#calculus — Public Fediverse posts

Alright, future engineers!
**Optimization** uses derivatives to find the max or min values of a function (e.g., max profit, min cost).
Ex: Set `f'(x)=0` to find critical points.
Pro-Tip: Always check endpoints and domain boundaries!
#Calculus #Optimization #STEM #StudyNotes

Alright, future engineers!
**The Integral** calculates the total accumulation of a quantity or the area under a curve.
Ex: Area under `y=f(x)` from `a` to `b` is `∫_a^b f(x) dx`.
Pro-Tip: If the derivative tells you the rate of change, the integral tells you the *total change* or *sum*!
#Calculus #Integration #STEM #StudyNotes

From @joannechocolat

#Maria_Gaetana_Agnesi (1718 – 1799) was an #Italian mathematician, philosopher, theologian, & humanitarian. She was the first woman to write a #mathematics handbook and the first woman appointed as a mathematics professor at a university.

Maria was recognized early on as a child prodigy; she could speak both Italian and French at five years of age. By her eleventh birthday, she had also learned Greek, Hebrew, Spanish, German, and Latin, and was referred to as the "Seven-Tongued Orator".

When she was nine years old, she composed & delivered an hour-long speech in #Latin to some of the most distinguished intellectuals of the day. The subject was "A Woman's Right to be Educated."

As an adolescent she suffered convulsions, which were attributed to excessive studying, but when it was noted that her health deteriorated when she was not allowed access to books, she was permitted to continue.

She never married, and although her father would not allow her to enter a convent, lived in semi-seclusion throughout most of her adult life, devoting herself to her studies. After her father's death in 1752 she devoted herself to the poor, giving away the gifts she had received.

She is credited with writing the first book discussing both differential and integral #calculus and was a member of the faculty at the University of Bologna, although she never served. In spite of her family's wealth, she died in poverty and was buried in a pauper's grave.

#CelebratingWomen

From @joannechocolat

#Maria_Gaetana_Agnesi (1718 – 1799) was an #Italian mathematician, philosopher, theologian, & humanitarian. She was the first woman to write a #mathematics handbook and the first woman appointed as a mathematics professor at a university.

Maria was recognized early on as a child prodigy; she could speak both Italian and French at five years of age. By her eleventh birthday, she had also learned Greek, Hebrew, Spanish, German, and Latin, and was referred to as the "Seven-Tongued Orator".

When she was nine years old, she composed & delivered an hour-long speech in #Latin to some of the most distinguished intellectuals of the day. The subject was "A Woman's Right to be Educated."

As an adolescent she suffered convulsions, which were attributed to excessive studying, but when it was noted that her health deteriorated when she was not allowed access to books, she was permitted to continue.

She never married, and although her father would not allow her to enter a convent, lived in semi-seclusion throughout most of her adult life, devoting herself to her studies. After her father's death in 1752 she devoted herself to the poor, giving away the gifts she had received.

She is credited with writing the first book discussing both differential and integral #calculus and was a member of the faculty at the University of Bologna, although she never served. In spite of her family's wealth, she died in poverty and was buried in a pauper's grave.

#CelebratingWomen

From @joannechocolat

#Maria_Gaetana_Agnesi (1718 – 1799) was an #Italian mathematician, philosopher, theologian, & humanitarian. She was the first woman to write a #mathematics handbook and the first woman appointed as a mathematics professor at a university.

Maria was recognized early on as a child prodigy; she could speak both Italian and French at five years of age. By her eleventh birthday, she had also learned Greek, Hebrew, Spanish, German, and Latin, and was referred to as the "Seven-Tongued Orator".

When she was nine years old, she composed & delivered an hour-long speech in #Latin to some of the most distinguished intellectuals of the day. The subject was "A Woman's Right to be Educated."

As an adolescent she suffered convulsions, which were attributed to excessive studying, but when it was noted that her health deteriorated when she was not allowed access to books, she was permitted to continue.

She never married, and although her father would not allow her to enter a convent, lived in semi-seclusion throughout most of her adult life, devoting herself to her studies. After her father's death in 1752 she devoted herself to the poor, giving away the gifts she had received.

She is credited with writing the first book discussing both differential and integral #calculus and was a member of the faculty at the University of Bologna, although she never served. In spite of her family's wealth, she died in poverty and was buried in a pauper's grave.

#CelebratingWomen

From @joannechocolat

#Maria_Gaetana_Agnesi (1718 – 1799) was an #Italian mathematician, philosopher, theologian, & humanitarian. She was the first woman to write a #mathematics handbook and the first woman appointed as a mathematics professor at a university.

Maria was recognized early on as a child prodigy; she could speak both Italian and French at five years of age. By her eleventh birthday, she had also learned Greek, Hebrew, Spanish, German, and Latin, and was referred to as the "Seven-Tongued Orator".

When she was nine years old, she composed & delivered an hour-long speech in #Latin to some of the most distinguished intellectuals of the day. The subject was "A Woman's Right to be Educated."

As an adolescent she suffered convulsions, which were attributed to excessive studying, but when it was noted that her health deteriorated when she was not allowed access to books, she was permitted to continue.

She never married, and although her father would not allow her to enter a convent, lived in semi-seclusion throughout most of her adult life, devoting herself to her studies. After her father's death in 1752 she devoted herself to the poor, giving away the gifts she had received.

She is credited with writing the first book discussing both differential and integral #calculus and was a member of the faculty at the University of Bologna, although she never served. In spite of her family's wealth, she died in poverty and was buried in a pauper's grave.

#CelebratingWomen

From @joannechocolat

#Maria_Gaetana_Agnesi (1718 – 1799) was an #Italian mathematician, philosopher, theologian, & humanitarian. She was the first woman to write a #mathematics handbook and the first woman appointed as a mathematics professor at a university.

Maria was recognized early on as a child prodigy; she could speak both Italian and French at five years of age. By her eleventh birthday, she had also learned Greek, Hebrew, Spanish, German, and Latin, and was referred to as the "Seven-Tongued Orator".

When she was nine years old, she composed & delivered an hour-long speech in #Latin to some of the most distinguished intellectuals of the day. The subject was "A Woman's Right to be Educated."

As an adolescent she suffered convulsions, which were attributed to excessive studying, but when it was noted that her health deteriorated when she was not allowed access to books, she was permitted to continue.

She never married, and although her father would not allow her to enter a convent, lived in semi-seclusion throughout most of her adult life, devoting herself to her studies. After her father's death in 1752 she devoted herself to the poor, giving away the gifts she had received.

