Hi all! I'm happy to announce a public availability of the BetterExplained Guide To Calculus. You can read it online:
This is a companion discussion topic for the original entry at http://betterexplained.com/articles/calculus-beta/
This is a companion discussion topic for the original entry at http://betterexplained.com/articles/calculus-beta/
I’m very excited for this course after reading your book. I went through (and enjoyed) a lot of higher math for my degree. Even with that background though, getting back to the basics the way you present it is awesome and fun. Makes me wish I’d had this stuff available when I was slogging through the hard parts the first time around.
Do you have a favorite go-to source for the historical stories you mix into the lessons (like Archimedes unrolling the rings of a circle to find the area)? I’ve been trying to get a good collection of those kinds of stories for my kids to use for intuition building as they learn math and science, but most of what I’ve found so far hasn’t been that great.
Thanks Erik – really glad the approach is clicking for you (and think it’s awesome you’re looking for similar ways to explain it to your kids).
You know, I’ve started to take a more biographical approach to math. So when we have some result (Limits are the theory that XYZ) I’ll usually look on Wikipedia for a quick overview of the history. Not the math, but just the people involved, what they were thinking, earlier versions, etc.
Eventually you wind up on ideas like the Archimedean Property (http://en.wikipedia.org/wiki/Archimedean_property) which is an ancient principle, but comes down to whether you “allow” there to be infinitely large (or small) elements in your numbers (that’s a decision?). Despite years of math classes, I’d never heard of that choice being spelled out, let along that it’s an ancient idea! It makes a lot of things snap into place for me. Some number systems allow them, some don’t. If you do, you can have infinitesimals (what Newton/Leibniz used), if you don’t, you get the theory of limits (1800s math formalizations).
I’d like to put in more footnotes to Wikipedia, many of the stories are things that seemed interesting as I poked around biographies.
For example, unrolling the triangle was in Archimedes treatise called “Measurement of a Circle”:
Why do we so often show what he found (pi*r^2) but not how he found it? Madness :).
[…] BetterExplained Calculus Course Now Available (Public Beta) […]
Am reviewing algebra (“Forgotten Algebra”) and will then tackle calculus for non-math majors (“Forgotten Calculus”). My goal is to learn higher math (not for college–I’m 64), just for me. What would you suggest after I go through your book? Any texts you are ardently fond of?
Hi Len! I haven’t looked closely at enough books, but I like the general approach of the “Cartoon guides to XYZ” or “The Manga guide to ABC”. My general philosophy is to get a blurry overview of a subject, building an intuition, then refine the details with the more formal / technical descriptions. Having a medium like a cartoon forces the author to use diagrams and visualizations :).
K–
In learning the Maclaurin series, I was struck by how the description of a whole curve–or at least the seeds of that description–could be contained in something as nebulous as the nth derivative. It reminded me of fractal geometry, where the concept of self-similarity can be found on every scale, from the smallest to the largest.
Hey Tim, thanks for the comment. Great point. There’s something neat that a curve holds so much information in the derivative – it’s almost like a hologram (or a cell in a body) where any tiny portion contains information about the greater whole. The equation for each derivative can be used to create the parent curve.
hi khalid…, i have thought of a small intuitive idea on “laplace transform” need your suggestions on it…
Laplace transform are basically used to solve differential equation. we know that derivative of e^x =e^x the same function… so if we can express a function f(t) in terms of e^x all the D operators will vanish… and we will get a simple picture … something like watching a function wearing exponential glasses … and how do we do that… by division…like how many 2 are there in 10 ??? =10/2=5 … likewise how much (scaling factor) of e^xt in f(t)…??= f(t)/e^xt… but since the function f(t) is extending in time ,so we have to integrate to get real expression of f(t) in e^xt. thus we get L{f(t)} =integration f(t)*e^-xt dt…
i also think the term “s” used to denote variable in laplace transform is misleading… it should be “sigma” (i don’t know how to write its symbol 
:D)…
my textbook uses s>o ,s<k like terms where s is complex variable…and i have learned that complex variables do not have ordering property (i.e they can’t be compared by equality or inequality ) and that confused me … i even had a argument with the teacher…
need some intuitive lessons on complex integration and analytic functions…help to rescue