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| | |.---.-..----.| |--..-----..----. | | |.-----..--.--.--..-----.
| || _ || __|| < | -__|| _| | || -__|| | | ||__ --|
|___|___||___._||____||__|__||_____||__| |__|____||_____||________||_____|
on Gopher (inofficial)
URI Visit Hacker News on the Web
COMMENT PAGE FOR:
URI Calculus for Mathematicians, Computer Scientists, and Physicists [pdf]
hintymad wrote 2 hours 28 min ago:
Honest question: what kind of rigor and abstraction can help us apply
maths to solve problems? Don't get me wrong: I enjoy studying abstract
maths and was pretty good at it in school. It's just that when it comes
to what to study to make one a more effective problem solver in
engineering, I was wonder I can best allocate time. For instance, I
find studying probability models more helpful than studying the measure
theory when it comes to applied data science or statistics. I also find
studying books like Mathematical Methods for Physics and Engineering,
which focuses a lot more on intuition and applications than rigor, is
more effective for me than going pure math books.
Surac wrote 2 hours 30 min ago:
Thanks for the pdf. I am more in numeric but this pdf is a nice
reference for things complete unreadably on Wikipedia.
SilentM68 wrote 4 hours 6 min ago:
Without minimizing the quality or your book, I actually like subject
matter books that encompass prerequisite knowledge into the text
without forcing the reader to read another book in parallel (e.g.
Calculus for Machine Learning by Jason Brownlee or No Bullshit Guide to
Math & Physics by Ivan Savov). Though not saying that these books are
better, they appeal to my learning style a bit more. Learning
institutions tend to force students to take too many courses in
parallel when they should find a way to join the subjects, whenever
possible, without having to break the instruction into multiple
semesters, just to sell more books.
hilbert42 wrote 5 hours 10 min ago:
So far, I've only had a brief look at this book and what I've seen I
like very much.
Many of usâespecially those of us who aren't mathematically
giftedâlearn mathematics in ways that mostly involve procedures,
rules and mechanical manipulation rather than through a rigorous
step-by-step theoretical framework (well, anyway that's how I leaned
the subject).
Somehow I absorbed those foundations more by osmosis than though a full
understanding as my early teachers were more concerned with bashing the
basics into my head. Sure, later on when confronted with advanced
topics I was forced into more rigorous thinking but it wasn't uniform
across the whole field.
What I really like about this book is that it confronts people like me
who've already learned mathematics to a reasonably advanced level to
review those fundamental concepts. The subject of 'What is Calculus?'
doesn't start until Chapter 6, p223, and 'Differentiation' at Ch 8,
p261. Those first 200 or so pages not only provide a comprehensive and
clearly explained overview of those basic fundamentals but they ensure
the reader has good understanding of them before the main subject is
introduced.
I'd highly recommend this book either as a refresher or as an adjunct
to one's current learning.
fud101 wrote 7 hours 44 min ago:
Book looks like it could be AI generated, nothing remarkable.
fnord77 wrote 8 hours 5 min ago:
What about the Stanford Math 51 book?
btilly wrote 10 hours 1 min ago:
When I saw it was for computer scientists, I briefly hoped that it
would take the Big-O, little-o approach as Knuth recommended in 1998.
See [1] for a repost of Knuth's letter on the topic.
Sadly, no. It just seems to start with a gentle version of real
analysis, leading into basic Calculus.
URI [1]: https://micromath.wordpress.com/2008/04/14/donald-knuth-calcul...
rramadass wrote 5 hours 19 min ago:
The book does introduce/use O-notation under the section "Order of
Vanishing" in the "Limits and Continuity" chapter.
svat wrote 9 hours 27 min ago:
Related to your comment, not the original post/book:
- On Knuthâs idea (which seems good to me), see [1] but also the
last two comments by David Speyer.
- And see also [2] and [3] that seem to have tried teaching along
somewhat similar (but different) lines.
- See [4] for a further formalization of O notation.
(These links via comments I left to myself at [5] )
URI [1]: http://quomodocumque.wordpress.com/2012/05/29/knuth-big-o-ca...
URI [2]: http://cornellmath.wordpress.com/2007/08/28/non-nonstandard-...
