> Let f ... be Riemann integrable and F ... differentiable.
What many people don't notice the first time they read this in the fundamental theorem of Calculus is that this is a double criteria. That f needs to be integrable seems like an extraneous point when F is differentiable. This holds also for the Lebesgue integral. The understanding is usually that if F is differentiable then its derivative is integrable, that is, people understand the integral as an anti-derivative but the Riemann/Lebesgue integral version of the fundamental theorem of calculus only proves that if the function you want the anti-derivative of is integrable, so you have this separate requirement to prove that f is integrable having already proven F to be differentiable (to f).
However, this theoretical (because if you aren't a mathematician you won't be bothered by this sticking point, you'll just insist that the integral is the anti-derivative when an anti-derivative exists) defect is ameliorated by the Henstock–Kurzweil integral which is (I feel) a lot easier to define and understand than the Lebesgue integral. It is practically the Riemann integral with just a minor tweak: the delta in the delta-epsilon proof is allowed to vary by location (essentially, as you approach non-integrable singularities, you tend the delta towards zero).
For the Henstock-Kurzweil integral, if F is differentiable then f is (Henstock-Kurzweil) integrable. This happens because not every derivative is Riemann or Lebesgue integrable, you need a stronger integral.
Sweet! I'm keen to learn about the basic fundamentals of calculus!
> For each subinterval ...(bunch of cool maths rendering I can't copy and paste because it's all comes out newline delimited on my clipboard) ... and let m<sub>k</sub> and M<sub>k</sub> denote the infimum and supremum of f on that subinterval...
Okay, guess it wasn't the kind of introduction I had assumed/hoped.
Very cool maths rendering though.
As someone who never passed high school or got a degree thanks to untreated ADHD, if anyone knows of an introduction to the basic fundamentals of calculus that a motivated but under educated maths gronk can grok, I would gratefully appreciate a link or ten.
You can ask for a syllabus first, then go through it.
It's interactive, and it covers in detail everything you don't get. You can ask infinite many practice material, exercises, flashcards, or anything you want.
If you dive into Analysis (the underlying theory behind calculus) this book - "How to Think About Analysis" by Lara Alcock is the book I wish I had when I studied it. Calculus by Spivak is the book I learnt from but it is probably not the easiest, it is very thorough though.
> if anyone knows of an introduction to the basic fundamentals of calculus that a motivated but under educated maths gronk can grok, I would gratefully appreciate a link or ten.
"The basic fundamentals of calculus" usually go under the name "real analysis".
I like the idea of the lean4 game, because if you do your work in lean you'll know whether you've made a mistake.
("Standard analysis" uses limiting behavior to ask what would happen if we were working with infinitely large or infinitesimally small values, even though of course we aren't really. "Nonstandard analysis" doesn't bother pretending and really uses infinitely large and infinitesimally small values. Other than the notational difference, they are the same, and a proof in one approach can be easily and mechanistically converted into the same proof in the other approach.)
Note that the ordinary course of study involves learning to do calculus problems first (in a "calculus" class), and studying the fundamentals second (in an "analysis" class). The textbook I linked is a "calculus" textbook, but there is a bit more focus on the theoretical backing because you can't rely on the student to learn about nonstandard analysis somewhere else.
Yeah, judging by the terseness, this is clearly aimed at undergrads. Then again, this is covered in literally every calculus class, so I'm not sure who this is supposed to be for.
For anyone interested in checking out the book, there is a PDF preview here[1] and printable concept maps[2], which should be useful no matter which book you're reading.
Here's my understanding: 1: In the 'olden days' the area A(x) under the graph f(x) used to be approximated as a Riemann sum. 2: Using limits, as the delta x in the Riemann sum->0, we'd call that an integral and set it to be the exact area A(x). 3: If we then look at some small change in A(x), we might notice f(x) = A'(x)... mind blown. 4: since we can now say A is an anti-derivative of f, we have A(x)=F(x)+C (we have to add the C because the derivative of a constant is 0). 5: Using logic and geometry we have C=-F(a) which leads to... 6: The area under the graph f between [a,b] is A = F(b)-F(a). 7: We don't have to cry anymore about pages of Riemann sum calculations.
I've studied the proofs before but there's still something mystical and unintuitive for me about the area under an entire curve being related to the derivative at only two points, especially for wobbly non monotonic functions.
