Calculus Made Easy (1910)
calculusmadeeasy.org
calculusmadeeasy.org
Learned something already!
In Medieval Latin, pars minuta prima "first small part" was used by mathematician Ptolemy for one-sixtieth of a circle, later of an hour (next in order was secunda minuta, which became second).
>I To deliver you from the Preliminary Terrors
>The preliminary terror [..] can be abolished once for all by simply stating what is the meaning–in common-sense terms–of the two principal symbols:
(1) d, which merely means “a little bit of.” Thus dx means a little bit of x; or du means a little bit of u.
(2) ∫, which is merely a long S, and may be called (if you like) “the sum of.”
As someone who taught Calculus, how I wish every book on the subject started like that!
If I ever have to do it again, I will use this book. Wish I had known about it earlier.
The book includes a very important chapter on compound interest, which is too often glossed over in texts used today. I wrote notes to remedy that (as an extra-credit reading project for the students), and was glad to find that the book has a similar approach:
http://romankogan.net/math/A_paper_of_interest/A_Paper_of_In...
First, why "d"? Well, "d" is for "difference". As in: as x changes from x_1 to x_2, the difference (x_2 - x_1) -- when it's very small.
But wait, there's more!
The commonly used symbols for finite difference like that is the Greek letter Delta: Δ
For a list of values x_1, x_2, x_3, x_4,.. we write Δx_i = (x_i - x_{i-1}). That is, Δx_i is the i'th change. (Side note: an airline had a marketing slogan Change is Delta, which some nerd must have been immensely proud of).
Ok, bear with me for a bit more!
The symbol we use for finite sums is Σ: we write Σy_i = y_1 + y_2 + ... + y_n
Why Σ? That will have to wait a little.
Sums and differences cancel out:
ΣΔy_i = (y_1 - y_0) + (y_2 - y_1) + (y_3 - y_2) + ... + (y_n - y_{n-1}) = y_n - y_0
That is, summing up small succesive changes gives you total change. Simple?
Now apply this to the situation where the small changes in the quantity you are looking at are proportional to changes in another:
Δy_i = Δx_i * f(x_i)
Say, y is position, x is time; then f(x_i) is the speed at time x_i: as time increases a little, so does your position; the ratio of the changes is the speed. Δy_i is how much you moved from time x_{i-1} to time x_i, which is proportional to change in time Δx_i.
Note that f(x_i) = Δy_i/Δx_i here (speed = change in position div. by change in time).
Now write:
y_n - y_0 = ΣΔy_i = Σf(x_i)Δx_i
Again, just summing up small changes to get the net chnage.
NOW, what you've been waiting for!
Imagine you took infinitely many measurements. The changes become infinitely small, and the sum becomes of infinitely many things.
We need new notation for this.
But let's keep it similar. Instead of using Greek letters, let's use the same letters... in Latin.
Δx becomes dx
ΣΔy becomes S dy
And, with some sloppy handwriting of the letter S, the net change equation becomes:
y_final - y_initial = ∫ f(x)dx
where
f(x) = dy/dx.
You now see that Σ and the sloppy S -- ∫ -- stand for Sum.
And that, my friend, is pretty much all there is to Calculus and its symbols, fundamentally[1].
[1]https://en.m.wikipedia.org/wiki/Fundamental_theorem_of_calcu...
How I wish there were books like this written for more topics!
—that's how I originally came across this book (reading Baez's recommended math texts for various subjects).
There's kind of a funny story/legacy behind the book too: Thompson originally published the book under "F.R.S." disguising his actual identity, but letting his peers know it was written by one of their own—a Fellow of the Royal Society—in spite of his knowing that they'd disapprove of the book.
I'd read somewhere, too (maybe in Gardner's preface?), that it remains a 'secret favorite' of many mathematicians who wouldn't welcome the social consequences of admitting this.
Thanks for making this resource available and giving it some exposure.
This is the revolution that software development is long overdue for.
If you are looking for vidoes, then check lectures by Herber Gross [0] on Youtube. These were recorded in 70s. They are in black & white, gives a feeling of watching some old beautifully shot movie. He goes into basics and gives you a taste of all derivations, by hand. Watch the first lecture by yourself [1] and you will immediately realise how good are these.
On a similar note, any similar resources like the one submitted, but for Linear Algebra? I am aware of Gilbert Strang's book [2] and vidoes [3], but I find them advanced for a beginner.
[0] - https://en.wikipedia.org/wiki/Herbert_Gross
[1] - https://www.youtube.com/watch?v=MFRWDuduuSw
[2] - https://ocw.mit.edu/courses/mathematics/18-06-linear-algebra...
