Fundamental Theorem of Calculus (david.alvarezrosa.com)
40 points by dalvrosa 11 hours ago
EdwardDiego 20 minutes ago
> 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.
chillax 12 minutes ago
You could se if it helps with https://betterexplained.com/calculus/lesson-1/ or https://youtu.be/WUvTyaaNkzM
dnemmers 16 minutes ago
mr_mitm 13 minutes ago
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.
Delphiza 3 minutes ago
"The No Bullshit Guide to Math and Physics"
homeonthemtn 17 minutes ago
https://en.wikipedia.org/wiki/Calculus_Made_Easy#:~:text=Cal...
1910 book, but actually does the job well
moi2388 18 minutes ago
Khan academy
mchinen an hour ago
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.
dalvrosa a minute ago
There is some geometric intuition in wikipedia page for this theorem you may like :)
magicalhippo 10 minutes ago
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.
ironSkillet an hour ago
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).
sambapa an hour ago
You meant antiderivative?
emacdona an hour ago
> 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
mjdv an hour ago
> 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.
emacdona an hour ago
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?
jfarmer 44 minutes ago
ironSkillet an hour ago
thaumasiotes an hour ago
jfarmer an hour ago
"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
bandrami an hour ago
"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.
sambapa an hour ago
"almost everywhere" can mean the curve has countably infinite number of discontinuities
Jaxan an hour ago
“Almost everywhere” is a mathematical term and can mean two things (I think):
- except finitely many, or
- except a set of measure zero.
mellosouls 2 hours ago
bikrampanda 2 hours ago
What is the font used on the site?
viscousviolin 2 hours ago
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
crispyambulance 2 hours ago
That font, and how it's integrated with the math looks amazing. Katex for the math?
viscousviolin 2 hours ago
Seems like Katex from the scripts getting loaded. I love the design too, kinda medieval-chic.
eru an hour ago
emmelaich an hour ago
Inspection suggests "https://online-fonts.com/fonts/alegreya"
thaumasiotes an hour ago
genezeta 2 hours ago
Alegreya
shmoil 23 minutes ago
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.