URI [3]: http://texnicalstuff.blogspot.in/2011/05/big-o-notation-for-...
URI [4]: https://terrytao.wordpress.com/2022/05/10/partially-specifie...
URI [5]: https://shreevatsa.wordpress.com/2014/03/13/big-o-notation-a...
JosephK wrote 11 hours 24 min ago:
>Calculus is an important part of the intellectual tradition handed
down
to us by the Ancient Greeks.
No it isn't? It was discovered by Newton and Leibnitz. If they're
talking about Archimedes and integrals, I seem to recall his work on
that was only rediscovered through a palimpsest in the last couple of
decades and it contributed nothing towards Newton and Leibnitz's work.
DroneBetter wrote 10 hours 1 min ago:
Archimedes had functionally developed a method of integration (which
was how he obtained results like volume/surface area of a sphere, or
centre of mass of a hemisphere) in a manuscript that got lost to time
and then rediscovered in a palimpsest (pasted and written over with a
religious text)
danielam wrote 7 hours 55 min ago:
From [0]:
"Laying the foundations for integral calculus and foreshadowing the
concept of the limit, ancient Greek mathematician Eudoxus of Cnidus
(c. 390â337 BC) developed the method of exhaustion to prove the
formulas for cone and pyramid volumes.
"During the Hellenistic period, this method was further developed
by Archimedes (c. 287 â c. 212 BC), who combined it with a
concept of the indivisiblesâa precursor to
infinitesimalsâallowing him to solve several problems now treated
by integral calculus. In 'The Method of Mechanical Theorems' he
describes, for example, calculating the center of gravity of a
solid hemisphere, the center of gravity of a frustum of a circular
paraboloid, and the area of a region bounded by a parabola and one
of its secant lines."
[0]
URI [1]: https://en.wikipedia.org/wiki/Calculus
zozbot234 wrote 11 hours 10 min ago:
Calculus was actually pioneered by the Kerala School of
mathematicians in India during the European Middle Ages, several
centuries prior to Newton and Leibniz popularizing it in Europe. The
Indian texts were also quite well known to Europeans by that time, it
was nowhere close to an independent discovery.
danielam wrote 7 hours 57 min ago:
From [0] (emphasis mine):
"BhÄskara II (c. 1114â1185) was acquainted with some ideas of
differential calculus and suggested that the "differential
coefficient" vanishes at an extremum value of the function.[18] In
his astronomical work, he gave a procedure that looked like a
precursor to infinitesimal methods. [...] In the 14th century,
Indian mathematicians gave a non-rigorous method, resembling
differentiation, applicable to some trigonometric functions.
Madhava of Sangamagrama and the Kerala School of Astronomy and
Mathematics stated components of calculus. They studied series
equivalent to the Maclaurin expansions of [redacted] more than
two hundred years before their introduction in Europe. [...]
however, were not able to 'combine many differing ideas under the
two unifying themes of the derivative and the integral, show the
connection between the two, and turn calculus into the great
problem-solving tool we have today.'"
[0]
URI [1]: https://en.wikipedia.org/wiki/Calculus
gbacon wrote 12 hours 48 min ago:
> Die Mathematiker sind eine Art Franzosen: Redet man zu ihnen, so
übersetzen sie es in ihre Sprache, und alsbald ist es etwas ganz
anderes. (Goethe)
belter wrote 14 hours 43 min ago:
This one is a hard pass. The book needs tighter editing and more
rigorous reviewing.
It tries to serve all at once, but ends up in an awkward middle ground.
Not deep enough to function as a real analysis text for Mathematicians,
yet full of proofs that Scientists and Engineers do not care about,
while failing to deliver the kind of practical rigor, those groups need
when using calculus as a tool.
anthk wrote 15 hours 7 min ago:
Get Zenlisp running too [1] and just have a look on
how the (intersection) function it's defined.
Now you'll get things in a much easier way, for both programming and
math.