She is credited with writing the first book discussing both differential and integral #calculus and was a member of the faculty at the University of Bologna, although she never served. In spite of her family's wealth, she died in poverty and was buried in a pauper's grave.

#CelebratingWomen

Alright, future engineers!
The **Derivative** measures a function's instantaneous rate of change – the slope of its tangent line.
Ex: `d/dx (x^3) = 3x^2` (Power Rule!)
Pro-Tip: A positive derivative means the function is increasing; negative means decreasing!
#Calculus #Derivatives #STEM #StudyNotes

@yaxu that's Weierstrass's Monster!

#maths #calculus

My calculus book dates from the seventies, and introduces derivatives at page 90.

#Calculus #Math #Maths

https://youtube.com/shorts/3o3zcMTBPZc

Differential Analysis on Complex Manifolds (Graduate Texts in Mathematics, 65) by Raymond O. Wells (PDF)
Author: Raymond O. Wells
File Type: PDF
Download at https://sci-books.com/differential-analysis-on-complex-manifolds-graduate-texts-in-mathematics-65-144192535x/
#Calculus, #RaymondO.Wells

Another one bites the dust...

I've tutored a couple of students who decided to not continue with me, but I think it is the class that is not suited to them more than me.

They had a handout that talked about the domain of a function in very general terms (no mention of avoiding division by zero, or square roots, etc.), but the exercises they had about domains were very specific (yep, we had to handle division by zero, square roots, and more exotic stuff). To put it differently, the exercises they had required much more facility with the material than the class handout they showed me.

Nominally, I was tutoring Calculus, but the material they had (both the handout and the exercises) felt like a revision that reached back to Algebra. And they were not completely at ease with this. 😬

I can't imagine what's going to happen if they stay in this Calculus class and start having to deal with limits, etc.

I should note that this is pretty rare that I get students who present me with a case like this. (Even though, I've just had a similar case with SAT. It is still rare.)

#tutoring #algebra #calculus

Another one bites the dust...

I've tutored a couple of students who decided to not continue with me, but I think it is the class that is not suited to them more than me.

They had a handout that talked about the domain of a function in very general terms (no mention of avoiding division by zero, or square roots, etc.), but the exercises they had about domains were very specific (yep, we had to handle division by zero, square roots, and more exotic stuff). To put it differently, the exercises they had required much more facility with the material than the class handout they showed me.

Nominally, I was tutoring Calculus, but the material they had (both the handout and the exercises) felt like a revision that reached back to Algebra. And they were not completely at ease with this. 😬

I can't imagine what's going to happen if they stay in this Calculus class and start having to deal with limits, etc.

I should note that this is pretty rare that I get students who present me with a case like this. (Even though, I've just had a similar case with SAT. It is still rare.)

#tutoring #algebra #calculus

Another one bites the dust...

I've tutored a couple of students who decided to not continue with me, but I think it is the class that is not suited to them more than me.

They had a handout that talked about the domain of a function in very general terms (no mention of avoiding division by zero, or square roots, etc.), but the exercises they had about domains were very specific (yep, we had to handle division by zero, square roots, and more exotic stuff). To put it differently, the exercises they had required much more facility with the material than the class handout they showed me.

Nominally, I was tutoring Calculus, but the material they had (both the handout and the exercises) felt like a revision that reached back to Algebra. And they were not completely at ease with this. 😬

I can't imagine what's going to happen if they stay in this Calculus class and start having to deal with limits, etc.

I should note that this is pretty rare that I get students who present me with a case like this. (Even though, I've just had a similar case with SAT. It is still rare.)

#tutoring #algebra #calculus

Another one bites the dust...

I've tutored a couple of students who decided to not continue with me, but I think it is the class that is not suited to them more than me.

They had a handout that talked about the domain of a function in very general terms (no mention of avoiding division by zero, or square roots, etc.), but the exercises they had about domains were very specific (yep, we had to handle division by zero, square roots, and more exotic stuff). To put it differently, the exercises they had required much more facility with the material than the class handout they showed me.

Nominally, I was tutoring Calculus, but the material they had (both the handout and the exercises) felt like a revision that reached back to Algebra. And they were not completely at ease with this. 😬

I can't imagine what's going to happen if they stay in this Calculus class and start having to deal with limits, etc.

I should note that this is pretty rare that I get students who present me with a case like this. (Even though, I've just had a similar case with SAT. It is still rare.)

#tutoring #algebra #calculus

The Calculi of Lambda-Conversion (AM-6), Volume 6 (Annals of Mathematics Studies) by Alonzo Church (PDF)
Author: Alonzo Church
File Type: PDF
Download at https://sci-books.com/the-calculi-of-lambda-conversion-am-6-volume-6-annals-of-mathematics-studies-b088pk9pc7/
#Calculus, #AlonzoChurch

🤯 Oh wow, 84 minutes to learn that "integral of a #diffusion model" is just fancy math-talk for "fancy noise-guessing game." DeepMind's secret sauce: taking baby steps in random noise-land until 🎯—voilà!—magic samples appear! 🍻 Cheers to #Sander for making #calculus look like it needs a therapist.
https://sander.ai/2026/05/06/flow-maps.html #DeepMind #Integral #Math #Noise #HackerNews #ngated

🤯 Oh wow, 84 minutes to learn that "integral of a #diffusion model" is just fancy math-talk for "fancy noise-guessing game." DeepMind's secret sauce: taking baby steps in random noise-land until 🎯—voilà!—magic samples appear! 🍻 Cheers to #Sander for making #calculus look like it needs a therapist.
https://sander.ai/2026/05/06/flow-maps.html #DeepMind #Integral #Math #Noise #HackerNews #ngated

🤯 Oh wow, 84 minutes to learn that "integral of a #diffusion model" is just fancy math-talk for "fancy noise-guessing game." DeepMind's secret sauce: taking baby steps in random noise-land until 🎯—voilà!—magic samples appear! 🍻 Cheers to #Sander for making #calculus look like it needs a therapist.
https://sander.ai/2026/05/06/flow-maps.html #DeepMind #Integral #Math #Noise #HackerNews #ngated

🤯 Oh wow, 84 minutes to learn that "integral of a #diffusion model" is just fancy math-talk for "fancy noise-guessing game." DeepMind's secret sauce: taking baby steps in random noise-land until 🎯—voilà!—magic samples appear! 🍻 Cheers to #Sander for making #calculus look like it needs a therapist.
https://sander.ai/2026/05/06/flow-maps.html #DeepMind #Integral #Math #Noise #HackerNews #ngated

