The Catenary (hanging chain), how it was first solved.
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- Опубликовано: 21 сен 2024
- The catenary is the mathematical shape of a hanging chain. Describing this shape is one of the famous original problems of calculus. I discuss the history of the problem, how it was determined that the curve was not a parabola, how to model the curve with a differential equation based on the balance of forces, and how the equation was first solved by Bernoulli, Leibniz, and Huygens. Bernoulli's solution involves the solution of another problem: the rectification of a parabola, or finding it's arc-length. This lecture illustrates most of the ideas in a second semester of Calculus.
This video is Episode 8 in a series I call Tricky Parts of Calculus, a series that covers the subtle and difficult topics in calculus that are usually missed in a calculus class. See the whole playlist here: • Tricky Parts of Calculus
Also check out my channel for general advice and opinions about math, as well as the Daniel Rubin Show podcast for interesting conversations about math and other topics: / @danielrubin1
I'm impressed you did all that writing so neatly, and without making any mistakes! Your mathematical and historical knowledge is superb. Low-tech tool perhaps, but a brilliant exposition of the topic. Great video.
Glad you enjoyed it!
Very interesting history of the problem. What an amazing guy Huygens was.
Awesome stuff. Really well researched. There is a lot to learn here both for experts and learners of calculus. Will be coming back to your channel a lot. Cheers!
Thanks a lot! Glad you enjoyed it!
Thanks you very much. I can't articulate my feelings better than existing comments, so I'll leave it at that.
I went and checked, my Calculus books (Larson and Hostetler, second edition from 79 (used in 89)) did have the catenary mentioned briefly on an example, only a few lines though (by the time the book went into trig and hyperbolic trig function arc length was already defined).
I expected your channel to be much more popular than it is! High Quality video and explanation keep up the good work
Thanks, will do!
This was a really good and informative video, I don't normally comment on things but this was really impressive!
I appreciate that!
Really nice to see the original way! What was yours source of the original Bernoulli solution. Looking forward to more of these!
Fantastic! Thank you for this effort!
bro I recently came across your calculus series. Could u pls suggest the books u used to develop those lectures, I really want to learn from books that give a historical aspect of the problem in addition to the motivation needed to solve it.
great video. how do you keep the paper so still? Do you put something on it ?
Thanks! My low-tech approach has just been to hold the paper in place with my left hand whenever I'm writing.
What about that brigde over the river cause this is a rigid body and the catenary is to minimize the potential energy.
very good
I take an exception to this video- such a waste of paper... use the same piece of paper as a backing when you write with a marker so you don't waste as much sheets, gaah. But I really, really do like the content- keep up the good work.
Hey what textbook do you recommend for multivariable calc? I've been using the standard for my uni, 8th edition of Early Transcendentals Calculus by Stewart, and I'm not really liking it much.
I taught a multivariable calculus class out of Stewart's book and I was satisfied with it. I think the most important objective for students in such a class, no matter what they go on to do, is to gain a really good facility for computation. I learned the subject from C.H. Edwards' Advanced Calculus of Several Variables, plus AP Physics. An alternative that covers the historical development (which I clearly favor) and a good deal of physics is Bressoud's Second Year Calculus: From Celestial Mechanics to Special Relativity.
@@DanielRubin1 looks like i should get more comfortable with calc 3 next semester then. will be taking a computational math methods class. i guess my biggest issue with multivariable calc is a lot of it requires me to have some faith, it feels very handwavy since a lot of the methods used are linear algebra and diff eqn concepts that most students havent even taken yet. e.g., i don't really understand the cross product until i understand the determinant from a linear algebra class.
Galileo actually recognize that this curve was different from parabola and wrote about it in his Two New Sciences.
Gaudi built La sagrada familia using this shape. I’m in amazement of these guys in the past. Nowadays it feels like cheating since they did the hard work
This is awesome, but the fact that you speak the equations before writing it makes it very hard to follow, since by the time you are writing one thing you are already speaking something different.
swoosh!
🗿👍
It's a bad habit to pronounce "sinh" like singe.
I thought it was fairly standard to pronounce sinh as "sinch."
"The basic hyperbolic functions are:
hyperbolic sine 'sinh' (/ˈsɪŋ, ˈsɪntʃ, ˈʃaɪn/),
hyperbolic cosine 'cosh' (/ˈkɒʃ, ˈkoʊʃ/)" ..."
from en.wikipedia.org/wiki/Hyperbolic_functions
really just a boring force diagram problem
It's true that if you understand the concept of a force diagram then it's not too hard to derive the equation of the catenary. The emphasis here is on solving the equation, and there was a time when finding expressions for the integrals that come up was not completely trivial.
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