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.
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).
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.
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.
>> the author’s 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?
> 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.
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?
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).
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)
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.
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.
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.
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!
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.
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
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)
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.
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.
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.
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.
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.
>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.
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.
"Bhāskara II (c. 1114–1185) 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.'"
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)
"Laying the foundations for integral calculus and foreshadowing the concept of the limit, ancient Greek mathematician Eudoxus of Cnidus (c. 390–337 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 – c. 212 BC), who combined it with a concept of the indivisibles—a precursor to infinitesimals—allowing 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."
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.
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).
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.
Thanks for the info on cheaper editions, not important to me but to others in USA it might be a big help.
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.
Spivak's book is still good too.
>> the author’s 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?
> Russian books doing this.
Mathematics: Its Content, Methods and Meaning by A.D. Aleksandrov, A.N. Kolomogorov, M.A. Lavrent’ev,.. https://www.goodreads.com/book/show/405880.Mathematics
It's still a masterpiece. Originally published in 1962 in 3 volumes. The English translation has all in one.
is that like a Landau Lifshitz for Math?
No, it's just a gentle overview.
> 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.
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?
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).
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.
Rigorous = a pain in the ass to learn from, but you gain imaginary points for the pain.
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:
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]: 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
You may also find the symmetric derivative too! 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
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 https://www.youtube.com/watch?v=oIhdrMh3UJw
Not necessarily proof-heavy, but at least with formally rigorous definitions and theorems.
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.
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.
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] https://easings.net/
[2] https://en.wikipedia.org/wiki/Jerk_%28physics%29
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.
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
While world class, Russian mathematics is not known for rigor, formality, or detail, so I'm dubious.
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!
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)
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 https://micromath.wordpress.com/2008/04/14/donald-knuth-calc... 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.
Related to your comment, not the original post/book:
- On Knuth’s idea (which seems good to me), see http://quomodocumque.wordpress.com/2012/05/29/knuth-big-o-ca... but also the last two comments by David Speyer.
- And see also http://cornellmath.wordpress.com/2007/08/28/non-nonstandard-... and http://texnicalstuff.blogspot.in/2011/05/big-o-notation-for-... that seem to have tried teaching along somewhat similar (but different) lines.
- See https://terrytao.wordpress.com/2022/05/10/partially-specifie... for a further formalization of O notation.
(These links via comments I left to myself at https://shreevatsa.wordpress.com/2014/03/13/big-o-notation-a...)
What about the Stanford Math 51 book?
> 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)
That's a pretty diverse audience. Is this .pdf supposed to be a one-size-fits-all effort?
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.
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.
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.
Usually engineering/math calc and then a much less rigorous business/arts&crafts calc for the rest.
Is there a hard copy to purchase? I can’t seem to find it anywhere.
How much math skills do you need to appreciate this book?
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.
My first impression, paging through it, is that it's at a somewhat higher level than the typical college calculus course.
Get Zenlisp running too https://www.t3x.org/zsp/index.html 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.
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.
>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.
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.
From [0] (emphasis mine):
"Bhāskara II (c. 1114–1185) 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] https://en.wikipedia.org/wiki/Calculus
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)
From [0]:
"Laying the foundations for integral calculus and foreshadowing the concept of the limit, ancient Greek mathematician Eudoxus of Cnidus (c. 390–337 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 – c. 212 BC), who combined it with a concept of the indivisibles—a precursor to infinitesimals—allowing 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] https://en.wikipedia.org/wiki/Calculus