🤯 Oh wow, 84 minutes to learn that "integral of a #diffusion model" is just fancy math-talk for "fancy noise-guessing game." DeepMind's secret sauce: taking baby steps in random noise-land until 🎯—voilà!—magic samples appear! 🍻 Cheers to #Sander for making #calculus look like it needs a therapist.
https://sander.ai/2026/05/06/flow-maps.html #DeepMind #Integral #Math #Noise #HackerNews #ngated

Classical and Stochastic Laplacian Growth (Advances in Mathematical Fluid Mechanics) 2014th Edition by Bjà¶rn Gustafsson (PDF)
Author: Björn Gustafsson
File Type: PDF
Download at https://sci-books.com/classical-and-stochastic-laplacian-growth-advances-in-mathematical-fluid-mechanics-2014th-edition-3319082868/
#Calculus, #BjörnGustafsson

“Something that doesn’t actually exist can still be useful”*…

Gregory Barber on ultrafinitism, a philosophy that rejects the infinite. Ultrafinitism has long been dismissed as mathematical heresy, but it is also producing new insights in math and beyond…

Doron Zeilberger is a mathematician who believes that all things come to an end. That just as we are limited beings, so too does nature have boundaries — and therefore so do numbers. Look out the window, and where others see reality as a continuous expanse, flowing inexorably forward from moment to moment, Zeilberger sees a universe that ticks. It is a discrete machine. In the smooth motion of the world around him, he catches the subtle blur of a flip-book.

To Zeilberger, believing in infinity is like believing in God. It’s an alluring idea that flatters our intuitions and helps us make sense of all sorts of phenomena. But the problem is that we cannot truly observe infinity, and so we cannot truly say what it is. Equations define lines that carry on off the chalkboard, but to where? Proofs are littered with suggestive ellipses. These equations and proofs are, according to Zeilberger — a longtime professor at Rutgers University and a famed figure in combinatorics — both “very ugly” and false. It is “completely nonsense,” he said, huffing out each syllable in a husky voice that seemed worn out from making his point.

As a matter of practicality, infinity can be scrubbed out, he contends. “You don’t really need it.” Mathematicians can construct a form of calculus without infinity, for instance, cutting infinitesimal limits out of the picture entirely. Curves might look smooth, but they hide a fine-grit roughness; computers handle math just fine with a finite allowance of digits. (Zeilberger lists his own computer, which he named “Shalosh B. Ekhad,” as a collaborator on his papers.) With infinity eliminated, the only thing lost is mathematics that was “not worth doing at all,” Zeilberger said.

Most mathematicians would say just the opposite — that it’s Zeilberger who spews complete nonsense. Not just because infinity is so useful and so natural to our descriptions of the universe, but because treating sets of numbers (like the integers) as actual, infinite objects is at the very core of mathematics, embedded in its most fundamental rules and assumptions.

At the very least, even if mathematicians don’t want to think about infinity as an actual entity, they acknowledge that sequences, shapes, and other mathematical objects have the potential to grow indefinitely. Two parallel lines can in theory go on forever; another number can always be added to the end of the number line.

Zeilberger disagrees. To him, what matters is not whether something is possible in principle, but whether it is actually feasible. What this means, in practice, is that not only is infinity suspect, but extremely large numbers are as well. Consider “Skewes’ number,” eee79. This is an exceptionally large number, and no one has ever been able to write it out in decimal form. So what can we really say about it? Is it an integer? Is it prime? Can we find such a number anywhere in nature? Could we ever write it down? Perhaps, then, it is not a number at all.

This raises obvious questions, such as where, exactly, we will find the end point. Zeilberger can’t say. Nobody can. Which is the first reason that many dismiss his philosophy, known as ultrafinitism. “When you first pitch the idea of ultrafinitism to somebody, it sounds like quackery — like ‘I think there’s a largest number’ or something,” said Justin Clarke-Doane, a philosopher at Columbia University.

“A lot of mathematicians just find the whole proposal preposterous,” said Joel David Hamkins, a set theorist at the University of Notre Dame. Ultrafinitism is not polite talk at a mathematical society dinner. Few (one might say an ultrafinite number) work on it. Fewer still are card-carrying members, like Zeilberger, willing to shout their views out into the void. That’s not just because ultrafinitism is contrarian, but because it advocates for a mathematics that is fundamentally smaller, one where certain important questions can no longer be asked.

And yet it gives Hamkins and others a good deal to think about. From one angle, ultrafinitism can be seen as a more realistic mathematics. It is math that better reflects the limits of what people can create and verify; it may even better reflect the physical universe. While we might be inclined to think of space and time as eternally expansive and divisible, the ultrafinitist would argue that these are assumptions that science has increasingly brought into question — much as, Zeilberger might say, science brought doubt to God’s doorstep.

“The world that we’re describing needs to be honest through and through,” said Clarke-Doane, who in April 2025 convened a rare gathering of experts to explore ultrafinitist ideas. “If there might only be finitely many things, then we’d better also be using a math that doesn’t just assume that there are infinitely many things at the get-go.” To him, “it sure seems like that should be part of the menu in the philosophy of math.”

For mathematicians to take it seriously, though, ultrafinitists first need to agree on what they’re talking about — to turn arguments that sound like “bluster,” as Hamkins puts it, into an official theory. Mathematics is steeped in formal systems and common frameworks. Ultrafinitism, meanwhile, lacks such structure.

It is one thing to tackle problems piecemeal. It is quite another to rewrite the logical foundations of mathematics itself. “I don’t think the reason ultrafinitism has been dismissed is that people have good arguments against it,” Clarke-Doane said. “The feeling is that, oh, well, it’s hopeless.”

That’s a problem that some ultrafinitists are still trying to address.

Zeilberger, meanwhile, is prepared to abandon mathematical ideals in favor of a mathematics that’s inherently messy — just like the world is. He is less a man of foundational theories than a man of opinions, of which he lists 195 on his website. “I cannot be a tenured professor without doing this crackpot stuff,” he said. But one day, he added, mathematicians will look back and see that this crackpot, like those of yore who questioned gods and superstitions, was right. “Luckily, heretics are no longer burned at the stake.”…

Read on for the history of ultrafinitism, the critical dialogue surrounding it, and its implications: “What Can We Gain by Losing Infinity?” from @gregbarber.bsky.social in @quantamagazine.bsky.social.

