Hacker News Board Hacker News, browsed like a classic message board
Live data from Hacker News

Calculus Made Easy by Silvanus P. Thompson (1910)

calculusmadeeasy.org

Post reply on HN
Print view View original

21–30 of 64 posts

Re: Calculus Made Easy by Silvanus P. Thompson (1910)

#21

Post by pfdietz »

dvt wrote: post #12
Qem wrote: post #3

Silvanus' book makes calculus simple by adopting an infinitesimal approach, like Newton and Leibniz did when they invented calculus. But that approach was shunned by mathematicians for a long time, because it was only made rigorous in the 60s. After Silvanus' book, I also recommend Elementary Calculus: An Infinitesimal Approach, by professor Jerome Keisler, for those interested in this alternative pathway to calculus…

I think I'd also prefer the infinitesimal version of calculus, but the idea of limits is applicable to many other areas of math, not just calculus (evaluating infinite series, for example). So learning limits is probably a better pathway to higher mathematics.

And non-standard analysis is useful elsewhere in math as well. Here's an example:

https://discreteanalysisjournal.com/article/87772-a-simple-c...

Re: Calculus Made Easy by Silvanus P. Thompson (1910)

#22

Post by xcjs »

I love projects like this, but I really wish instead of converting text resources to web sites that these projects would produce epub outputs. It's great for distribution, offline reading, and scaling to different display sizes, aspect ratios, and resolutions.

Re: Calculus Made Easy by Silvanus P. Thompson (1910)

#23

Post by btilly »

finite_depth wrote: post #17
btilly wrote: post #7

Earlier quoted context omitted.

I have mixed feelings about this. I've been through nonstandard analysis. My response was, "We shouldn't need the axiom of choice to define the derivative." But I think that it is extremely important to understand that the infinitesimal notation really MEANS something. Here is some Python to demonstrate. # d is a functor. It takes a function and returns a second function. # The second function captures the change in…

Most calculus students don't need the full formal power of rigorous analysis. Calculus, taken alone and with the elementary properties of the real numbers assumed and a few elementary properties of infinitesimals (0 <<< infinitesimal^2 <<< infinitesimal <<< any positive real), can get you a lot of power for very little formal work.

Absolutely true. However this comes at the cost of having to not think too hard about issues like "what is a function".

You generally don't run into trouble with 1, x, 1/x, sin(x) and the like. But when you push past the analytic functions, you wind up having to unlearn a lot of ideas so that you can learn an entirely different foundation.

Re: Calculus Made Easy by Silvanus P. Thompson (1910)

#24

Post by marai2 »

wallflower wrote: post #16

My calculus teacher tried to talk me out of taking the AP Calculus AB exam because I wasn’t doing that well in his class. “Calculus the Easy Way” which teaches in the setting of a mythical fantasy world where applied problems are solved with calculus unlocked the secret of calculus for me. I ended up scoring a “4” which was a big triumph for me, as it placed me out of taking calculus with all the people who did not w…

Douglas Downing's Trigonometry Made Easy is also a really great book for non-maths people. Same approach of situating Trigonometry in a mythical fantasy land. I remember Trigonometry as a chore of trying to memorize double-angle formulas and such, but this book really helped connect trig in an intuitive way.

I didn't know he had a calculus book, I'll have to check it out now.

Re: Calculus Made Easy by Silvanus P. Thompson (1910)

#25

Post by ykonstant »

dvt wrote: post #12
Qem wrote: post #3

Silvanus' book makes calculus simple by adopting an infinitesimal approach, like Newton and Leibniz did when they invented calculus. But that approach was shunned by mathematicians for a long time, because it was only made rigorous in the 60s. After Silvanus' book, I also recommend Elementary Calculus: An Infinitesimal Approach, by professor Jerome Keisler, for those interested in this alternative pathway to calculus…

I think I'd also prefer the infinitesimal version of calculus, but the idea of limits is applicable to many other areas of math, not just calculus (evaluating infinite series, for example). So learning limits is probably a better pathway to higher mathematics.

