Visual calculus
en.wikipedia.org
en.wikipedia.org
What this kind of thing is really good at is giving good intuition for understanding a proved concept or thinking about a potential solution to an unproven one. It doesn't actually replace having to then do the math behind it to see if it really makes sense. Even a bog standard classroom calculus textbook will show a visual representation of e.g. Simpson's rule before dumping the actual equations and derivations on you.
"Birth of Calculus" (1986)
Edit: I don't think the video leaves itself enough time to do a good job covering this point. But there's a clearer description of the history at Wolfson (2001) "The Crooked Made Straight: Roberval and Newton on Tangents" AMM 108(3): 206–216 https://jstor.org/stable/2695381
Those are visual in nature, but the rules must be rigorously followed, if they are the resulting proofs are reliable. The steps of the proof may also be written out textually, but that's merely a translation, the construction itself is a proof. It's how we can tell the difference between a diagram which looks like a trisected angle, and a construction of a trisected angle, which has been proved impossible in the general case.
Part of the point I was making is that proof by construction isn't really a "visual proof" at all, it's a proof system where the steps in the proof are added to a diagram, according to exact rules which must be followed at every step.
This is the furthest from encyclopedic language you can possibly get. Vague, unsourced, flowery and subjective.
I found these courses allowed me to do calculations (e.g. contour integrals, conformal transformations) that were useful but always felt like black magic. I don't have an intuitive feel for the subject. I am wondering if this book will help me with that?
- https://www.goodreads.com/book/show/58059196-make (geometry)
http://web.archive.org/web/20010421091011/http://www.cco.cal...
> This problem can be easily solved by the methods developed in this chapter. The reader may wish to try solving it before reading the chapter.
That's a great way to motivate the methods and generate appreciation for them.
The area of a tangent sweep is equal to the area of its tangent cluster, regardless of the shape of the original curve."
Intuition tells me that there's something definitely there...
"If you want to look smart at a maths conference, just wait for the presenter to say something that everyone reacts to, then raise your hand ask 'Yes, but does it generalize?'"
And no, there is no intent on my part to suggest that you are doing the equivalent of that here!
The model of taking the same swept shapes and rearranging them into another shape whose area or volume is easy to calculate is the same approach though.