I feel similar about the trace of a matrix being equal to the sum of eigenvalues.
Probably this means I should sit with it more until it is obvious, but I also kind of like this feeling.
It is not determined by the derivative, it's the antiderivative, as someone else mentioned. The derivative is the rate of change of a function. The "area under a curve" of the graph of a function measures how much the function is "accumulating", which is intuitively a sum of rates of change (taken to an infinitesimal limit).
> there's still something mystical and unintuitive for me about the area under an entire curve being related to the derivative
the discrete version is much clearer to me. Suppose you have a function f(n) defined at integer positions n. Its "derivative" is just the difference of consecutive values
f'(n) = f(n+1) - f(n)
Then the fundamental theorem is just a telescopic sum:
If I tell you I have function f with f(a) = 10 and on it's path from a to b, the graph first
increaes by 5 units then by another 10, and then later on drops by 25 units, you can immediately deduce that
f(b) = f(a) + (+5 +10 -25) = 0. The fundamental theorem of calculus uses the same concept:
To see why \int_a^b f(x) dx = F(b) - F(a) with F'(x) = f(x),
we replace f with f' (and hence F with f) and get
\int_a^b f'(x) dx = f(b) - f(a).
Re-arranging terms, we get
f(b) = f(a) + \int_a^b f'(x) dx.
The last line just says: The value of function f at point b is is the value at point a plus the sum of all the infinitely many changes the function goes through on its path from a to b.
The antiderivative at x is defined as the area under the curve from 0 to x, which the Riemann sum gives a nice intuition for how you can get from the derivative.
So to get the area under the curve between a and b, you calculate the area under the curve from 0 to b (antiderivative at b) and subtract the area under the curve from 0 to a (antiderivative at a).
I took calculus in high school and college, and I don't think any of my instructors explained the intuition as well. So sleep-deprived or not, it's a great one!
> f is Riemann integrable iff it is bounded and continuous almost everywhere.
FWIW, I think this is the same as saying "iff it is bounded and has finite discontinuities". I like that characterization b/c it seems more precise than "almost everywhere", but I've heard both.
I mention that because when I read the first footnote, I thought this was a mistake:
> boundedness alone ensures the subinterval infima and suprema are finite.
But it wasn't. It does, in fact, insure that infima and suprema are finite. It just does NOT ensure that it is Riemann integrable (which, of course the last paragraph in the first section mentions).
Thanks for posting. This was a fun diversion down memory lane whilst having my morning coffee.
If anyone wants a rabbit hole to go down:
Think about why the Dirichlet function [1], which is bounded -- and therefore has upper and lower sums -- is not Riemann integrable (hint: its upper and lower sums don't converge. why?)
Then, if you want to keep going down the rabbit hole, learn how you _can_ integrate it (ie: how you _can_ assign a number to the area it bounds) [2]
> FWIW, I think this is the same as saying "iff it is bounded and has finite discontinuities".
It is not: for example, the piece-wise constant function f: [0,1] -> [0,1] which starts at f(0) = 0, stays constant until suddenly f(1/2) = 1, until f(3/4) = 0, until f(7/8) = 1, etc. is Riemann integrable.
"Continuous almost everywhere" means that the set of its discontinuities has Lebesgue measure 0. Many infinite sets have Lebesgue measure 0, including all countable sets.
The indicator function of the Cantor set is Riemann integrable. Like you said, though, the Dirichlet function (which is the indicator function of the rationals) is not Riemann integrable.
The reason is because the Dirchlet function is discontinuous everywhere on [0,1], so the set of discontinuities has measure 1. The Cantor function is discontinuous only on the Cantor set.
"Almost everywhere" is precisely defined, and it is broader than that. E.g. the real numbers are almost everywhere normal, but there are uncountably many non-normal numbers between any two normal reals.
I love this -- I'll have to do something like that for my site. I always liked the big initials on the start of a paragraph. Though it feels a bit more prose-applicable than for non-fiction writing.
Technically "it depends on the browser settings," but the body font Alegreya is served directly by the site, so I think it would be the one used in almost all cases.
The math fonts used in the formulas are just the ones provided by KaTeX, which I think are just TeX's default math fonts.