More importantly for this audience, the idea of limits lies at the heart of numerical analysis. Explicating the quantifiers of the definition of a limit is the first step in obtaining control over any estimator of your data. This, among other things, is why I am perpetually baffled at the people in this audience who say that limits is something "they will never need". Limits, their algebra (which subsumes all of high school algebra and inequalities) and their delicate analysis is what makes a ton of numerical algorithms work; Higham's classic has it all, and is perfectly clear about it all.

Re: Calculus Made Easy by Silvanus P. Thompson (1910)

#27

Post by btilly »

dvt wrote: post #12
Qem wrote: post #3

Silvanus' book makes calculus simple by adopting an infinitesimal approach, like Newton and Leibniz did when they invented calculus. But that approach was shunned by mathematicians for a long time, because it was only made rigorous in the 60s. After Silvanus' book, I also recommend Elementary Calculus: An Infinitesimal Approach, by professor Jerome Keisler, for those interested in this alternative pathway to calculus…

I think I'd also prefer the infinitesimal version of calculus, but the idea of limits is applicable to many other areas of math, not just calculus (evaluating infinite series, for example). So learning limits is probably a better pathway to higher mathematics.

I think that Big O / little o are both more approachable and provide a richer understanding than limits. See, for example, https://micromath.wordpress.com/2008/04/14/donald-knuth-calc... to see Donald Knuth agreeing.

As an example of the conceptual richness, pick up a Calculus book and flip to the problem section for L'Hôpital's rule. Without using any special rules at all, attempt to write them out in o-notation and observe that you generally don't need L'Hôpital's rule to work them out. It is possible to produce examples that can be calculated by L'Hôpital's rule, but not by simply understanding o-notation. But it isn't easy, and you're unlikely to find them in textbooks.

It is probably true that as you go on, limits are more useful in higher mathematics than o-notation. But o-notation is far more useful in most subjects that use mathematics. Given how easy it is to master limits if you know o-notation, why not teach o-notation first?

Re: Calculus Made Easy by Silvanus P. Thompson (1910)

#28

Post by ezekiel68 »

btilly wrote: post #23
finite_depth wrote: post #17

Earlier quoted context omitted.

Most calculus students don't need the full formal power of rigorous analysis. Calculus, taken alone and with the elementary properties of the real numbers assumed and a few elementary properties of infinitesimals (0 <<< infinitesimal^2 <<< infinitesimal <<< any positive real), can get you a lot of power for very little formal work.

Absolutely true. However this comes at the cost of having to not think too hard about issues like "what is a function". You generally don't run into trouble with 1, x, 1/x, sin(x) and the like. But when you push past the analytic functions, you wind up having to unlearn a lot of ideas so that you can learn an entirely different foundation.

You're right. But then again, lots of scaffolding gets discarded when an arch gets constructed also.

Re: Calculus Made Easy by Silvanus P. Thompson (1910)

#29

Post by MrBlueIncognito »

tarkin2 wrote: post #26

Has anyone been through the calculus courses on khan academy? What did you think?

I went through them a long time ago. It’s not the most in-depth resource, but you will learn enough calculus for when you need to actually apply it or even just pass tests. Also learning math on KA is really fun, there’s something they just get right. I’d definitely recommend giving it a try.

If you don’t feel satisfied after going through the courses, you can always pick up a book afterwards to dig deeper.

Re: Calculus Made Easy by Silvanus P. Thompson (1910)

#30

Post by MrBlueIncognito »

xcjs wrote: post #22

I love projects like this, but I really wish instead of converting text resources to web sites that these projects would produce epub outputs. It's great for distribution, offline reading, and scaling to different display sizes, aspect ratios, and resolutions.

I wonder if there will be a time when textbooks will be created in digital-first format, instead of being mere replicas of what print books are. It doesn’t have to be static text and images on A4 pages.
Post reply on HN