Calculus for Beginners and Artists (2003)
www-math.mit.edu
www-math.mit.edu
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
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.
But it is weird not being shown the proofs for things like generalization for other situations. I studied Calculus while in engineering at college and we always got the proofs for the tricks we were using.
I don’t remember many proofs that I did, but I’m happy to know the maths that I still remember really works.
On the topic of infinitesmals, Gardner addresses their existence/utility as historically controversial. By providing both sides of the argument I was more able to understand how both parties were correct, and it has given me better insight as to why and when small numbers are ghosts and when they corporealise.
https://arxiv.org/abs/1811.03459
I certainly wouldn’t recommend this as a sole introductory textbook; it’s obviously produced on a shoestring budget (a LaTeX file with the default template sent over to a print shop) and is somewhat limited in many aspects compared to established calculus textbooks. But the pedagogical idea of focusing on differentials seems sound. I haven’t ever taught an introductory calculus course, but I think it seems entirely plausible that this approach would save some confusion for many students (this is something which could be tested empirically, if any math-ed researcher has the time and budget for a study).
It’s not clear why the guy’s ideas about evolutionary theory (which I know nothing about) have anything to do with his ideas about derivatives. From what I understand he’s a computer programmer and math teacher without extensive training in biology; I wouldn’t expect him to have any insight into evolutionary theory.
Seems like a good path to learn high level abstractions. As you progress in understanding you dig deeper.
Maybe I'm even still too much a sprite—but when I first learned anything about computers the first program I wrote was in a high level, simpler language. I wasn't moving bits around with explicit knowledge of where they were going.
Then again, maybe it's not a fair comparison on my part?
BTW computers have much cleaner abstractions than mathematics. e.g. the JLS defines Java independently of hardware; IEEE 754 is similar for fp arithmetic. There are specifications all over the place.
But my experience with mathematics is completely different - you have to understand the lower level to understand the next level.
In my personal journey, I started off with your perspective of just learning the higher levels that I directly needed. It was very difficult, but after heroic efforts, I made breakthroughs! After a while, I noticed these "breakthroughs" were mostly entirely to do with material from lower levels... So I went back to them. This happened again and again, going lower and lower. Now I'm basically re-doing high school maths.
Your point about specifications makes sense. I think that was a point that made some aspects of maths harder for me— contextual differences in notation all over the place. Thanks for the input.
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.)
> 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.
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.
I guess this is the one?
Gilbert Strang’s book is wonderful! Some will find it to be the “best book” to learn calculus, others won’t. No need to poo-poo others’ work just to value signal. (I’m sure the authors would appreciate any constructive feedback though!)
... the question is who? Is it them? Is it me? Is it you?
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.
Many of them use calculus in one way or another.
Check out the Bridges conference, http://bridgesmathart.org/
But it depends on the audience and more the depth of result required.
If I was to introduce calculus to non-mathematicians, I'd use almost no maths at at. Not even symbols! I'd use area and slope, with specific examples - probably just distance, velocity and acceleration.
I think you'd be able to reveal the magic relationship between slope and area with just that.
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.
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.
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.
(I take no responsability for any formatting errors. Math seems to render correctly, but probably some links are broken)
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.
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
Umm... whut?
This has me completely lost. I don't see how this can be geared towards beginners and artists.
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)
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
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.
x-b x-a
f(x) = A --- + B ---
a-b b-a
However, I don't find this kind of guide any good for learning math...
f(0) = A, f(1) = B,
then the unique solution is
f(t) = (1-t)A + tB
The more general version with a and b is just composing this with a transformation of [a, b] into [0, 1]
Relevant keywords: linear interpolation, affine combination, barycentric coordinates.
There are two variations of "linear function". To quote https://en.wikipedia.org/wiki/Linear_function:
> In mathematics, the term linear function refers to two distinct but related notions:
> - In calculus and related areas, a linear function is [...] a polynomial function of degree one or zero.
> - In linear algebra, mathematical analysis, and functional analysis, a linear function is a linear map.
I personally prefer the terminology "linear function" for the former definition (polynomials of degree 1 or 0), and "linear transform" for the latter definition (a linear map).
This is not “wrong”, but just a different convention.
g(t) = (1-t) f(a) + t f(b)
g(0) = (1) f(a) + (0) f(b) = f(a)
g(1) = (0) f(a) + (1) f(b) = f(b).
Now, if we want to use x instead of t, where x goes from a to b, we need to convert:
x = a <=> t = 0
x = b <=> t = 1
x-a
t = ---
b-a
f(x) = (1 - (x-a)/(b-a)) f(a) + ((x-a)/(b-a)) f(b)
= (1 + (x-a)/(a-b)) f(a) + ((x-a)/(b-a)) f(b) //since -(b-a) = a-b
= ((a-b)/(a-b) + (x-a)/(a-b)) f(a) + ((x-a)/(b-a)) f(b) //since 1 = (a-b)/(a-b)
= ((a-b+x-a)/(a-b)) f(a) + ((x-a)/(b-a)) f(b) //combining terms with common denominator
= ((x-b)/(a-b)) f(a) + ((x-a)/(b-a)) f(b)
* g(x) is something different: g(x) = (1-x) f(a) + x f(b), just by substituting in.
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.
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.
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.
I'm not sure. I'll rather learn and hope the dots will connect later than ignore the subject altogether.
Sorry if you already knew this, but it seemed like you were making a connection that wasn't there.
[0]:Could be called "numerical calculus" or "calculus of integration and differentiation", perhaps.
For example, The predicate calculus is invaluable for writing correct programs with mind-bogglingly large input domains, such as the set of all C++ programs.
Just "calculus" is short for "differential and integral calculus," which is about the local behavior of functions and the measure of objects in continuous spaces. Many people study this in high school (and earlier), since it's the underlying mathematical technology for much of the physical sciences. "Lambda calculus" is a logical system for manipulating symbols representing functions, and is indeed equivalent to Turing's symbol-manipulation system. So they both have to do with functions, but that's about where the similarity ends!
I hope I'm not coming off as brusque or anything, I just don't want you to spend loads of time on something that isn't what you think it is! That said, lots of programmers are surprisingly weak in (differential/integral) calculus, and could use a good dose of it anyway :)
Integral and differential calculus is not useful at all [1]. Compilers are not very computation-heavy, and where they do require grinding through math, the interesting problems are discrete math rather than continuous variable problems, which means that integral and differential calculus very rarely offer insights.
The math you usually want is algebra. Even then, the elementary algebra you learned in high school (or earlier) is usually sufficient. More complex domains in math are usually used for theoretical analysis and are less common in practice. Many dataflow optimizations can be described as monotonic functions over a semilattice, but they are rarely implemented as such. Type systems in functional languages can be understood more richly as applied category theory. Linear algebra is useful when multidimensional loop nests come into play.
[1] Unless maybe you're building a neural network compiler.
I saw someone reading this at a coffeeshop, it’s a new one:
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.
Here's an example, demonstrating how Kepler's Second Law is an expression of the conservation of angular momentum.
https://www.math.upenn.edu/~ghrist/calculus.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)
“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.
“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.
If you want a video course
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.
Now I only wish there were similar resources for Linear Algebra and Probability/Statistics.
Any good recommendations?
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.
> 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".