#calculus — Public Fediverse posts
Live and recent posts from across the Fediverse tagged #calculus, aggregated by home.social.
-
My calculus book dates from the seventies, and introduces derivatives at page 90.
-
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 -
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 -
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).
#calculus #culture #DoronZeilberger #GregoryBarber #history #infinitesimalCalculus #infinity #IsaacBarrow #IsaacNewton #Leibniz #Mathematics #philosophy #Science #ultrafinitism -
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 -
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 -
`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 -
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 -
Alright, future engineers!
**The Chain Rule** helps differentiate composite functions (functions within functions).
Ex: If `y = sin(x^2)`, then `y' = cos(x^2) * 2x`.
Pro-Tip: Derivative of the outside, times derivative of the inside!
#Calculus #Derivatives #STEM #StudyNotes -
Alright, future engineers!
The **Product Rule** differentiates a product of two functions.
Formula: `(fg)' = f'g + fg'`
Pro-Tip: 'First D Second + Second D First' is a classic mnemonic! -
Alright, future engineers!
The **Product Rule** differentiates a product of two functions.
Formula: `(fg)' = f'g + fg'`
Pro-Tip: 'First D Second + Second D First' is a classic mnemonic! -
Alright, future engineers!
The **Product Rule** differentiates a product of two functions.
Formula: `(fg)' = f'g + fg'`
Pro-Tip: 'First D Second + Second D First' is a classic mnemonic! -
Alright, future engineers!
The **Product Rule** differentiates a product of two functions.
Formula: `(fg)' = f'g + fg'`
Pro-Tip: 'First D Second + Second D First' is a classic mnemonic! -
Alright, future engineers!
The **Product Rule** differentiates a product of two functions.
Formula: `(fg)' = f'g + fg'`
Pro-Tip: 'First D Second + Second D First' is a classic mnemonic! -
Alright, future engineers!
The **Fundamental Theorem of Calculus (FTC)** links differentiation & integration. It states: If `F(x)=∫_a^x f(t)dt`, then `F'(x)=f(x)`.
Ex: If `F(x)=∫_0^x cos(t)dt`, `F'(x)=cos(x)`.
Pro-Tip: It's the ultimate 'undo' button between derivatives & integrals!
#Calculus #FTC #STEM #StudyNotes -
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 -
Alright, future engineers!
The **Mean Value Theorem (MVT)** guarantees that a function's instantaneous rate of change equals its average rate over an interval.
Ex: `f'(c) = (f(b)-f(a))/(b-a)` for some `c` in `(a,b)`.
Pro-Tip: It's all about guaranteeing a *specific point* where slopes match!
#Calculus #MVT #STEM #StudyNotes -
**Critical Points**: Where `f'(x)=0` or `f'(x)` is undefined. These are spots where local max/min *might* occur.
Ex: `f(x)=x^2`, `f'(x)=2x`. `2x=0` means `x=0` is a crit. pt.
Pro-Tip: Always test critical pts (and endpoints!) for absolute extrema!
#Calculus #Optimization #STEM #StudyNotes -
Alright, future engineers!
A **Limit** is the value a function approaches as its input gets arbitrarily close to a specific point.
Ex: `lim(x->2) (x^2 - 4)/(x - 2) = 4`.
Pro-Tip: Always simplify first if you hit 0/0! -
Alright, future engineers!
**U-Substitution:** Simplifies integrals by changing variables.
Ex: For `∫2x cos(x^2) dx`, let `u=x^2`, so `du=2x dx`.
Pro-Tip: Find a function & its derivative in the integrand! -
Alright, future engineers!
A **Critical Point** is where a function's derivative is zero or undefined. Ex: For `f(x)=x^3-3x`, `f'(x)=3x^2-3=0` at `x=±1`. Pro-Tip: They're candidates for local max/min. Use 1st/2nd Derivative Tests!
-
Alright, future engineers!
**Optimization** is finding the maximum or minimum value of a function to solve a real-world problem.
Ex: To maximize profit P(x), set P'(x)=0 & check critical points.
Pro-Tip: It’s how we design for peak performance or minimal cost!
#Calculus #Optimization #STEM #StudyNotes -
Alright, future engineers!
**Taylor Series:** Approximates a function with an infinite sum of polynomial terms. Ex: `e^x approx 1 + x + x^2/2!` (around x=0). Pro-Tip: They let you model complex functions with simple polynomials!
-
Alright, future engineers!
**Mean Value Theorem:** If f is continuous on [a,b] & differentiable on (a,b), then `f'(c) = (f(b)-f(a))/(b-a)` for some c in (a,b).
Pro-Tip: Guarantees an instant rate of change matches the average. Think average speed vs. speedometer!
#Calculus #MVT #STEM #StudyNotes -
Alright, future engineers!
The **Derivative** `f'(x)` is the instantaneous rate of change of a function, or the slope of its tangent line. Ex: If `f(x)=x^3`, `f'(x)=3x^2`. Pro-Tip: It's crucial for optimization & understanding how systems respond instantly!
#Calculus #Derivatives #STEM #StudyNotes -
Alright, future engineers!