* Ian Stewart (whose point was somewhat different from Zeilberger’s :-), Infinity: A Very Short Introduction

###

As we engage the endless, we might spare a thought for a man whose work touched on the infinitesimal, Isaac Barrow; he died on this date in 1677. A theologian and mathematician, he played a key role in the development of infinitesimal calculus (in particular, for a proof of the fundamental theorem of calculus). Barrow was the inaugural holder of the prestigious Lucasian Professorship of Mathematics at the University of Cambridge, a post later held by his student, Isaac Newton (who, of course, shares primary credit for the development of calculus with Gottfried Wilhelm Leibniz).

source

“Something that doesn’t actually exist can still be useful”*…

Gregory Barber on ultrafinitism, a philosophy that rejects the infinite. Ultrafinitism has long been dismissed as mathematical heresy, but it is also producing new insights in math and beyond…

Doron Zeilberger is a mathematician who believes that all things come to an end. That just as we are limited beings, so too does nature have boundaries — and therefore so do numbers. Look out the window, and where others see reality as a continuous expanse, flowing inexorably forward from moment to moment, Zeilberger sees a universe that ticks. It is a discrete machine. In the smooth motion of the world around him, he catches the subtle blur of a flip-book.

To Zeilberger, believing in infinity is like believing in God. It’s an alluring idea that flatters our intuitions and helps us make sense of all sorts of phenomena. But the problem is that we cannot truly observe infinity, and so we cannot truly say what it is. Equations define lines that carry on off the chalkboard, but to where? Proofs are littered with suggestive ellipses. These equations and proofs are, according to Zeilberger — a longtime professor at Rutgers University and a famed figure in combinatorics — both “very ugly” and false. It is “completely nonsense,” he said, huffing out each syllable in a husky voice that seemed worn out from making his point.

As a matter of practicality, infinity can be scrubbed out, he contends. “You don’t really need it.” Mathematicians can construct a form of calculus without infinity, for instance, cutting infinitesimal limits out of the picture entirely. Curves might look smooth, but they hide a fine-grit roughness; computers handle math just fine with a finite allowance of digits. (Zeilberger lists his own computer, which he named “Shalosh B. Ekhad,” as a collaborator on his papers.) With infinity eliminated, the only thing lost is mathematics that was “not worth doing at all,” Zeilberger said.

Most mathematicians would say just the opposite — that it’s Zeilberger who spews complete nonsense. Not just because infinity is so useful and so natural to our descriptions of the universe, but because treating sets of numbers (like the integers) as actual, infinite objects is at the very core of mathematics, embedded in its most fundamental rules and assumptions.

At the very least, even if mathematicians don’t want to think about infinity as an actual entity, they acknowledge that sequences, shapes, and other mathematical objects have the potential to grow indefinitely. Two parallel lines can in theory go on forever; another number can always be added to the end of the number line.

Zeilberger disagrees. To him, what matters is not whether something is possible in principle, but whether it is actually feasible. What this means, in practice, is that not only is infinity suspect, but extremely large numbers are as well. Consider “Skewes’ number,” eee79. This is an exceptionally large number, and no one has ever been able to write it out in decimal form. So what can we really say about it? Is it an integer? Is it prime? Can we find such a number anywhere in nature? Could we ever write it down? Perhaps, then, it is not a number at all.

This raises obvious questions, such as where, exactly, we will find the end point. Zeilberger can’t say. Nobody can. Which is the first reason that many dismiss his philosophy, known as ultrafinitism. “When you first pitch the idea of ultrafinitism to somebody, it sounds like quackery — like ‘I think there’s a largest number’ or something,” said Justin Clarke-Doane, a philosopher at Columbia University.

“A lot of mathematicians just find the whole proposal preposterous,” said Joel David Hamkins, a set theorist at the University of Notre Dame. Ultrafinitism is not polite talk at a mathematical society dinner. Few (one might say an ultrafinite number) work on it. Fewer still are card-carrying members, like Zeilberger, willing to shout their views out into the void. That’s not just because ultrafinitism is contrarian, but because it advocates for a mathematics that is fundamentally smaller, one where certain important questions can no longer be asked.

And yet it gives Hamkins and others a good deal to think about. From one angle, ultrafinitism can be seen as a more realistic mathematics. It is math that better reflects the limits of what people can create and verify; it may even better reflect the physical universe. While we might be inclined to think of space and time as eternally expansive and divisible, the ultrafinitist would argue that these are assumptions that science has increasingly brought into question — much as, Zeilberger might say, science brought doubt to God’s doorstep.

“The world that we’re describing needs to be honest through and through,” said Clarke-Doane, who in April 2025 convened a rare gathering of experts to explore ultrafinitist ideas. “If there might only be finitely many things, then we’d better also be using a math that doesn’t just assume that there are infinitely many things at the get-go.” To him, “it sure seems like that should be part of the menu in the philosophy of math.”

For mathematicians to take it seriously, though, ultrafinitists first need to agree on what they’re talking about — to turn arguments that sound like “bluster,” as Hamkins puts it, into an official theory. Mathematics is steeped in formal systems and common frameworks. Ultrafinitism, meanwhile, lacks such structure.

It is one thing to tackle problems piecemeal. It is quite another to rewrite the logical foundations of mathematics itself. “I don’t think the reason ultrafinitism has been dismissed is that people have good arguments against it,” Clarke-Doane said. “The feeling is that, oh, well, it’s hopeless.”

That’s a problem that some ultrafinitists are still trying to address.

Zeilberger, meanwhile, is prepared to abandon mathematical ideals in favor of a mathematics that’s inherently messy — just like the world is. He is less a man of foundational theories than a man of opinions, of which he lists 195 on his website. “I cannot be a tenured professor without doing this crackpot stuff,” he said. But one day, he added, mathematicians will look back and see that this crackpot, like those of yore who questioned gods and superstitions, was right. “Luckily, heretics are no longer burned at the stake.”…

Read on for the history of ultrafinitism, the critical dialogue surrounding it, and its implications: “What Can We Gain by Losing Infinity?” from @gregbarber.bsky.social in @quantamagazine.bsky.social.

* Ian Stewart (whose point was somewhat different from Zeilberger’s :-), Infinity: A Very Short Introduction

###

As we engage the endless, we might spare a thought for a man whose work touched on the infinitesimal, Isaac Barrow; he died on this date in 1677. A theologian and mathematician, he played a key role in the development of infinitesimal calculus (in particular, for a proof of the fundamental theorem of calculus). Barrow was the inaugural holder of the prestigious Lucasian Professorship of Mathematics at the University of Cambridge, a post later held by his student, Isaac Newton (who, of course, shares primary credit for the development of calculus with Gottfried Wilhelm Leibniz).

source

“Something that doesn’t actually exist can still be useful”*…

Gregory Barber on ultrafinitism, a philosophy that rejects the infinite. Ultrafinitism has long been dismissed as mathematical heresy, but it is also producing new insights in math and beyond…

Doron Zeilberger is a mathematician who believes that all things come to an end. That just as we are limited beings, so too does nature have boundaries — and therefore so do numbers. Look out the window, and where others see reality as a continuous expanse, flowing inexorably forward from moment to moment, Zeilberger sees a universe that ticks. It is a discrete machine. In the smooth motion of the world around him, he catches the subtle blur of a flip-book.

