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Fundamental Theorem of Calculus

david.alvarezrosa.com

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Re: Fundamental Theorem of Calculus

#51

Post by ivansavz »

Delphiza wrote: post #34
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"

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...

Re: Fundamental Theorem of Calculus

#52

Post by bmacho »

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…

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 for an unlimited amount practice material, exercises, flashcards, or anything you want.

Re: Fundamental Theorem of Calculus

#53

Post by BlackFly »

> 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.

Re: Fundamental Theorem of Calculus

#54

Post by thaumasiotes »

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…

> 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... (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.

Re: Fundamental Theorem of Calculus

#55

Post by bmacho »

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.

I have internalised that in mathematics nice things come in bouquets. If there is a thing defined with properties A, B, C, and there is an other thing defined with properties D, E, F, then chances are that those 2 things are the same thing, because there are only so few nice concepts.

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.
Also someone mentioned discrete functions, partial sums and difference series are indeed easier. Say, F is your gross money and f is your monthly salary, or F is gross amount of rain and f is daily rain. Summing a series or taking differences between 2 consecutive data points are each other's inverses.

> 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)

Re: Fundamental Theorem of Calculus

#56

Post by ghighi7878 »

BlackFly wrote: post #53

> 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…

Henstock-Kurzweil is a neat teaching trick. Often also because it shows that definition of riemann integration is not the only possible one. It leads a good motivation for lebesque later but also to of importance of spaces.

Re: Fundamental Theorem of Calculus

#57

Post by nimonian »

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.

If you think of it as being an accumulator function it can feel a bit more natural - the _definition_ of this accumulator is that, F(x) is the area from 0 to x

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

Re: Fundamental Theorem of Calculus

#58

Post by globalnode »

chadcmulligan wrote: post #46
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…

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.

Table of contents look really helpful to understanding.

Re: Fundamental Theorem of Calculus

#59

Post by viscousviolin »

ghighi7878 wrote: post #56
BlackFly wrote: post #53

> 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…

Henstock-Kurzweil is a neat teaching trick. Often also because it shows that definition of riemann integration is not the only possible one. It leads a good motivation for lebesque later but also to of importance of spaces.

Does it usually get taught in the undergrad maths curriculum?

Re: Fundamental Theorem of Calculus

#60

Post by mchinen »

nimonian wrote: post #57
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.

If you think of it as being an accumulator function it can feel a bit more natural - the _definition_ of this accumulator is that, F(x) is the area from 0 to x 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

Thank you for wording it better than I could. On the computational side, the integral as a function that measures how the another function accumulates intuitively lets you measure the area under the curve between two points only requires two variables.

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.

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