The general Stoke's theorem via differential forms.
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- Опубліковано 7 сер 2024
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This has been sitting as a draft on the backend of the channel for a while and I kind of forgot about it. Well, maybe it is time to post it!
Could you please show an example of some geometric calculus using differential forms? Is there a way to find a form in G(3,0,1) for a multivector valued function.
I think it's a topic that is not well understood among many students.
maybe like, Dr Penn you could define terms, ex. n-chain, pullbacks, etc etc
@@charleyhoward4594 True, me thinks that the notation of differential forms is mathematical wizardry designed to obscure lack of true understanding.
thank you so much for posting this king
As a non-mathematician, the generalized Stokes theorem is one of the most elegant formula I've seen.
Me before watching this video: "Well, I learned Stoke's theorem in Calculus and used in Fluid Mechanics in my Engineering school. Let's watch the video and learn something new!"
Me after watching this video: "Hmm... I know some of these words."
If you watch the previous 23 videos in this series all will be revealed.
The idea that the solution on the boundary gives the solution for the whole volume is quite cool. Like the derivative of the volume of the sphere with respect to radius is the surface area
i remember having a panic attack during my calc 3 lecture when this was presented, even just hearing "spivak" sends chills down my spine.... great video as always tho!
this is done in 3rd grade in russia
@@TurboGamasek228We do this in the womb on Mars.
Should be filed in the differential forms playlist.
So happy this one is here now!
I’m so glad you’re coming back to this series!
This series is back! I remeber it very fondly because it made my quarantine more bearable back then!
I've been waiting for this for so long. Was wondering what happened to the diff forms playlist.
I was hoping for the series on differential forms to be continued some day. And here it is! 😊🎉🎉
thank you so much for not abandoning this series!!! cheers
By the way I think a clarification is needed when you impose the condition that j runs from 1 to n at 6:00. I believe that it is rather ambiguous as one can interpret that as a double sum with j = 1 to n on the outside and i_j = 1 to N on the inside, when in fact it extends from the definition of evaluating an integral of a two-form in one of the earlier videos in this series where a double sum is used to index the increments in the two parameters involved. Therefore what j being bounded between 1 and n really represents is n summations of i_1 = 1 to N up to i_n = 1 up to N. Otherwise, the "distribution" of the j = 1 telescoping series at 12:46 wouldn't make much sense.
Also another minor (but important) detail is that the multi-index I is not the usual indexing for elementary differential m-forms that we are used to seeing (which is that they were strictly increasing natural numbers up to m), but rather (i_1,...,i_n) being an n-tuple of integers with each index running from 0 to N, independent of one another. This allows us to define an arbitrary coordinate within the uniform partitioning of the unit n-hypercube which was glossed over at the start, and hence even allows us to apply the converse mean-value theorem, because x_I* is chosen to be such that its coordinates are within the interval or geometrically lying within an infinitesimal unit n-hypercube slice defined at x_I.
So far I felt that this was the most difficult instalment of the differential forms series to comprehend.
Oh snap, I guess I gotta go re-watch the entire series from 3 years ago!
I'm stoked!
This is how I learned Stoke’s Theorem! In Shurman’s Calculus and Analysis in Euclidean Space. Made learning homology in grad school easier, since I was already familiar with chains and whatnot
lol I went the other way. As an undergrad I did a directed study in algebraic topology and learned homology, then I found this playlist (and am about to start grad school in the fall).
Long await now comes to an end! Thank you sir!
I had the years ago in a calculus class (very honors! -- of 10 people in class at least 3 became professional academic mathematicians.) from Spivak's Calculus on Manifolds. The result on a cube ( or if you'd rather, a simplex) is really just a straightforward calculation, although setting it up (with a bunch of definitions) is somewhat lengthy.
Please add this video to the differential forms playlist! I almost missed this video.
you're a hero
Any guesses on when we're going to see the video on pull-backs and the full treatment of the generalised Stokes's Theorem?
It’s about time
Came back on the series after 3 years
16:06
Is that a Rieman integral definition on manifolds? In our course the proof was kinda different, don't remember this definition of integral
super coolness
Need a link to definition of ^
Why do we put in vectors ei/N and so on? Why divide by N?
I'm kind of guessing, but in the past I think the approximate tangent vectors he used were the lattice displacement vectors, so after dividing the big hypercube by N in each dimension, each of the mini hypercubes have side lengths of 1/N (where the "lattice points" are vertices of the minicubes here).
….hyper cube….
Absolutely incomprehensible for me. But I DO appreciate the effort to show new things
the word cube is there just as an easier transition for us
hypercube basically means the same properties of a cube but for higher dimensions-much like going from a square to a cube
holy shit is it my birthday let’s goooooo
what's wedge? did I miss that in the beginning?
Obviously check out the whole series
There is a playlist on his channel called “differential forms.” This is the final video of the playlist (should be, wasn’t filed in the playlist) but in that playlist he gave all the background for this. In short, dx in differential forms are not your typical dx you learned in calculus, but are rather operators, and the wedge product is a way of combining these operators to form a new operator.
3 years late isn't so bad for an academic
We should talk to UA-cam manager to change or. Add the value of video __valuable_v ; to the structure of video data type, we can’t compare this great video with some bull..ship unusefull mega ton of video on UA-cam, we should change this : because many scientists noticed that science is declining very than we predicted, and this start just after 1945, but accelerated after internet spread, why??? `I remember when i was young, hero is not footballer or dancer, or… , is Einstein, Feynman , Kasparov(chess), Ali (in sport), … and many good celibrity,
This. Why should these kind of video never evaluated as, normal video. Time To Change it.
First
first
That's a physicist demo ....
😐😐😐
The man's name is Sir George Gabriel Stokes, not Sir George Gabriel "Stoke". I mean, Jesu's, for someone who knows many proofs of Pythagora's Theorem, one would hope that spelling "Stokes" would not be too hard. =)
Lmao, do you have nothing better to do?
I may be wrong but ... I think physicists have a neater less abstract and complicated way of explaining it?
And another opinion: is math really done by providing definitions?
Or is it better done by providing background and necessity for such definitions to exist in first place?
Vision justifies and embodies small print or small print justifies and embodies vision?
Can it be whole without both?
Basis: math is easy no matter how complicated mathematicians wish to make it 🙂
Doing math is not easy, but I agree a lot of higher level concepts are relatively graspable for non-mathematicians. Actually doing math with those concepts is a whole different story and is quite hard.