Good job, David. Have a lollipop. Now learn & write up the proof that the Henstock-Kurzweil integral integrates _every_ derivative. This is what we had in my calculus class on top of the outdated Riemann integral.
> Let f ... be Riemann integrable and F ... differentiable.
What many people don't notice the first time they read this in the fundamental theorem of Calculus is that this is a double criteria. That f needs to be integrable seems like an extraneous point when F is differentiable. This holds also for the Lebesgue integral. The understanding is usually that if F is differentiable then its derivative is integrable, that is, people understand the integral as an anti-derivative but the Riemann/Lebesgue integral version of the fundamental theorem of calculus only proves that if the function you want the anti-derivative of is integrable, so you have this separate requirement to prove that f is integrable having already proven F to be differentiable (to f).
However, this theoretical (because if you aren't a mathematician you won't be bothered by this sticking point, you'll just insist that the integral is the anti-derivative when an anti-derivative exists) defect is ameliorated by the Henstock–Kurzweil integral which is (I feel) a lot easier to define and understand than the Lebesgue integral. It is practically the Riemann integral with just a minor tweak: the delta in the delta-epsilon proof is allowed to vary by location (essentially, as you approach non-integrable singularities, you tend the delta towards zero).
For the Henstock-Kurzweil integral, if F is differentiable then f is (Henstock-Kurzweil) integrable. This happens because not every derivative is Riemann or Lebesgue integrable, you need a stronger integral.
> This post introduces the Riemann integral
Sweet! I'm keen to learn about the basic fundamentals of calculus!
> For each subinterval ...(bunch of cool maths rendering I can't copy and paste because it's all comes out newline delimited on my clipboard) ... and let m<sub>k</sub> and M<sub>k</sub> denote the infimum and supremum of f on that subinterval...
Okay, guess it wasn't the kind of introduction I had assumed/hoped.
Very cool maths rendering though.
As someone who never passed high school or got a degree thanks to untreated ADHD, if anyone knows of an introduction to the basic fundamentals of calculus that a motivated but under educated maths gronk can grok, I would gratefully appreciate a link or ten.
ChatGPT.
You can ask for a syllabus first, then go through it.
It's interactive, and it covers in detail everything you don't get. You can ask infinite many practice material, exercises, flashcards, or anything you want.
If you dive into Analysis (the underlying theory behind calculus) this book - "How to Think About Analysis" by Lara Alcock is the book I wish I had when I studied it. Calculus by Spivak is the book I learnt from but it is probably not the easiest, it is very thorough though.
3Blue1Brown has an excellent video series that introduces calculus using very intuitive animations and explanations: https://www.youtube.com/playlist?list=PLZHQObOWTQDMsr9K-rj53...
https://calculusmadeeasy.org/1.html
You could se if it helps with https://betterexplained.com/calculus/lesson-1/ or https://youtu.be/WUvTyaaNkzM
> if anyone knows of an introduction to the basic fundamentals of calculus that a motivated but under educated maths gronk can grok, I would gratefully appreciate a link or ten.
"The basic fundamentals of calculus" usually go under the name "real analysis".
You have many options for studying it.
MIT OpenCourseWare: https://ocw.mit.edu/courses/18-100a-real-analysis-fall-2020/
Free calculus-through-nonstandard-analysis textbook: https://people.math.wisc.edu/~hkeisler/calc.html
Lean4 game implementing Alex Kontorovich's undergrad course: https://adam.math.hhu.de/#/g/alexkontorovich/realanalysisgam...
I like the idea of the lean4 game, because if you do your work in lean you'll know whether you've made a mistake.
("Standard analysis" uses limiting behavior to ask what would happen if we were working with infinitely large or infinitesimally small values, even though of course we aren't really. "Nonstandard analysis" doesn't bother pretending and really uses infinitely large and infinitesimally small values. Other than the notational difference, they are the same, and a proof in one approach can be easily and mechanistically converted into the same proof in the other approach.)
Note that the ordinary course of study involves learning to do calculus problems first (in a "calculus" class), and studying the fundamentals second (in an "analysis" class). The textbook I linked is a "calculus" textbook, but there is a bit more focus on the theoretical backing because you can't rely on the student to learn about nonstandard analysis somewhere else.