To Zeilberger, believing in infinity is like believing in God. It’s an alluring idea that flatters our intuitions and helps us make sense of all sorts of phenomena. But the problem is that we cannot truly observe infinity, and so we cannot truly say what it is. Equations define lines that carry on off the chalkboard, but to where? Proofs are littered with suggestive ellipses. These equations and proofs are, according to Zeilberger — a longtime professor at Rutgers University and a famed figure in combinatorics — both “very ugly” and false. It is “completely nonsense,” he said, huffing out each syllable in a husky voice that seemed worn out from making his point.

As a matter of practicality, infinity can be scrubbed out, he contends. “You don’t really need it.” Mathematicians can construct a form of calculus without infinity, for instance, cutting infinitesimal limits out of the picture entirely. Curves might look smooth, but they hide a fine-grit roughness; computers handle math just fine with a finite allowance of digits. (Zeilberger lists his own computer, which he named “Shalosh B. Ekhad,” as a collaborator on his papers.) With infinity eliminated, the only thing lost is mathematics that was “not worth doing at all,” Zeilberger said.

Most mathematicians would say just the opposite — that it’s Zeilberger who spews complete nonsense. Not just because infinity is so useful and so natural to our descriptions of the universe, but because treating sets of numbers (like the integers) as actual, infinite objects is at the very core of mathematics, embedded in its most fundamental rules and assumptions.

At the very least, even if mathematicians don’t want to think about infinity as an actual entity, they acknowledge that sequences, shapes, and other mathematical objects have the potential to grow indefinitely. Two parallel lines can in theory go on forever; another number can always be added to the end of the number line.

Zeilberger disagrees. To him, what matters is not whether something is possible in principle, but whether it is actually feasible. What this means, in practice, is that not only is infinity suspect, but extremely large numbers are as well. Consider “Skewes’ number,” eee79. This is an exceptionally large number, and no one has ever been able to write it out in decimal form. So what can we really say about it? Is it an integer? Is it prime? Can we find such a number anywhere in nature? Could we ever write it down? Perhaps, then, it is not a number at all.

This raises obvious questions, such as where, exactly, we will find the end point. Zeilberger can’t say. Nobody can. Which is the first reason that many dismiss his philosophy, known as ultrafinitism. “When you first pitch the idea of ultrafinitism to somebody, it sounds like quackery — like ‘I think there’s a largest number’ or something,” said Justin Clarke-Doane, a philosopher at Columbia University.

“A lot of mathematicians just find the whole proposal preposterous,” said Joel David Hamkins, a set theorist at the University of Notre Dame. Ultrafinitism is not polite talk at a mathematical society dinner. Few (one might say an ultrafinite number) work on it. Fewer still are card-carrying members, like Zeilberger, willing to shout their views out into the void. That’s not just because ultrafinitism is contrarian, but because it advocates for a mathematics that is fundamentally smaller, one where certain important questions can no longer be asked.

And yet it gives Hamkins and others a good deal to think about. From one angle, ultrafinitism can be seen as a more realistic mathematics. It is math that better reflects the limits of what people can create and verify; it may even better reflect the physical universe. While we might be inclined to think of space and time as eternally expansive and divisible, the ultrafinitist would argue that these are assumptions that science has increasingly brought into question — much as, Zeilberger might say, science brought doubt to God’s doorstep.

“The world that we’re describing needs to be honest through and through,” said Clarke-Doane, who in April 2025 convened a rare gathering of experts to explore ultrafinitist ideas. “If there might only be finitely many things, then we’d better also be using a math that doesn’t just assume that there are infinitely many things at the get-go.” To him, “it sure seems like that should be part of the menu in the philosophy of math.”

For mathematicians to take it seriously, though, ultrafinitists first need to agree on what they’re talking about — to turn arguments that sound like “bluster,” as Hamkins puts it, into an official theory. Mathematics is steeped in formal systems and common frameworks. Ultrafinitism, meanwhile, lacks such structure.

It is one thing to tackle problems piecemeal. It is quite another to rewrite the logical foundations of mathematics itself. “I don’t think the reason ultrafinitism has been dismissed is that people have good arguments against it,” Clarke-Doane said. “The feeling is that, oh, well, it’s hopeless.”

That’s a problem that some ultrafinitists are still trying to address.

Zeilberger, meanwhile, is prepared to abandon mathematical ideals in favor of a mathematics that’s inherently messy — just like the world is. He is less a man of foundational theories than a man of opinions, of which he lists 195 on his website. “I cannot be a tenured professor without doing this crackpot stuff,” he said. But one day, he added, mathematicians will look back and see that this crackpot, like those of yore who questioned gods and superstitions, was right. “Luckily, heretics are no longer burned at the stake.”…

Read on for the history of ultrafinitism, the critical dialogue surrounding it, and its implications: “What Can We Gain by Losing Infinity?” from @gregbarber.bsky.social in @quantamagazine.bsky.social.

* Ian Stewart (whose point was somewhat different from Zeilberger’s :-), Infinity: A Very Short Introduction

###

As we engage the endless, we might spare a thought for a man whose work touched on the infinitesimal, Isaac Barrow; he died on this date in 1677. A theologian and mathematician, he played a key role in the development of infinitesimal calculus (in particular, for a proof of the fundamental theorem of calculus). Barrow was the inaugural holder of the prestigious Lucasian Professorship of Mathematics at the University of Cambridge, a post later held by his student, Isaac Newton (who, of course, shares primary credit for the development of calculus with Gottfried Wilhelm Leibniz).

source

“Something that doesn’t actually exist can still be useful”*…

Gregory Barber on ultrafinitism, a philosophy that rejects the infinite. Ultrafinitism has long been dismissed as mathematical heresy, but it is also producing new insights in math and beyond…

Doron Zeilberger is a mathematician who believes that all things come to an end. That just as we are limited beings, so too does nature have boundaries — and therefore so do numbers. Look out the window, and where others see reality as a continuous expanse, flowing inexorably forward from moment to moment, Zeilberger sees a universe that ticks. It is a discrete machine. In the smooth motion of the world around him, he catches the subtle blur of a flip-book.