URI [1]: https://www.t3x.org/zsp/index.html
mathattack wrote 15 hours 12 min ago:
Seems like a lot of different audiences. My observation is this is
trying to cover 2 of the 3 common tracks:
1 - Proof based calculus for math majors
2 - Technique based calculus for hard science majors
3 - Watered down calculus for soft science and business majors (yes,
there are a few schools that are exceptions to this)
If he can pull off unifying 1 and 2, good for him!
lanstin wrote 14 hours 14 min ago:
I don't think they are unifiable, the aims and methods one needs to
learn are just too different. Limits of covering boxes and scaling
your epsilons and so on, stuff from Tao's class on analysis is far
away from being able to deal either non-trivial differential
equations or stability analysis. You can prove all sorts of things
about dense subspaces of Hilbert space and still get totally lost in
multiple scale analysis, and vice versa. (Ed: epsilon was spelled
espikon)
zkmon wrote 15 hours 23 min ago:
>> the authorâs wish to present ... mathematics, as intuitively and
informally as possible, without compromising logical rigor
The books in the West in general kept getting less rigorous, with time.
I don't see Asian or Russian books doing this. The audience getting
less receptive to rigor and wishing for more visuals and informal talk.
When they get to higher studies and research, would they be able to
cope with steep curve of more formalism and rigor?
matheusmoreira wrote 1 hour 22 min ago:
There's an old blog that addresses this topic: [1] It correlates
student loans with the destruction of academic integrity. The idea is
school administrators want to capture as many student loan dollars as
possible, and that means maximizing the number of enrolled students.
To that end, complexity, rigor and difficulty are all reduced as much
as possible. Students are prevented from failing, since if they fail
they might drop out, reducing profits. I even remember one article
which draws comparisons with russian education.
URI [1]: https://professorconfess.blogspot.com/
cyberax wrote 7 hours 4 min ago:
There are less formal math books in Russian. My absolute favorite
calculus textbook is Fikhtenholts's "A Course of Differential and
Integral Calculus". It is a bit less formal than many modern texts,
but somehow much more approachable.
My pet peeve about calculus books is that they almost always overlook
the importance of continuity. In some extreme cases, they even start
with infinitely small sequences, with some rather gnarly theorems
like BolzanoâWeierstrass theorem about converging subsequences.
I think this is a mistake. It's much easier to start with continuous
functions and build from there. Modern readers then can visualize the
epsilon-delta formulation of limits as "zooming in" on the function.
The "epsilon" is the height of the screen, and the "delta" is the
"zoom level" at which the function fragment fits on the screen.
And once you "get" the idea of continuity and function limits, the
other limit theorems just fall out naturally.
bmitc wrote 11 hours 18 min ago:
While world class, Russian mathematics is not known for rigor,
formality, or detail, so I'm dubious.
zozbot234 wrote 13 hours 45 min ago:
If you care about getting all the nitty gritty details of a
"rigorous" proof, maybe the quicker approach is to install Lean on
your computer and step through a machine-checked proof from Mathlib.
What you get from even the most heavyweight math books is still quite
far from showing you all the steps involved.
actinium226 wrote 14 hours 30 min ago:
> The books in the West in general kept getting less rigorous, with
time.
I wonder if it's because more people are going to college who would
have otherwise gone to a vocational or trade school? If the audience
expands to include people who might not have studied calculus had
they not chosen to go to college, I feel like textbooks have to
change to accommodate that.
nabla9 wrote 14 hours 43 min ago:
> Russian books doing this.
Mathematics: Its Content, Methods and Meaning by
A.D. Aleksandrov, A.N. Kolomogorov, M.A. Lavrentâev,.. [1] It's
still a masterpiece. Originally published in 1962 in 3 volumes. The
English translation has all in one.
URI [1]: https://www.goodreads.com/book/show/405880.Mathematics
wirrbel wrote 11 hours 30 min ago:
is that like a Landau Lifshitz for Math?
rramadass wrote 4 hours 57 min ago:
It is very good and has a succinct coverage of a broad range of
topics from Mathematics; just the right understandable level of
rigor without being overwhelming.
Published by Dover Publications and hence quite affordable. See
ToC at
URI [1]: https://store.doverpublications.com/products/97804864091...
nextos wrote 10 hours 42 min ago:
No, it's just a gentle overview.
elcapitan wrote 14 hours 52 min ago:
This may be a stupid question, but what do people usually mean when
they refer to a mathematical text as being "rigorous"? Does it mean
that everything is strictly proof-based rather than
application-oriented?
zorked wrote 10 hours 55 min ago:
Rigorous = a pain in the ass to learn from, but you gain imaginary
points for the pain.
godelski wrote 8 hours 21 min ago:
It certainly is more painful, but it is more beneficial. It is
also harder to teach, but I stand by my claim.
