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Calculus Made Easy (1910)

calculusmadeeasy.org

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Re: Calculus Made Easy (1910)

#61

Post by xelxebar »

aeneasmackenzie wrote: post #23

I like Calculus Made Easy because it uses informal infinitesimals. You can make these fully rigorous if you want and they're a much more intuitive technique than epsilon-delta.

Is it really all that intuitive, though? I mean, where does (dx)^2 = 0 come from?? Usually people say that, well, since dx is already small, then (dx)^2 is really really small, so for magical reasons it's okay to pretend that it's zero. I mean, if we're willy-nilly ignoring small things, why can't we ignore the already "infinitely small" dx?

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.

Re: Calculus Made Easy (1910)

#62

Post by mettamage »

I notice that I stumble over math over small but important details. I understand the big ideas, but then at chapter 4 in the book it says:

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}

Re: Calculus Made Easy (1910)

#63

Post by tobmlt »

jcoffland wrote: post #48
tobmlt wrote: post #22

Earlier quoted context omitted.

Do us a favor and delete your comment. This kind of talk can discourage and embarrass people who would otherwise be focused on learning. 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 sti…

Wow, that was reactionary. I'm not trying to discourage or embarrass anyone. My personal opinion is that the book is too dumbed down. It goes into the material way to slowly which makes it more difficult for me to stay focused on. That may not be other people's experience but I'm pretty sure some would agree. It's a matter of preference and I think I should be able to state mine without it being such a big deal.

Well, here you are defending the way you expressed your personal reaction to the book. Your reaction itself is of course fine. -But earlier you expressed your reaction to the book as if your particular experience of it were an absolute truth. Obviously (to both of us I have no doubt), the book is not anything in absolute terms, but you did not put it that way in your original comment. The original statement says flatly the book is "too dumbed down." This puts an implicit value judgement on anyone who might like this style of exposition. And a new learner is often _vulnerable_. So thank you for returning to clarify here.

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)

Re: Calculus Made Easy (1910)

#64

Post by gspetr »

mettamage wrote: post #62

I notice that I stumble over math over small but important details. I understand the big ideas, but then at chapter 4 in the book it says: 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…

4. Try to look for a different (one you might understand better) explanation of the same concept in different sources.

These can be youtube videos, other books, math.stackexchange.com, math forums, etc.

Re: Calculus Made Easy (1910)

#65

Post by mettamage »

gspetr wrote: post #64
mettamage wrote: post #62

I notice that I stumble over math over small but important details. I understand the big ideas, but then at chapter 4 in the book it says: 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…

4. Try to look for a different (one you might understand better) explanation of the same concept in different sources. These can be youtube videos, other books, math.stackexchange.com, math forums, etc.

Haha I'm now looking at Khan Academy.

I went to https://tutorme.com/ and went on a free trial.

Re: Calculus Made Easy (1910)

#66

Post by jackkinsella »

avinassh wrote: post #52

This is amazing! The fonts are easy on eyes and page renders beautifully. 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…

YouTube math god 3Blue1Brown has a lovely series of videos that visualize linear algebra: https://www.youtube.com/watch?v=fNk_zzaMoSs&list=PLZHQObOWTQ...

Re: Calculus Made Easy (1910)

#68

Post by dopkew »

In chapter 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.

Re: Calculus Made Easy (1910)

#69

Post by dopkew »

dopkew wrote: post #68

In chapter 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, ra…

And in chapter 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

Re: Calculus Made Easy (1910)

#70

Post by boatwoodwater »

dopkew wrote: post #69
dopkew wrote: post #68

In chapter 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, ra…

And in chapter 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

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.

Check https://en.wikipedia.org/wiki/Long_and_short_scales for more

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