I've often thought that an interesting treatment would start with differences and sums of integers as approximations, demonstrate their errors and then introduce reals and limits as a tool for making better theoretical models using the infinite "zoom button" continuity property of the reals.
http://www.nlc-bnc.ca/eppp-archive/100/201/300/cdn_medical_a...
eg Something like 'limit(f(x),x->a) = L' is just 'f(x) ≈ L when x ≈ a'
Personally, I always found hand-waving such an infinitesimal explanation to be much more frustrating than simply building the darn things from pieces I already understand.
Well done!
y+dy = (x+dx)^-2
is equal to
x^−2 * (1 + dx/x)^−2
[1]
To me (not that strong at math) this isn't apparent at all.
I have a couple of options here:
1. Spend a couple of hours fiddling around and trying to figure out the answer.
2. Hopefully find some app.
3. Ask a friend.
Regarding the options: I don't have a friend and I don't have an app. If you wouldn't know how to solve this, then what other strategies for understanding this are there?
[1] The LaTeX version:
y+dy &= (x+dx)^{-2} \\ &= x^{-2} \left(1 + \frac{dx}{x}\right)^{-2}
These can be youtube videos, other books, math.stackexchange.com, math forums, etc.
I went to https://tutorme.com/ and went on a free trial.
As for the expression, you just have to expand it:
y+dy = (x+dx)^-2
= 1 / (x+dx)^2
= 1 / (x^2 + 2xdx + d^2x)
= 1 / [x^2 * (1 + 2dx/x + d^2x / x^2)]
= 1 / [x^2 * (1 + dx/x)^2]
= [x^2 * (1 + dx/x)^2]^-1
= x^-2 * (1 + dx/x)^-2
> Let us think of x as a quantity that can grow by a small amount so as to become x+dx, where dx is the small increment added by growth. The square of this is x2+2x⋅dx+(dx)^2. The second term is not negligible because it is a first-order quantity; while the third term is of the second order of smallness, being a bit of, a bit of x^2.
It seems to me that the third term is actually a bit of a bit of x, rather than of x^2.
>Now if, for such a purpose, we regard 1/1,000,000 (or one millionth) as a small quantity, then 1/1,000,000 of 1/1,000,000, that is 1/1,000,000,000,000 (or one billionth) ..
1/1,000,000,000,000 is actually one trillionth
Check https://en.wikipedia.org/wiki/Long_and_short_scales for more
1,000,000,000, i.e. one thousand million, or 109 (ten to the ninth power), as defined on the short scale. This is now the meaning in both British and American English.
Historically, in British English, 1,000,000,000,000, i.e. one million million, or 1012 (ten to the twelfth power), as defined on the long scale. This is one thousand times larger than the short scale billion, and equivalent to the short scale trillion.
Bring back the milliard!
[0]: https://en.wikipedia.org/wiki/Long_and_short_scales [1]: http://calculusmadeeasy.org/2.html
I know it’s a quibble and no fault of the author’s that time has passed, but smooothing out litttle bumps get more people deeper into the content.
> About this edition & thanks > The text is based on the PDF version from Project Gutenberg converted to html by hand.
> Thanks to Paula Appling, Don Bindner, Chris Curnow, Andrew > D. Hwang and Project Gutenberg Online Distributed Proofreading Team for preparing the original PDF.
> The theme is borrowed from Dive Into HTML5 by Mark Pilgrim released under the CC-BY-3.0 license.
Besides that, there is an awful lot of literature that you must be unable of reading if you find any reference to historical norms of past times to be offputting, which is incredibly dangerous. What do you think would have happened to human progress if muslim or renaissance christian scholars felt like you about the texts from classical antiquity that they learnt so much from?
https://ocw.mit.edu/ans7870/resources/Strang/Edited/Calculus...
Calculus Made Easy seems too dumbed down to me.
Please don't insult people earlier in their education than you with slurs like "dumbed down". Education is not a competition, you don't need to elevate yourself by calling others inferior.
Just link to the book, which is good, and save us your little puffing yourself up bit. I am sorry but I cannot sit by and watch someone belittle people who would want to learn.
edit: sorry, I am a bit high strung today. Defending tomorrow afternoon. Whatever though. the above is still true. We have an epidemic of "make those who would try hard feel stupid" and it needs to end.
Seriously, +1 on the above -- as someone who has been defeated several times by the calculus terrors preliminary and otherwise, I can say with some confidence that the last thing calc-shy students need is being made to feel dumber.
Down-to-earth books like the one in question are a boon.
To anyone struggling through calculus for the first time: Use what works! For all we know, Strang himself might of learned from Calculus Made Easy. He'd be in good company if so, though it seems like RPF was rather free with the calc books, if ya know what I mean. (see the other thread)