Fundamental Theorem of Calculus(david.alvarezrosa.com) |
Fundamental Theorem of Calculus(david.alvarezrosa.com) |
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.
It can take you from knowing your multiplication tables all the way to all the math you need for an engineering and/or CS degree.
"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... (also includes videos of the course lectures)
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.
1910 book, but actually does the job well
If you want to get into calculus, I strongly recommend you ensure you're adept at algebra.
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 for an unlimited amount practice material, exercises, flashcards, or anything you want.
"The No Bullshit Guide to Math and Physics"
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...
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.
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
--edit: The font used for those initials is called Goudy Initialen: https://www.dafont.com/goudy-initialen.font
The math fonts used in the formulas are just the ones provided by KaTeX, which I think are just TeX's default math fonts.
The source code of the website is open if you wanna check it out!
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)
f(b) - f(a) = \sum_a^b f'(n)The fact that the derivative of this accumulator function is equal to the original function, this is the fundamental theorem of calculus, and I violently agree with you that this part is shockingly, unexpectedly beautiful
Where it gets interesting is that if I were to naively construct this accumulator in a non-computational, infinitesimal manner, I would probably start using the idea that the infinitesimal accumulation is related to the instantaneous slope and using that somehow with the previous accumulation. But this is going the opposite direction of derivatives that the fundamental theorem uses.
I understand there are a few other types of integrals that slice things different ways that might be more intuitive. I vaguely recall some (non fundamental theorem) proof that made Riemann integrals click for a while.
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.
+5 +10 +0 -25 = -15
+5 +12 -2 -25 = -15
They have different graphs if you consider the values above sampling points, this is what parent is asking.
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.
There are many types of examples, and many different reasons why I don't find a particular connection or connection type surprising. So I can concentrate on memorising them, and building intuition.
For the Fundamental Theorem of Calculus:
- int f' = f: the sum of the change is the thing itself. E.g. pour water in the bathtub, if you sum the rate you pour, that's the total water in the bathtub
- int f' = f(t2) - f(t1) : same but water differences between 2 times.
- (int f)' = f: the rate of the sum is the function itself. If you go and integrate your function f, the integrate function's change rate at x is f(x)
- and so on.
> the area under an entire curve being related to the derivative at only two points
This is a very wrong sentence. The area under f on [a,b] is not related to the derivative of f at a and b. The area under f on [0,x] is a real function F(x) by definition, and there is nothing surprising that the area of f on [a,b] is F(b)-F(a). Simple interval arithmetic. Granted F, the integral function, is related to f: F' = f.
tl;dr : in the "fundamental theorem of calculus" there are 2 main observations:
- the operators 'sum of' and 'change rate' are each other's inverse and commute:
(int f)' = int (f') = f
F' = f <==> int f = F
- from interval arithmetics:
S(a,b) = S(0,b) - S(0,a)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.
"iff it is bounded and has countable discontinuities"?
Or, are there some uncountable sets which also have Lebesgue measure 0?
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
let x_i be in (a,b) with any i drawn from [0,N] and x_0=a and x_N=b
then
int_a^b d/dx F = (F(x_1) - F(x_0))+ (F(x_i+1) - F(x_i))+ (F(x_n) - F(x_n-1)) = F(x_N) - F(x_0)
the derivative produces a ratio of differences and the integral is a weighted sum of those same differences
every middle term is added and subtracted. and only the endpoints remain with opposite sign
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...