Yeah, judging by the terseness, this is clearly aimed at undergrads. Then again, this is covered in literally every calculus class, so I'm not sure who this is supposed to be for.
https://minireference.com/
"The No Bullshit Guide to Math and Physics"
Thx for the plug @Delphiza
For anyone interested in checking out the book, there is a PDF preview here[1] and printable concept maps[2], which should be useful no matter which book you're reading.
[1] https://minireference.com/static/excerpts/noBSmathphys_v5_pr...
[2] https://minireference.com/static/conceptmaps/math_and_physic...
Fair, sorry about that
https://en.wikipedia.org/wiki/Calculus_Made_Easy#:~:text=Cal...
1910 book, but actually does the job well
Here's my understanding: 1: In the 'olden days' the area A(x) under the graph f(x) used to be approximated as a Riemann sum. 2: Using limits, as the delta x in the Riemann sum->0, we'd call that an integral and set it to be the exact area A(x). 3: If we then look at some small change in A(x), we might notice f(x) = A'(x)... mind blown. 4: since we can now say A is an anti-derivative of f, we have A(x)=F(x)+C (we have to add the C because the derivative of a constant is 0). 5: Using logic and geometry we have C=-F(a) which leads to... 6: The area under the graph f between [a,b] is A = F(b)-F(a). 7: We don't have to cry anymore about pages of Riemann sum calculations.
I recommend Math Academy + Mathematica + YouTube + ChatGPT, Gemini, or Claude Opus and a LOT of motivation.
Khan academy
I've studied the proofs before but there's still something mystical and unintuitive for me about the area under an entire curve being related to the derivative at only two points, especially for wobbly non monotonic functions.
I feel similar about the trace of a matrix being equal to the sum of eigenvalues.
Probably this means I should sit with it more until it is obvious, but I also kind of like this feeling.
It is not determined by the derivative, it's the antiderivative, as someone else mentioned. The derivative is the rate of change of a function. The "area under a curve" of the graph of a function measures how much the function is "accumulating", which is intuitively a sum of rates of change (taken to an infinitesimal limit).
Thanks for bringing some intuition!
> there's still something mystical and unintuitive for me about the area under an entire curve being related to the derivative
the discrete version is much clearer to me. Suppose you have a function f(n) defined at integer positions n. Its "derivative" is just the difference of consecutive values
Then the fundamental theorem is just a telescopic sum:Very. Very nice
If I tell you I have function f with f(a) = 10 and on it's path from a to b, the graph first increaes by 5 units then by another 10, and then later on drops by 25 units, you can immediately deduce that f(b) = f(a) + (+5 +10 -25) = 0. The fundamental theorem of calculus uses the same concept:
To see why \int_a^b f(x) dx = F(b) - F(a) with F'(x) = f(x),
we replace f with f' (and hence F with f) and get
\int_a^b f'(x) dx = f(b) - f(a).
Re-arranging terms, we get
f(b) = f(a) + \int_a^b f'(x) dx.
The last line just says: The value of function f at point b is is the value at point a plus the sum of all the infinitely many changes the function goes through on its path from a to b.
The antiderivative at x is defined as the area under the curve from 0 to x, which the Riemann sum gives a nice intuition for how you can get from the derivative.
So to get the area under the curve between a and b, you calculate the area under the curve from 0 to b (antiderivative at b) and subtract the area under the curve from 0 to a (antiderivative at a).
At least that's my sleep deprived take.
I took calculus in high school and college, and I don't think any of my instructors explained the intuition as well. So sleep-deprived or not, it's a great one!
There is some geometric intuition in wikipedia page for this theorem you may like :)
You meant antiderivative?
https://en.wikipedia.org/wiki/Fundamental_theorem_of_calculu...
> f is Riemann integrable iff it is bounded and continuous almost everywhere.
FWIW, I think this is the same as saying "iff it is bounded and has finite discontinuities". I like that characterization b/c it seems more precise than "almost everywhere", but I've heard both.
I mention that because when I read the first footnote, I thought this was a mistake:
> boundedness alone ensures the subinterval infima and suprema are finite.
But it wasn't. It does, in fact, insure that infima and suprema are finite. It just does NOT ensure that it is Riemann integrable (which, of course the last paragraph in the first section mentions).
Thanks for posting. This was a fun diversion down memory lane whilst having my morning coffee.