To Zeilberger, believing in infinity is like believing in God. It’s an alluring idea that flatters our intuitions and helps us make sense of all sorts of phenomena. But the problem is that we cannot truly observe infinity, and so we cannot truly say what it is. Equations define lines that carry on off the chalkboard, but to where? Proofs are littered with suggestive ellipses. These equations and proofs are, according to Zeilberger — a longtime professor at Rutgers University and a famed figure in combinatorics — both “very ugly” and false. It is “completely nonsense,” he said, huffing out each syllable in a husky voice that seemed worn out from making his point.

As a matter of practicality, infinity can be scrubbed out, he contends. “You don’t really need it.” Mathematicians can construct a form of calculus without infinity, for instance, cutting infinitesimal limits out of the picture entirely. Curves might look smooth, but they hide a fine-grit roughness; computers handle math just fine with a finite allowance of digits. (Zeilberger lists his own computer, which he named “Shalosh B. Ekhad,” as a collaborator on his papers.) With infinity eliminated, the only thing lost is mathematics that was “not worth doing at all,” Zeilberger said.

Most mathematicians would say just the opposite — that it’s Zeilberger who spews complete nonsense. Not just because infinity is so useful and so natural to our descriptions of the universe, but because treating sets of numbers (like the integers) as actual, infinite objects is at the very core of mathematics, embedded in its most fundamental rules and assumptions.

At the very least, even if mathematicians don’t want to think about infinity as an actual entity, they acknowledge that sequences, shapes, and other mathematical objects have the potential to grow indefinitely. Two parallel lines can in theory go on forever; another number can always be added to the end of the number line.

Zeilberger disagrees. To him, what matters is not whether something is possible in principle, but whether it is actually feasible. What this means, in practice, is that not only is infinity suspect, but extremely large numbers are as well. Consider “Skewes’ number,” eee79. This is an exceptionally large number, and no one has ever been able to write it out in decimal form. So what can we really say about it? Is it an integer? Is it prime? Can we find such a number anywhere in nature? Could we ever write it down? Perhaps, then, it is not a number at all.

This raises obvious questions, such as where, exactly, we will find the end point. Zeilberger can’t say. Nobody can. Which is the first reason that many dismiss his philosophy, known as ultrafinitism. “When you first pitch the idea of ultrafinitism to somebody, it sounds like quackery — like ‘I think there’s a largest number’ or something,” said Justin Clarke-Doane, a philosopher at Columbia University.

“A lot of mathematicians just find the whole proposal preposterous,” said Joel David Hamkins, a set theorist at the University of Notre Dame. Ultrafinitism is not polite talk at a mathematical society dinner. Few (one might say an ultrafinite number) work on it. Fewer still are card-carrying members, like Zeilberger, willing to shout their views out into the void. That’s not just because ultrafinitism is contrarian, but because it advocates for a mathematics that is fundamentally smaller, one where certain important questions can no longer be asked.

And yet it gives Hamkins and others a good deal to think about. From one angle, ultrafinitism can be seen as a more realistic mathematics. It is math that better reflects the limits of what people can create and verify; it may even better reflect the physical universe. While we might be inclined to think of space and time as eternally expansive and divisible, the ultrafinitist would argue that these are assumptions that science has increasingly brought into question — much as, Zeilberger might say, science brought doubt to God’s doorstep.

“The world that we’re describing needs to be honest through and through,” said Clarke-Doane, who in April 2025 convened a rare gathering of experts to explore ultrafinitist ideas. “If there might only be finitely many things, then we’d better also be using a math that doesn’t just assume that there are infinitely many things at the get-go.” To him, “it sure seems like that should be part of the menu in the philosophy of math.”

For mathematicians to take it seriously, though, ultrafinitists first need to agree on what they’re talking about — to turn arguments that sound like “bluster,” as Hamkins puts it, into an official theory. Mathematics is steeped in formal systems and common frameworks. Ultrafinitism, meanwhile, lacks such structure.

It is one thing to tackle problems piecemeal. It is quite another to rewrite the logical foundations of mathematics itself. “I don’t think the reason ultrafinitism has been dismissed is that people have good arguments against it,” Clarke-Doane said. “The feeling is that, oh, well, it’s hopeless.”

That’s a problem that some ultrafinitists are still trying to address.

Zeilberger, meanwhile, is prepared to abandon mathematical ideals in favor of a mathematics that’s inherently messy — just like the world is. He is less a man of foundational theories than a man of opinions, of which he lists 195 on his website. “I cannot be a tenured professor without doing this crackpot stuff,” he said. But one day, he added, mathematicians will look back and see that this crackpot, like those of yore who questioned gods and superstitions, was right. “Luckily, heretics are no longer burned at the stake.”…

Read on for the history of ultrafinitism, the critical dialogue surrounding it, and its implications: “What Can We Gain by Losing Infinity?” from @gregbarber.bsky.social in @quantamagazine.bsky.social.

* Ian Stewart (whose point was somewhat different from Zeilberger’s :-), Infinity: A Very Short Introduction

###

As we engage the endless, we might spare a thought for a man whose work touched on the infinitesimal, Isaac Barrow; he died on this date in 1677. A theologian and mathematician, he played a key role in the development of infinitesimal calculus (in particular, for a proof of the fundamental theorem of calculus). Barrow was the inaugural holder of the prestigious Lucasian Professorship of Mathematics at the University of Cambridge, a post later held by his student, Isaac Newton (who, of course, shares primary credit for the development of calculus with Gottfried Wilhelm Leibniz).

source

“Something that doesn’t actually exist can still be useful”*…

Gregory Barber on ultrafinitism, a philosophy that rejects the infinite. Ultrafinitism has long been dismissed as mathematical heresy, but it is also producing new insights in math and beyond…

Doron Zeilberger is a mathematician who believes that all things come to an end. That just as we are limited beings, so too does nature have boundaries — and therefore so do numbers. Look out the window, and where others see reality as a continuous expanse, flowing inexorably forward from moment to moment, Zeilberger sees a universe that ticks. It is a discrete machine. In the smooth motion of the world around him, he catches the subtle blur of a flip-book.

To Zeilberger, believing in infinity is like believing in God. It’s an alluring idea that flatters our intuitions and helps us make sense of all sorts of phenomena. But the problem is that we cannot truly observe infinity, and so we cannot truly say what it is. Equations define lines that carry on off the chalkboard, but to where? Proofs are littered with suggestive ellipses. These equations and proofs are, according to Zeilberger — a longtime professor at Rutgers University and a famed figure in combinatorics — both “very ugly” and false. It is “completely nonsense,” he said, huffing out each syllable in a husky voice that seemed worn out from making his point.