I'll quote Poincare:
Math is not about the study of numbers, but the relationships
between them.
The difficulty and benefit of the rigor is the abstraction. Math
is all about abstraction.
The abstraction makes it harder to understand how to apply these
rules, but if one breaks through this barrier one is able to
apply the rules far more broadly.
----
Let's take the Fundamental Theorem of Calculus as an example[0]:
f'(x) = lim_{h->0} {f(x + h) - f(x)} / {h}
Take a moment here and think about it's form. Are there
equivalent ones? What do each of these symbols mean?
If you actually study this, you may realize that there are an
infinite number of equations that allow us to describe a secant
line. So why this one? Is there something special? (hint: yes)
Let's call that the "forward derivative". Do you notice that
through the secant line explanation that the "backward
derivative" also works? That is
f'(x) = lim_{h->0} {f(x) - f(x - h)} / {h}
You may also find the symmetric derivative too!
f'(x) = lim_{h->0} {f(x + h) - f(x - h)} / {2h}
In fact, you see these in computational programs all the time!
The symmetric derivative even has the added advantage of error
converging at an O(n^2) rate instead of O(n)! Yet, are these the
same? (hint: no)
Or tell me about the general case of
f'(x) = lim_{h->0} {f(x + ah) - f(x + bh)}/{(a-b)h}
I'm betting that most classes that went through deriving the
derivative did not answer these questions for you (or you don't
remember). Yet, had you, you would have instantly known how to do
numerical differentiation and understand the limits, pitfalls,
and other subjects like FEM (Finite-Element Methods) or
Computational Methods would be much easier for those who take
them.
----
Yet, I still will say that this is much harder to teach. Math is
about abstraction, and abstraction is simply not that easy. But
abstraction is incredibly powerful, as I hope every programmer
can intuitively understand. After all, all we do is deal with
abstractions. One can definitely be overly abstract and it will
make a program uninterpretable for most, but one also can make a
program have too little abstraction, which in that case we end up
writing a million variations of the same thing, taking far more
lines to write/read, and making the program too complex. There is
a balance, but I'd argue that if one is able to understand
abstraction that it is far easier to reduce abstraction than it
is to abstract.
This is just a tiny taste of what rigor holds. You are absolutely
right to be frustrated and annoyed, but I hope you understand
your conclusion is wrong. Unless you're Ramanujan, every
mathematician has spent hours banging their head against a
literal or metaphorical wall (or both!). The frustration and pain
is quite real! But it is absolutely worth it.
[0] Linking an EpsilonDelta video that covers this exact example
in more detail
URI [1]: https://www.youtube.com/watch?v=oIhdrMh3UJw
zorked wrote 1 hour 49 min ago:
You are arguing for rigor, not for its didactics. Those are
different.
> had you, you would have instantly known how to do numerical
differentiation and understand the limits, pitfalls, and other
subjects like FEM
No, you wouldn't. You would also learn things out of order. You
would be exposed to things without understanding why you are
learning them. People who argue this usually learn things the
intuitive way (whether from rigorous material or not - what
goes on in their mind isn't rigorous), and then they go back
and reassess the rigor in the light of that. Then they pretend
that they learned from the rigorous exposition. No, they
didn't.
It is totally fine to iterate. Learn non-rigorously. Go back to
it and iterate on rigor later. As it becomes necessary, and if
it ever becomes necessary for your field.
> Unless you're Ramanujan, every mathematician has spent hours
banging their head against a literal or metaphorical wall
Particularly if you are learning from "rigorous" material. But
then you go watch some YouTube videos to make up for the
absence of didactics in your textbook.
I mean, why don't we just throw Bourbaki books at freshmen and
let them sort it out without classes? They are maximally
rigorous, therefore maximally great to learn from, right?
godelski wrote 1 hour 1 min ago:
> I mean, why don't we just throw Bourbaki books at freshmen
and let them sort it out without classes?
The Bourbaki group was quite famous for wanting to
restructure math education. Teaching many things that are
considered advanced to children. Despite not sticking around
we see elements of resurgence and effectiveness.
