite,
surely; a tiny thing is not infinite. So how can a tiny-time change be simpler
than a finite-time change, when it is itself a finite-time change?

Errors of this sort lost my faith early on. I hope they’re corrected, because
the promise of the article is appealing.

~~~
bardworx
I’m my opinion, your comment is very self serving and reminds me of a quote
from Theodore Roosevelt:

“It is not the critic who counts; not the man who points out how the strong
man stumbles, or where the doer of deeds could have done them better. The
credit belongs to the man who is actually in the arena, whose face is marred
by dust and sweat and blood; who strives valiantly; who errs, who comes short
again and again, because there is no effort without error and shortcoming; but
who does actually strive to do the deeds; who knows great enthusiasms, the
great devotions; who spends himself in a worthy cause; who at the best knows
in the end the triumph of high achievement, and who at the worst, if he fails,
at least fails while daring greatly, so that his place shall never be with
those cold and timid souls who neither know victory nor defeat.”

If you really cared about the subject you could have sent the
creator/department a message with your constructive criticism. Instead, you
posted on the form how a few errors invalidates the entire work that took
significant amount of time.

------
jedahan
I want the inverse so badly. Call it 'Art for Beginners and Mathists'.

~~~
barry-cotter
If you want a book _Drawing On the Right Hand Side of the Brain_

[http://drawright.com/](http://drawright.com/)

If you want a video course

[http://drawabox.com/](http://drawabox.com/)

------
theNJR
Just last night I was complaining to someone about wanting this very thing. I
suppose I should now accuse Hacker News or listening in on my phone.

------
itronitron
Khan Academy is a better resource for being introduced to calculus.

The title of the course is unfortunately perpetuating the mistaken stereotype
that artists are not mathematically inclined. The incoming class my freshman
year of the art school at the university I attended had the highest median
Math SAT score of any of the other schools (i.e engineering, science) in the
university.

------
rayalez
Great link!

Now I only wish there were similar resources for Linear Algebra and
Probability/Statistics.

Any good recommendations?

~~~
zrobotics
For linear algebra, the YouTube channel 3blue1brown has a truly excellent
series on linear algebra; I found it was the best intro to the subject
avaliable.

------
nothis
Artists?

~~~
theoh
This comment is like a red rag to a bull for me, because why wouldn't some
artists want a basic grasp of calculus?

Bear in mind that a secondary education, for most people, involves learning
the basics of calculus. You might be thinking that all artists are bad at math
or too stupid to grasp calculus. You might be thinking that there are no
applications of calculus in art. I don't know. But given the role artists are
supposed to play in society, it seems pretty reasonable to expect that some of
them will have an interest in learning calculus to a high school level like a
significant fraction of the non-specialist public.

A concrete example of this might be an artist who is interested in systems
theory from an ecological perspective. Once you start talking about stocks and
flows (of fish or minerals or greenhouse gases) then calculus and differential
equations are really the next thing to tackle.

~~~
thethirdone
I interpreted the vague "Artists?" very differently from you. I thought it was
saying "What about this makes it appropriate for artists? It is not showing
off the beauty and simplicity of Calculus."

> You might be thinking that all artists are bad at math or too stupid to
> grasp calculus

I think labeling it as "for Beginners and Artists" is actually doing some of
that. I certainly would not make a book labeled for artists unless I thought
it would be particularly good for artists and from what I have seen this book
does not touch on creativity or beauty. Therefore it must be labeled as for
Artists because they can't learn calculus from existing books which might
imply "all artists are bad at math or too stupid to grasp calculus".

~~~
nothis
Thank you!

------
thisguyuknow
I noticed the use of jQWidgets and I wondered what people had to say about
that.

~~~
boikom
It's a UI Component library with a good quality at a good price