As a matter of practicality, infinity can be scrubbed out, he contends. “You don’t really need it.” Mathematicians can construct a form of calculus without infinity, for instance, cutting infinitesimal limits out of the picture entirely. Curves might look smooth, but they hide a fine-grit roughness; computers handle math just fine with a finite allowance of digits. (Zeilberger lists his own computer, which he named “Shalosh B. Ekhad,” as a collaborator on his papers.) With infinity eliminated, the only thing lost is mathematics that was “not worth doing at all,” Zeilberger said.

Most mathematicians would say just the opposite — that it’s Zeilberger who spews complete nonsense. Not just because infinity is so useful and so natural to our descriptions of the universe, but because treating sets of numbers (like the integers) as actual, infinite objects is at the very core of mathematics, embedded in its most fundamental rules and assumptions.

At the very least, even if mathematicians don’t want to think about infinity as an actual entity, they acknowledge that sequences, shapes, and other mathematical objects have the potential to grow indefinitely. Two parallel lines can in theory go on forever; another number can always be added to the end of the number line.

Zeilberger disagrees. To him, what matters is not whether something is possible in principle, but whether it is actually feasible. What this means, in practice, is that not only is infinity suspect, but extremely large numbers are as well. Consider “Skewes’ number,” eee79. This is an exceptionally large number, and no one has ever been able to write it out in decimal form. So what can we really say about it? Is it an integer? Is it prime? Can we find such a number anywhere in nature? Could we ever write it down? Perhaps, then, it is not a number at all.

This raises obvious questions, such as where, exactly, we will find the end point. Zeilberger can’t say. Nobody can. Which is the first reason that many dismiss his philosophy, known as ultrafinitism. “When you first pitch the idea of ultrafinitism to somebody, it sounds like quackery — like ‘I think there’s a largest number’ or something,” said Justin Clarke-Doane, a philosopher at Columbia University.

“A lot of mathematicians just find the whole proposal preposterous,” said Joel David Hamkins, a set theorist at the University of Notre Dame. Ultrafinitism is not polite talk at a mathematical society dinner. Few (one might say an ultrafinite number) work on it. Fewer still are card-carrying members, like Zeilberger, willing to shout their views out into the void. That’s not just because ultrafinitism is contrarian, but because it advocates for a mathematics that is fundamentally smaller, one where certain important questions can no longer be asked.

And yet it gives Hamkins and others a good deal to think about. From one angle, ultrafinitism can be seen as a more realistic mathematics. It is math that better reflects the limits of what people can create and verify; it may even better reflect the physical universe. While we might be inclined to think of space and time as eternally expansive and divisible, the ultrafinitist would argue that these are assumptions that science has increasingly brought into question — much as, Zeilberger might say, science brought doubt to God’s doorstep.

“The world that we’re describing needs to be honest through and through,” said Clarke-Doane, who in April 2025 convened a rare gathering of experts to explore ultrafinitist ideas. “If there might only be finitely many things, then we’d better also be using a math that doesn’t just assume that there are infinitely many things at the get-go.” To him, “it sure seems like that should be part of the menu in the philosophy of math.”

For mathematicians to take it seriously, though, ultrafinitists first need to agree on what they’re talking about — to turn arguments that sound like “bluster,” as Hamkins puts it, into an official theory. Mathematics is steeped in formal systems and common frameworks. Ultrafinitism, meanwhile, lacks such structure.

It is one thing to tackle problems piecemeal. It is quite another to rewrite the logical foundations of mathematics itself. “I don’t think the reason ultrafinitism has been dismissed is that people have good arguments against it,” Clarke-Doane said. “The feeling is that, oh, well, it’s hopeless.”

That’s a problem that some ultrafinitists are still trying to address.

Zeilberger, meanwhile, is prepared to abandon mathematical ideals in favor of a mathematics that’s inherently messy — just like the world is. He is less a man of foundational theories than a man of opinions, of which he lists 195 on his website. “I cannot be a tenured professor without doing this crackpot stuff,” he said. But one day, he added, mathematicians will look back and see that this crackpot, like those of yore who questioned gods and superstitions, was right. “Luckily, heretics are no longer burned at the stake.”…

Read on for the history of ultrafinitism, the critical dialogue surrounding it, and its implications: “What Can We Gain by Losing Infinity?” from @gregbarber.bsky.social in @quantamagazine.bsky.social.

* Ian Stewart (whose point was somewhat different from Zeilberger’s :-), Infinity: A Very Short Introduction

###

As we engage the endless, we might spare a thought for a man whose work touched on the infinitesimal, Isaac Barrow; he died on this date in 1677. A theologian and mathematician, he played a key role in the development of infinitesimal calculus (in particular, for a proof of the fundamental theorem of calculus). Barrow was the inaugural holder of the prestigious Lucasian Professorship of Mathematics at the University of Cambridge, a post later held by his student, Isaac Newton (who, of course, shares primary credit for the development of calculus with Gottfried Wilhelm Leibniz).

source

Last week, I went down a huge rabbit hole and tried to come up with a mathematical generalization of dealing with errors in circuits.

I am not a mathematician, but I tried to come up with a proof as rigorous as possible. However, other eyes are very much appreciated. I'd be happy about a boost and/or feedback.

And of course, I'd be glad if this was useful to anyone else :)

https://www.maxgenson.de/blog/demystifying-tolerances/

#electricalengineering #errorhandling #math #calculus #mathproblems #feedback #helpwanted

Last week, I went down a huge rabbit hole and tried to come up with a mathematical generalization of dealing with errors in circuits.

I am not a mathematician, but I tried to come up with a proof as rigorous as possible. However, other eyes are very much appreciated. I'd be happy about a boost and/or feedback.