So I'm not sure your argument of "out of order" is accurate.
The order is what we make. There's no clear optimal way to
teach math. Your argument hinges on that. You might argue
that the current status quo is working, so why disrupt it,
and I'll point around asking if you really think it's so
effective when many demonstrate a lack of understanding all
around us. That so many struggle with calculus is evidence
itself. We need not even acknowledge that there are many
children who learn this (and let's certainly not admit that
it's far more common for them to learn it in unconventional
ways).
> let them sort it out without classes?
To suggest I'm arguing for the elimination of educators is
beyond silly. I'd hope the caliber of your arguments would
match that of your diction.
layer8 wrote 12 hours 42 min ago:
Not necessarily proof-heavy, but at least with formally rigorous
definitions and theorems.
actinium226 wrote 14 hours 32 min ago:
Generally that's what it means. And also when proofs are presented,
a rigorous book will go through it fully, whereas a less rigorous
one might just sketch out the main ideas of the proof and leave out
some of the nitty gritty details (i.e. it's less rigorous to talk
about "continuity" as "you can draw it without lifting the pen" as
compared to the epsilon-delta definition, but epsilon-delta is
pretty detailed and for intro calculus for non-mathematicians you
don't really need it).
mike50 wrote 10 hours 12 min ago:
This is the reason that everyone at my university said to just
take the Applied version of Calculus 1 and 2 t avoid the proofs.
kardianos wrote 15 hours 9 min ago:
I agree with this. But I don't see the students rejecting this, but
the education degreed peoples who choose texts and the publishers
want to make all learning for all people. This is foolish. Most
people don't need to know calculus. And if you do learn it, do so
with rigor so you actually learn it and not just the appearance of
it, which is much much worse.
rramadass wrote 5 hours 42 min ago:
> Most people don't need to know calculus.
People should have at the minimum a conceptual idea of Calculus. A
good motivation is Everyday Calculus: Discovering the Hidden Math
All around Us by Oscar E. Fernandez - [1] > And if you do learn it,
do so with rigor so you actually learn it
This is not strictly necessary for everybody. The conceptual ideas
are what are important; else you are merely doing "plug-and-chug"
Maths without any understanding. You need to focus on rigor only
based on your needs and at your own pace. Concepts come first
Formalism comes second.
A good example; In the Principia Newton actually uses the phrase
Quantity of Motion for what we define today as Momentum. The phrase
is evocative and beautifully captures the main concept instead of
the bland p = m x v definition which though correct and needed for
calculations conveys no mental imagery.
In Mathematics one should always approach a concept/idea from
multiple perspectives including (but not limited to) Imagination,
Conceptual, Graphical, Symbolic, Relationship, Applications,
Definition/Theorem/Proof.
URI [1]: https://press.princeton.edu/books/paperback/9780691175751/...
godelski wrote 8 hours 5 min ago:
> Most people don't need to know calculus
I don't like this line of argument. It applies to many things, many
of which we'd laugh at for suggesting.
Most people don't need to know how to read. Most people don't need
to know how to add. Most people don't need to know how to use a
computer. The foolishness of these statements are all subjective
and based on what one believes one "needs". Yet, I have no doubt
all of these things can improve peoples lives.
I'd argue the same with calculus. While I don't compute derivatives
and integrals every day[0], I certainly use calculus every day.
That likely sounds weird, but it is only because one thinks that
math and computation is the same. When I drive I use calculus as
I'm thinking about my rates of change, not only my velocity.
Understanding different easing functions[1] I am able to create a
smoother ride, be safer, drive faster, and save fuel. All at the
same time!
The magic of the rigor is often lost, but the magic is abstraction.
That's what we've done here with the car example. I don't need to
compute numbers to "do math", I only need to have an abstract
formulation. To understand that multiple variables are involved and
there are relationships between them, and understanding that there
are concepts like a rate of change, the rate of change of the rate
of change, and even the rate of change of the rate of change of the
rate of change! (the jerk!)[2].
That's still math. It may not be as rigorous, but a rigorous
foundation gives you a greater ability to be less rigorous at times
and take advantage of the lessons.