An **Integral** calculates the total accumulation of a quantity, often representing the area under a curve. Ex: `∫[a,b] f(x) dx` is the area from `a` to `b`. Pro-Tip: Think of it as summing up infinitely tiny slices!
-
Alright, future engineers!
A **Derivative** is the instantaneous rate of change of a function, or the slope of its tangent line. Ex: For `f(x) = x^2`, `f'(x) = 2x`. Pro-Tip: It's how fast things *are changing* right now!
-
Alright, future engineers! A **Limit** describes the value a function approaches as its input gets closer to some value. Ex: `lim (x->0) (sin x)/x = 1`. Pro-Tip: It's key for understanding continuity and the foundation of derivatives!
#Calculus #Limits #STEM #StudyNotes -
Alright, future engineers!
A **Definite Integral** calculates the *net area* under a function's curve between two points. Ex: Area = ∫[a,b] f(x) dx. Pro-Tip: Think of it as summing infinitely many tiny rectangles!
-
A **Derivative** measures a function's *instantaneous* rate of change. Ex: `d/dx(x^n) = nx^(n-1)`. Pro-Tip: Think slope of the tangent line! It's crucial for analyzing motion, rates, & optimization.
#Calculus #Derivatives #STEM #StudyNotes -
The other day I had a fairly popular post talking about how mathematicians easily and often admit that they don't know things or don't understand things. Today at work a real-life example came up!
Original post linked to in the next toot since apparently I can't post a link and have an image at the same time ... wtf?!
I was helping a student with Calc I in my office. The question gave a function and asked for values of x where the tangent line was horizontal. The function is the first in the image.
This requires taking the derivative with the product rule. The result of this is the second in the image. Since the second term has a denominator (other than 1 of course) we need to combine the two terms so we can set the numerator to 0 and solve.
The result of this operation is the third in the image. Fractions are 0 when their numerators are 0, so the fourth line shows the equation to be solved.
The student got this far without any help but was unable to solve the equation. This is commonplace. After all, the hardest part of calculus is algebra. But I couldn't see how to solve it either, so I told the student this.
At this moment two of my colleagues were talking in the hall outside my office so I told the student I'd ask them about it. Neither knew how to solve it and told me as much. So I told the student, who was actually thrilled that none of us could solve it either.
So I asked Wolfram Alpha, which gave a solution using the Lambert W, aka the productlog function. I'm a combinatorial topologist -- I do graph theory of various kinds. I've heard of this function but otherwise know nothing at all about it. And I'm happy to admit it! Anyway, that's how mathematicians roll.
ETA: Of course this problem shouldn't have appeared in an introductory calculus text since no undergraduate at that level would be able to solve it, so its inclusion was a mistake of the author or the editor.
#Calculus #LambertW #HorizontalTangent #DifferentialCalculus #Math #Mathematics #Mathematicians
-
Alright, future engineers!
A **Limit** is the value a function *approaches* as its input gets infinitesimally close to a specific point. Ex: `lim(x->0) sin(x)/x = 1`. Pro-Tip: Evaluate from both sides to confirm the limit exists!
-
The derivative measures a function's instantaneous rate of change (its slope!). Ex: `d/dx(x^3) = 3x^2`. Pro-Tip: Crucial for optimization – find max/min points when `f'(x)=0`.
#Calculus #Derivatives #STEM #StudyNotes -
The derivative measures a function's instantaneous rate of change (slope of tangent). Ex: d/dx(x^n) = nx^(n-1). Pro-Tip: Think slope! For position, its derivative is velocity; for velocity, it's acceleration.
-
Chain Rule helps differentiate composite functions (function of a function). Formula: d/dx [f(g(x))] = f'(g(x)) * g'(x). Ex: d/dx [(x^2+1)^3] = 3(x^2+1)^2 * (2x). Pro-Tip: Always think Outside-Inside! Differentiate the outer function first, then multiply by the inner's derivative.
#Calculus #Differentiation #STEM #StudyNotes -
Chain Rule helps differentiate composite functions (function of a function). Formula: d/dx [f(g(x))] = f'(g(x)) * g'(x). Ex: d/dx [(x^2+1)^3] = 3(x^2+1)^2 * (2x). Pro-Tip: Always think Outside-Inside! Differentiate the outer function first, then multiply by the inner's derivative.
#Calculus #Differentiation #STEM #StudyNotes -
Chain Rule helps differentiate composite functions (function of a function). Formula: d/dx [f(g(x))] = f'(g(x)) * g'(x). Ex: d/dx [(x^2+1)^3] = 3(x^2+1)^2 * (2x). Pro-Tip: Always think Outside-Inside! Differentiate the outer function first, then multiply by the inner's derivative.
#Calculus #Differentiation #STEM #StudyNotes -
Chain Rule helps differentiate composite functions (function of a function). Formula: d/dx [f(g(x))] = f'(g(x)) * g'(x). Ex: d/dx [(x^2+1)^3] = 3(x^2+1)^2 * (2x). Pro-Tip: Always think Outside-Inside! Differentiate the outer function first, then multiply by the inner's derivative.
#Calculus #Differentiation #STEM #StudyNotes