And of course, I'd be glad if this was useful to anyone else :)

https://www.maxgenson.de/blog/demystifying-tolerances/

#electricalengineering #errorhandling #math #calculus #mathproblems #feedback #helpwanted

Alright, future engineers!
The **Fundamental Theorem of Calculus (FTC)** bridges diff. & int., showing they're inverse ops!
Ex: `Int_a^b f'(x)dx = f(b) - f(a)`
Pro-Tip: This is WHY you use antiderivatives to solve definite integrals!
#Calculus #FTC #STEM #StudyNotes

Alright, future engineers!
A **Derivative** measures the instantaneous rate of change of a function or the slope of its tangent line.
Ex: Power Rule: `d/dx (x^n) = n*x^(n-1)` (e.g., `d/dx(x^3) = 3x^2`).
Pro-Tip: Think 'slope'! It tells you how fast something is changing *at that exact moment*.
#Calculus #Derivatives #STEM #StudyNotes

**Fundamental Theorem of Calculus (FTC):** The vital connection between differentiation & integration.
Ex: `Int(f(x)dx) from a to b = F(b) - F(a)` (where `F'(x) = f(x)`).
Pro-Tip: It's *why* antiderivatives calculate definite areas! Master this core concept.
#Calculus #FTC #STEM #StudyNotes

Alright, future engineers!
A **Limit** describes the value a function approaches as its input gets closer to a specific point.
Ex: `lim (x->0) sin(x)/x = 1`.
Pro-Tip: Limits are the bedrock of derivatives & continuity! Understand them well.
#Calculus #Limits #STEM #StudyNotes

Alright, future engineers!
A **Limit** describes the value a function approaches as its input gets closer to a specific point.
Ex: `lim (x->0) sin(x)/x = 1`.
Pro-Tip: Limits are the bedrock of derivatives & continuity! Understand them well.
#Calculus #Limits #STEM #StudyNotes

Ah, another riveting revelation that "areas" need definitions! 🤔 This mind-blowing journey through #rectangles and partitions will surely redefine your life—assuming, of course, that you've never heard of #calculus before. 🙄 Bravo for connecting #integrals and #derivatives, an insight only centuries old! 🎉
https://david.alvarezrosa.com/posts/fundamental-theorem-of-calculus/ #areas #definitions #HackerNews #ngated

Ah, another riveting revelation that "areas" need definitions! 🤔 This mind-blowing journey through #rectangles and partitions will surely redefine your life—assuming, of course, that you've never heard of #calculus before. 🙄 Bravo for connecting #integrals and #derivatives, an insight only centuries old! 🎉
https://david.alvarezrosa.com/posts/fundamental-theorem-of-calculus/ #areas #definitions #HackerNews #ngated

Ah, another riveting revelation that "areas" need definitions! 🤔 This mind-blowing journey through #rectangles and partitions will surely redefine your life—assuming, of course, that you've never heard of #calculus before. 🙄 Bravo for connecting #integrals and #derivatives, an insight only centuries old! 🎉
https://david.alvarezrosa.com/posts/fundamental-theorem-of-calculus/ #areas #definitions #HackerNews #ngated

Ah, another riveting revelation that "areas" need definitions! 🤔 This mind-blowing journey through #rectangles and partitions will surely redefine your life—assuming, of course, that you've never heard of #calculus before. 🙄 Bravo for connecting #integrals and #derivatives, an insight only centuries old! 🎉
https://david.alvarezrosa.com/posts/fundamental-theorem-of-calculus/ #areas #definitions #HackerNews #ngated

Ah, another riveting revelation that "areas" need definitions! 🤔 This mind-blowing journey through #rectangles and partitions will surely redefine your life—assuming, of course, that you've never heard of #calculus before. 🙄 Bravo for connecting #integrals and #derivatives, an insight only centuries old! 🎉
https://david.alvarezrosa.com/posts/fundamental-theorem-of-calculus/ #areas #definitions #HackerNews #ngated

Fundamental Theorem of Calculus

https://david.alvarezrosa.com/posts/fundamental-theorem-of-calculus/

#HackerNews #FundamentalTheoremOfCalculus #Calculus #Math #Education #Learning #HackerNews

`Various software efforts embrace the idea that object oriented programming enables a convenient implementation of the chain rule, facilitating so-called automatic differentiation via backpropagation. Such frameworks have no mechanism for simplifying the expressions (obtained via the chain rule) before evaluating them. As we illustrate below, the resulting errors tend to be unbounded.`

https://arxiv.org/abs/2305.03863

#calculus #software #numericalAnalysis #automaticDifferentiation #uncertainty

Ah, the riveting world of approximating functions we barely remember from #calculus class! 🎉 Who knew cramming "fast" and "tanh" in the same sentence could be a thing? 🤔 Next, they'll reveal the secrets of speed-knitting hyperbolic scarves. 🧶✨
https://jtomschroeder.com/blog/approximating-tanh/ #functionApproximation #fun #hyperbolicScarf #HackerNews #HackerNews #ngated

"A free course offering the core concept of Calculus, with a visuals-first approach aimed at making you feel like you could have discovered the subject yourself."

by Grant Sanderson: https://www.youtube.com/playlist?list=PLZHQObOWTQDMsr9K-rj53DwVRMYO3t5Yr

#learn #learning #calculus #maths #dataViz #course #onlineCourse #tutorial

A **Definite Integral** calculates the signed area under a curve between two points.
Ex: `int(f(x)dx)` from `a` to `b` = `F(b)-F(a)`.
Pro-Tip: Think total accumulation! It's used for total displacement, volume, or work done.
#Calculus #Integrals #STEM #StudyNotes

A **Definite Integral** calculates the signed area under a curve between two points.
Ex: `int(f(x)dx)` from `a` to `b` = `F(b)-F(a)`.
Pro-Tip: Think total accumulation! It's used for total displacement, volume, or work done.
#Calculus #Integrals #STEM #StudyNotes

A **Definite Integral** calculates the signed area under a curve between two points.
Ex: `int(f(x)dx)` from `a` to `b` = `F(b)-F(a)`.
Pro-Tip: Think total accumulation! It's used for total displacement, volume, or work done.
#Calculus #Integrals #STEM #StudyNotes

A **Definite Integral** calculates the signed area under a curve between two points.
Ex: `int(f(x)dx)` from `a` to `b` = `F(b)-F(a)`.
Pro-Tip: Think total accumulation! It's used for total displacement, volume, or work done.
#Calculus #Integrals #STEM #StudyNotes

A **Definite Integral** calculates the signed area under a curve between two points.
Ex: `int(f(x)dx)` from `a` to `b` = `F(b)-F(a)`.
Pro-Tip: Think total accumulation! It's used for total displacement, volume, or work done.
#Calculus #Integrals #STEM #StudyNotes

Alright, future engineers!
A **Partial Derivative** measures how a multi-variable function changes when *only one* variable shifts, keeping others fixed.
Ex: If `f(x,y) = x^2y`, then `∂f/∂x = 2xy`.
Pro-Tip: Treat other variables like constants while differentiating!
#MultivariableCalc #Calculus #STEM #StudyNotes

Differential Analysis on Complex Manifolds (Graduate Texts in Mathematics, 65) by Raymond O. Wells (PDF)
Author: Raymond O. Wells
File Type: PDF
Download at https://sci-books.com/differential-analysis-on-complex-manifolds-graduate-texts-in-mathematics-65-144192535x/
#Calculus, #RaymondO.Wells