So yes, most people "don't need calculus" but learning it can give
them a lot of power in how to think. This is true for much of
mathematics. You may argue that this is not how it is taught, but
with that I'll agree. The inefficiency of how it is taught is
orthogonal to the utility of its lessons.
[0] Is a physicist not doing math just when they do symbol
manipulation? I can tell you with great confidence, and experience,
that much of their job is doing math without the use of numbers. It
is about deriving formulations. Relationship! [1]
URI [1]: https://easings.net/
URI [2]: https://en.wikipedia.org/wiki/Jerk_%28physics%29
altmanaltman wrote 4 hours 10 min ago:
I get the argument you're making but that's a bit like saying
cavemen used to do calculus as they hunt, which is a valid way of
looking at this maybe but they didn't really "use calculus" just
intuition. Simillarly, when learning calculus, most people do not
do so at a driving course, they do it in the classroom.
If you're willing to stretch the definition of what "using" maths
is then it can apply to everything and that devalues the concept
as a whole. I'm not on the toilet, I'm doing calculus!
godelski wrote 3 hours 53 min ago:
I understand that interpretation but it's different from what I
meant.
The difference may be in two different cavemen. One throws his
spear on intuition alone. The other is thinking about the speed
he throws, how the animal moves, the wind, and so on. There is
a formulation, though not as robust as you'd see in a physics
book.
> the definition of what "using" maths is then it can apply
to everything
In a sense yes.
Math is a language, or more accurately a class of languages. If
you're formulating your toilet activities, then it might be
math. But as you might gather, there's nuance here.
I quoted Poincaré in another comment but I'll repeat here as I
think it may help reduce confusion (though may add more)
Math is not the study of numbers, but the relationships
between them.
Or as the category people say "the study of dots and arrows".
Anything can be a dot, but you need the arrows
altmanaltman wrote 3 hours 23 min ago:
Yeah, I do understand your point of view. I'm just doubting
if it applies universally, like you may superimpose that
assumption on the thinking caveman, but is the thinking
caveman really doing the same?
Yes, technique is one thing, but being really good at
throwing spears doesn't make you really good at math, is my
argument. And most people will encounter maths in a formal
setting while lacking the broader perspective that everything
is technically "math".
Yet, we need to see the argument from the common person's
view, if we're talking about calculus and learning in the
traditional sense. The view you stated is quite esoteric and
doesn't generalize well in this setting imo.
It's like a musician saying they see music in every action,
but to most non-musicians (even if the stated thing is kind
of true) that doesn't make a lot of sense etc.
godelski wrote 1 hour 18 min ago:
> but is the thinking caveman really doing the same?
Are you projecting a continuous space onto a binary one?
You'll need to be careful about your threshold and I'm
pretty sure it'll just make everything I said complete
nonsense. If you must use a discrete space then allocate
enough bins to recognize that I clearly stated there's a
wide range of rigor. Obviously the caveman example is on
the very low end of this.
> It's like a musician saying they see music in every
action, but to most non-musicians (even if the stated thing
is kind of true) that doesn't make a lot of sense etc.
Exactly. So ask why the musician, who is certainly more
expert than the non-musician has a wider range? They have
expertise in the matter, are you going to just ignore that
simply because you do not understand? Or are you going to
try to understand?
The musician, like the mathematician, understands that
every sound is musical. If you want to see this in action
it's quite enlightening[0]. I'm glad you brought up that
comparison because I think it can help you understand what
I really mean. There is depth here. Every human has access
to the sounds but the training is needed to put them
together and make these formulations. Benn here isn't
exactly being formal writing his music using a keyboard and
formalizing it down to musical notes on a sheet (though
this is something I know he is capable of).
But maybe I should have quoted Picasso instead of Poincaré
Learn the rules like a pro, so you can break them like an
artist.
His abstract nature to a novice looks like something they
could do (Jackson Pollocks is a common example) but he
would have told you he couldn't have done this without
first mastery of the formal art first.
I know this is confusing and I wish I could explain it
better. But at least we can see that regardless of the
field of expertise we find similar trains of thought. Maybe
a bridge can be created by leveraging your own domain of
expertise
Maybe I can put it this way: gibberish is more intelligible
when crafted by someone who can already speak.
