Hacker News Board Hacker News, browsed like a classic message board
Live data from Hacker News

Fundamental Theorem of Calculus

david.alvarezrosa.com

Post reply on HN
Print view View original

31–40 of 70 posts

Re: Fundamental Theorem of Calculus

#31

Post by chillax »

EdwardDiego wrote: post #26

> 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 k and M k 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…

You could se if it helps with https://betterexplained.com/calculus/lesson-1/ or https://youtu.be/WUvTyaaNkzM

Re: Fundamental Theorem of Calculus

#32

Post by magicalhippo »

mchinen wrote: post #12

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.

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.

Re: Fundamental Theorem of Calculus

#33

Post by [deleted] »

jfarmer wrote: post #23
emacdona wrote: post #19

Earlier quoted context omitted.

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…

[deleted]

Re: Fundamental Theorem of Calculus

#34

Post by Delphiza »

EdwardDiego wrote: post #26

> 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 k and M k 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…

https://minireference.com/

"The No Bullshit Guide to Math and Physics"

Re: Fundamental Theorem of Calculus

#35

Post by dalvrosa »

mchinen wrote: post #12

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.

There is some geometric intuition in wikipedia page for this theorem you may like :)

Re: Fundamental Theorem of Calculus

#36

Post by UltraSane »

EdwardDiego wrote: post #26

> 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 k and M k 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…

I recommend Math Academy + Mathematica + YouTube + ChatGPT, Gemini, or Claude Opus and a LOT of motivation.

Re: Fundamental Theorem of Calculus

#38

Post by dalvrosa »

Jaxan wrote: post #16
emacdona wrote: post #11

> 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…

“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

Re: Fundamental Theorem of Calculus

#39

Post by dalvrosa »

thaumasiotes wrote: post #21
emacdona wrote: post #19

Earlier quoted context omitted.

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?

The Cantor set is uncountable and has Lebesgue measure 0.

Great example

Re: Fundamental Theorem of Calculus

#40

Post by dalvrosa »

EdwardDiego wrote: post #26

> 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 k and M k 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…

Fair, sorry about that
Post reply on HN