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Visual calculus
- analognoise 3y agoThis is too cool thank you
- revskill 3y agoYou mean cold ?
- throwoutway 3y agoUnfortunately, all the links/images in a primary source that the wiki uses are broken: http://www.cco.caltech.edu/~mamikon/calculus.html http://www.cco.caltech.edu/~mamikon/calculus.html
- Infernal 3y agoSpot checking a few, the images/links here seem to have been archived. http://web.archive.org/web/20010421091011/http://www.cco.caltech.edu/~mamikon/calculus.html http://web.archive.org/web/20010421091011/http://www.cco.cal...
- Sniffnoy 3y agoDoes Mamikon's theorem have any good generalizations, say to higher dimensions?
- mindcrime 3y agoI have no idea, but your comment does remind me of a joke I heard once: "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!
- eru 3y agoThat might be especially fun at a conference on 'generalised abstract nonsense', ie category theory.
- Sharlin 3y agoShame that nobody can come up with a meta-category theory because category theory is already its own metatheory.
- bordercases 3y agoDoes it scale though?
- mindcrime 3y agoMongoCalculus™ is Web Scale!
- Someone 3y agoThere’s “Volumes of Solids Swept Tangentially Around General Surfaces” (https://forumgeom.fau.edu/FG2015volume15/FG201504.pdf https://forumgeom.fau.edu/FG2015volume15/FG201504.pdf) by Tom M. Apostol and Mamikon A. Mnatsakanian (the “Mamikon” of this theorem)
- jameshart 3y agoThis Mathologer video talks about applying a similar approach to some 3D calculus problems - although the tangent swept shapes are arcs not triangles. https://www.youtube.com/watch?v=5q_sfXY-va8&t=0s https://www.youtube.com/watch?v=5q_sfXY-va8&t=0s 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.
- lupire 3y agoI came to post this. Great channel. More in depth than 3B1B but less fancy animations. But he makes his videos in PowerPoint! Not a bullet point in sight, so Edward Tufte would allow it
- JKCalhoun 3y agoReminds me of the simpler geometric example where you show that the area of a parallelogram is base X height, the same as a rectangle, by making a cut in the parallelogram perpendicular to the base and sliding/joining the severed piece to the other edge to show that you can create a rectangle with the same area.
- n_plus_1_acc 3y agoThat's what de did in fifth grade. I think it's very intuitive because using paper, you can see the area doesn't change.
- Sharlin 3y agoAlso the similar one with a bit more calculus, where you slice a circle (well, a disk to be precise) into an even number n of identical sectors and reorder them to form a parallelogram-ish shape (with "bumpy" bases) that approaches a rectangle with side lengths r and πr as n → ∞. This is an easy way to visualize why the enclosed area of a circle is πr². (You do have to take as a given that the circumference equals 2πr though.)
- gogurt2 3y agoFans of this should absolutely check out "Visual Complex Analysis" by Tristan Needham. No other book out there like it.
- jacobolus 3y agoThese are both neat, but entirely unrelated.
- bainsfather 3y agoI've been considering reading this book - can you tell me what I might get from it in addition to what I learned in undergraduate maths/physics courses? 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?
- auggierose 3y agoThere is now also "Visual Differential Geometry and Forms", also by Needham.
- billfruit 3y agoThere was something in that book on the relationship between Taylor series and Fourier series which I haven't seen anywhere else, however I'm unable to recollect the details. Do you know what is the relation between these two structures as described in the book?
- knightoffaith 3y agoI'm reminded of this professor explaining the gradient geometrically and how it leads to really elegant solutions to some problems: https://www.youtube.com/watch?v=uP8V-O8hncI https://www.youtube.com/watch?v=uP8V-O8hncI
- JKolios 3y ago"Many problems that would otherwise seem quite difficult yield to the method with hardly a line of calculation." This is the furthest from encyclopedic language you can possibly get. Vague, unsourced, flowery and subjective.
- mofunnyman 3y agoThat's what they don't get paid for.
- auggierose 3y agoWhat I like about the book referenced is that each chapter starts like this: > 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.
- fnordsensei 3y agoA skill in the fantastic game Disco Elysium: https://discoelysium.fandom.com/wiki/Visual_Calculus https://discoelysium.fandom.com/wiki/Visual_Calculus
- ogogmad 3y agoLooks like a special case of an integral substitution, but done visually.
- lupire 3y agoYes. Nearly every integration technique is some kind of substitutions.
- WillAdams 3y agoA series of books which builds (literally --- the books describe the use of 3D printing and Lego bricks) up to this might be: - https://www.goodreads.com/book/show/58059196-make https://www.goodreads.com/book/show/58059196-make (geometry) - https://www.goodreads.com/book/show/123127774-make https://www.goodreads.com/book/show/123127774-make - https://www.goodreads.com/book/show/61739368-calculus https://www.goodreads.com/book/show/61739368-calculus
- zamadatix 3y agoThe caution with visual proofs being just because it looks right doesn't mean it is right. A classic example being the "missing square puzzle". More on point examples would be a curve which ever so slightly changes to concave up for a short portion of a shallow concave down region in an overall wavy function, a curve that looks like it converges at a limit but actually doesn't, or something undefined at only 1 point that otherwise looks continuous. 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.
- photochemsyn 3y agoAnother example is the use of the parallelogram method to calculate tangents, which works sometimes and fails in other cases. "Birth of Calculus" (1986) https://youtu.be/ObPg3ki9GOI?t=334 https://youtu.be/ObPg3ki9GOI?t=334
- jacobolus 3y agoIs there a more complete explanation? They state that this method fails for the quadratrix, but the chosen vectors drawn in the picture in the video seem clearly nonsensical, so it's not clear to me precisely what procedure was being followed. 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 https://jstor.org/stable/2695381
- samatman 3y agoNote that this doesn't apply to proofs by construction, as in Euclidean geometry. 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.
- photon_lines 3y agoI made a little write-up on getting an visual intuition behind calculus / the derivative here, although I'm not sure that it's really that well done and I should have maybe provided a few more examples: https://photonlines.substack.com/p/visualizing-the-derivative https://photonlines.substack.com/p/visualizing-the-derivativ...
- peter_d_sherman 3y ago>"Mamikon's theorem: 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...