[0]
URI [1]: https://m.youtube.com/shorts/ZLPCGEbHoDI
DrSAR wrote 12 hours 59 min ago:
Not sure I agree with 'appearance [...] is much worse'.
Given the choice between a class room of first years who believe
(in themselves and) an appearance of calculus knowledge or a room
of scared undergrads that recoil from any calculus-inspired
argument they 'have never learnt it properly', I'll take the
former. I can work with that much more easily. Sure, some things
might break - but what's the worst that can happen?
We'll sort out the rigour later while we patch the bruises of
overextending some analogies.
mjburgess wrote 15 hours 13 min ago:
Nope, but mathematics research is one of the most rarefied fields
being extremely difficlt to get into, hard to get money, etc. --
(this is my understanding, at least). Progress is made here by people
who, aged 10 are already showing signs of capability.
There's not much need for a large amount of PhD places, and funding,
for pure mathematics research.
Likewise, on the applied side, "calculus" now as a pure thing has
been dead alone time. Gradients are computed with algorithms and
numerical approximations, that are better taught -- with the formal
stuff maintained via intuition.
I'm much more open to the idea that the west has this wrong, and we
should be more focused on developing the applied side after spending
the last century overly focused on the pure
kalx wrote 15 hours 37 min ago:
How much math skills do you need to appreciate this book?
nightshift1 wrote 14 hours 29 min ago:
the Postscript at the end says:
While not every student is expected to read the book sequen-
tially cover to cover, it is important to have the details in one
place.
Calculus is not a subject that can be learned in one pass. Indeed,
this
book nearly assumes readers have already had a year of calculus, as
had
the students of MAT 157Y. I hope this book will grow with its
readers,
remaining both readable and informative over multiple traversals, and
that it provides a useful bridge between current calculus texts and
more
advanced real analysis texts.
analog31 wrote 15 hours 23 min ago:
My first impression, paging through it, is that it's at a somewhat
higher level than the typical college calculus course.
garyfirestorm wrote 15 hours 40 min ago:
Is there a hard copy to purchase? I canât seem to find it anywhere.
CamperBob2 wrote 15 hours 42 min ago:
That's a pretty diverse audience. Is this .pdf supposed to be a
one-size-fits-all effort?
analog31 wrote 15 hours 31 min ago:
I'm probably dating myself, but at my college, there was one calculus
course for everybody. But also, a lot of the students in those areas
had overlapping or double majors. For instance I majored in math and
physics.
Perhaps the bigger question is whether it's at the right level of
difficulty for the audience.
anikom15 wrote 15 hours 20 min ago:
I think there are usually two: Calculus for scientists and
engineers which is analytical and has lots of symbols, and Calculus
for everyone else which is more practical.
Math majors might have their own. I also know they end up taking
complex Calculus.
beezle wrote 14 hours 0 min ago:
Usually engineering/math calc and then a much less rigorous
business/arts&crafts calc for the rest.
analog31 wrote 15 hours 7 min ago:
Thinking about it, ours was a small college -- 2500 students. So
there may have been a practical reason for everybody taking the
same math courses. They were taught more as "service" courses for
the sciences and engineering than as theoretical math courses.
And the students who didn't need calculus typically satisfied
their math requirement with a statistics course.
Complex analysis and real analysis were among the higher-level
courses, attended mostly by math majors, with the proviso that
there were a lot of double majors. That was where it got
interesting.
The requirements for the physics major were only a handful of
math credits shy of the math major.
dogmatism wrote 7 hours 15 min ago:
>The requirements for the physics major were only a handful of
math credits shy of the math major.
lol, that's how I ended up with a math major. Got lost in the
physics (realized I had no intuition for what was actually
happening, just manipulating equations) took a couple extra
courses, and boom! Math!
qntty wrote 15 hours 55 min ago:
Writing a calculus book that's more rigorous than typical books is hard
because if you go too hard, people will say that you've written a real
analysis book and the point of calculus is to introduce certain
concepts without going full analysis. This book seems to have at least
avoided the trap of trying to be too rigorous about the concept of
convergence and spending more time on introducing vocabulary to talk
about functions and talking about intersections with linear algebra.
impendia wrote 3 hours 41 min ago:
> The point of calculus is...
