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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 This is what we had in my calculus class on top of the outdated Riemann integral.
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 This is what we had in my calculus class on top of the outdated Riemann integral.
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 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) d…
+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.
Earlier quoted context omitted.
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?
> 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…
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.
Earlier quoted context omitted.
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) d…
how does this work for these two different paths which arrive at the same endpoint: +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.
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
> 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 want to get into calculus, I strongly recommend you ensure you're adept at algebra.
> 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…