If anyone wants a rabbit hole to go down:
Think about why the Dirichlet function [1], which is bounded -- and therefore has upper and lower sums -- is not Riemann integrable (hint: its upper and lower sums don't converge. why?)
Then, if you want to keep going down the rabbit hole, learn how you _can_ integrate it (ie: how you _can_ assign a number to the area it bounds) [2]
[1] One of my favorite functions. It seems its purpose in life is to serve as a counter example. https://en.wikipedia.org/wiki/Dirichlet_function
[2] https://en.wikipedia.org/wiki/Lebesgue_integral
> FWIW, I think this is the same as saying "iff it is bounded and has finite discontinuities".
It is not: for example, the piece-wise constant function f: [0,1] -> [0,1] which starts at f(0) = 0, stays constant until suddenly f(1/2) = 1, until f(3/4) = 0, until f(7/8) = 1, etc. is Riemann integrable.
"Continuous almost everywhere" means that the set of its discontinuities has Lebesgue measure 0. Many infinite sets have Lebesgue measure 0, including all countable sets.
Ah, thanks for the clarification! Would it have been accurate then to have said:
"iff it is bounded and has countable discontinuities"?
Or, are there some uncountable sets which also have Lebesgue measure 0?
No, it's really sets of measure zero. The Cantor set is an example of an uncountable set of measure 0: https://en.wikipedia.org/wiki/Cantor_set
The indicator function of the Cantor set is Riemann integrable. Like you said, though, the Dirichlet function (which is the indicator function of the rationals) is not Riemann integrable.
The reason is because the Dirchlet function is discontinuous everywhere on [0,1], so the set of discontinuities has measure 1. The Cantor function is discontinuous only on the Cantor set.
Likewise, the indicator function of a "fat Cantor set" (a way of constructing a Cantor-like set w/ positive measure) is not Riemann integrable: https://en.wikipedia.org/wiki/Smith%E2%80%93Volterra%E2%80%9...
No that's not true either. A quick Google will reveal many examples, in particular the "Cantor set".
The Cantor set is uncountable and has Lebesgue measure 0.
Great example
"Almost everywhere" means "everywhere except on a set of measure 0", in the Lebesgue measure sense.
Here's an example of a Riemann integrable function w/ infinitely many discontinuities: https://en.wikipedia.org/wiki/Thomae%27s_function
Anyone interested in this should check out the Prologue to Lebesgue's 1901 paper: http://scratchpost.dreamhosters.com/math/Lebesgue_Integral.p...
It gives several reasons why we "knew" the Riemann integral wasn't capturing the full notion of integral / antiderivative
"Almost everywhere" is precisely defined, and it is broader than that. E.g. the real numbers are almost everywhere normal, but there are uncountably many non-normal numbers between any two normal reals.
"almost everywhere" can mean the curve has countably infinite number of discontinuities
“Almost everywhere” is a mathematical term and can mean two things (I think):
- except finitely many, or
- except a set of measure zero.
Here is used in the Lebesgue measure theory sense
What is the font used on the site?
Today I learned there's a CSS property for styling the first letter of a paragraph, neat. (https://css-tricks.com/almanac/properties/i/initial-letter/)
--edit: The font used for those initials is called Goudy Initialen: https://www.dafont.com/goudy-initialen.font
I love this -- I'll have to do something like that for my site. I always liked the big initials on the start of a paragraph. Though it feels a bit more prose-applicable than for non-fiction writing.
Technically "it depends on the browser settings," but the body font Alegreya is served directly by the site, so I think it would be the one used in almost all cases.
The math fonts used in the formulas are just the ones provided by KaTeX, which I think are just TeX's default math fonts.
That font, and how it's integrated with the math looks amazing. Katex for the math?
Seems like Katex from the scripts getting loaded. I love the design too, kinda medieval-chic.
Looks more early modern to me. :)
Already replied :)
The source code of the website is open if you wanna check it out!
Inspection suggests "https://online-fonts.com/fonts/alegreya"
Also https://online-fonts.com/fonts/goudy-initialen
Alegreya
Good job, David. Have a lollipop. Now learn & write up the proof that the Henstock-Kurzweil integral integrates _every_ derivative. This is what we had in my calculus class on top of the outdated Riemann integral.