As a math professor who has taught calculus many times, I'd say there
are many different things one could hope to learn from a calculus
course. I don't think the subject distills well to a single point.
One unusual feature of calculus is that it's much easier to
understand at a non-rigorous level than at a rigorous level. I
wouldn't say this is true of all of math. For example, if you want to
understand why the quadratic formula is true, an informal explanation
and a rigorous proof would amount to approximately the same thing.
But, when teaching or learning calculus, if you're willing to say
that "the derivative is the instantaneous rate of change of a
function", treat dy/dx as the fraction which it looks like (the chain
rule gets a lot easier to explain!), and so on, you can make a lot of
progress.
In my opinion, the issue with most calculus books is that they don't
commit to a rigorous or to a non-rigorous approach. They are usually
organized around a rigorous approach to the subject, but then watered
down a lot -- in anticipation that most of the audience won't care
about the rigor.
I believe it's best to choose a lane and stick to it. Whether that's
rigorous or non-rigorous depends on your tastes and interests as a
learner. This book won't be for everybody, but I'd call that a
strength rather than a weakness.
msla wrote 28 min ago:
The rigorous form of the non-rigorous version is non-standard
analysis: There really are tiny little numbers we can manipulate
algebraically and we don't need the epsilon-delta machinery to do
"real math". It's so commonsensical that both Newton and Leibniz
invented it in that form before rigor became the fashion, and the
textbook "Calculus Made Easy" was doing it that way in 1910, a
half-century before Robinson came along and showed us it was
rigorous all along. [1]
URI [1]: https://calculusmadeeasy.org/
URI [2]: https://en.wikipedia.org/wiki/Calculus_Made_Easy
zozbot234 wrote 12 min ago:
> The rigorous form of the non-rigorous version is non-standard
analysis
This is quite overstated. There are other approaches to
infinitesimals such as synthetic differential geometry (SDG aka.
smooth infinitesimal analysis) that are probably more intuitive
in some ways and less so in others. SDG infinitesimals lose the
ordering of hyperreals in non-standard analysis and force you to
use some non-classical logic (intuitively, smooth infinitesimals
are "neither equal nor non-equal to 0"), but in return you gain
nilpotency (d^n = 0 for any infinitesimal d) which is often
regarded as a desirable feature in informal reasoning.
JJMcJ wrote 14 hours 27 min ago:
Anyway you've already got Apostol - if it's just calculus as such get
an older edition. Modern ones have extra goodies like linear algebra
but have modern text book pricing (cries softly in $150/volume).
throwaway81523 wrote 10 hours 38 min ago:
Spivak's book is still good too.
tzs wrote 13 hours 8 min ago:
Getting an old enough edition of Apostol's "Calculus" to not
include linear algebra might be a bit challenging. Linear algebra
was added to both volumes in their second editions, which came out
in 1967 for volume 1 and 1969 for volume 2.
The second editions are still the current edition, so no worry that
you might be missing out on something if you go with used copies.
If you do want new copies (maybe you can't find used copies or they
are in bad shape) take a look at international editions.
A new copy of the international edition for India from a seller in
India on AbeBooks is around $15 per volume plus around $19 shipping
to the US. Same contents as the US edition but paperback instead of
hardback, smaller pages, and rougher paper. (International editions
also often replace color with grayscale but that's not relevant in
this case because Apostol does not use color)).
You can also find US sellers on AbeBooks that has imported an
international edition. That will be around $34 but usually with
free shipping.
SanjayMehta wrote 9 hours 8 min ago:
Indian editions sometimes have different question sets to prevent
students from using them in other countries' coursework.
They also have a hologram sticker alongside a printed warning
that they are not for sale or export outside of India, Nepal and
a couple of other countries.
JJMcJ wrote 6 hours 34 min ago:
I think those restrictions apply only to retail sellers in
those countries, not to purchasers or used stores.
JJMcJ wrote 12 hours 37 min ago:
Thanks for the info on cheaper editions, not important to me but
to others in USA it might be a big help.
marai2 wrote 5 hours 55 min ago:
i bought a compilers book that was an Indian edition. The paper
and print quality was so bad (like smudgy) that I could not
read it and I didnât think I was particularly picky about
this. Not sure if I just got unlucky or if this is generally
true?
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