I studied mathematics and had various courses on differential geometry and aspects of lie-theory. Feel like i am now getting a real understanding of some of this stuff for the first time - great content, keep going!
You’re putting out some of the best mathematical content on UA-cam right now! I really struggled with my first manifolds course since we never received any intuitive explanations.. so this series is amazing.
Cosmic Dissonance Cosmic Dissonance Well everyone is at different levels of education and I think it’s fine that some people may overestimate their expertise on a subject. It’s an easy trap to fall into. If they’re really serious about pursuing maths they will find out sooner or later that their knowledge is lacking. I agree that their are a lot of channels who do present more challenging topics but their presentation is just awful! Michael’s clarity in these videos is awesome. Personally, I wouldn’t say maths is taught extremely poorly but then everyone has had different experiences. I’ve had many professors who have been able to convey their content well. On the flip side though, you will get professors who are just bad teachers. I usually find that they will write down and prove many definitions and theorems that are second nature to them, but then this leaves the whole class not being able to remember anything, or not being able to connect anything, especially without imagery. I think visuals really help internalise an idea which most academics have already done and perhaps they don’t realise that students haven’t.
Oh my! After so much time spent on form in order to study general relativity and electromagnetism, I finally get the intuitive mental imagery of form contrast to the algebraic expression which I didn't understand, thank you, signed area explanation was brilliant
10:34 It's obvious, but just in case anyone is confused, he's not correcting himself to say it's not 2 but 3 vectors (like it sounds). He's just calling those vectors "3-vectors." 18:12 "recall" is referring to lecture 1, ua-cam.com/video/PaWj0WxUxGg/v-deo.html
I loved to study this back at uni. I like this forms much more because they helped me understand what a gradient, rotor amd divergence are in higher dimensions, Also Green's theorem and Stokes' become much simpler to remember. Awsome video, keep it up.☺
I'm watching these for a better understanding of "n-forms", because they keep coming up, but I can't help but think that a bivector(/trivector/quadvector/etc.) formulation might be conceptually simpler (at least to me). I guess I've just never had much appreciation for this distinction between vector spaces and their duals.
For others confused near 16:30, as to the nature of dx and dy (are they no longer components of vectors? He's taking the wedge product of them?): Here's what I think is going on. EDIT: 18:42 describes exactly what I was saying here... oops. I think we're supposed to, in this context, think of dx as the 1-form that maps a vector (v1,v2) to the first component: v1. And think of dy as a 1-form that maps a vector(v1,v2) to the 2nd component: v2. In other words, in the language he has used, dx takes (dx,dy) to just dx (abusive notation here: using dx as the name of a function and as one of the input vectors to that function). And (again with abusive notation) dy takes (dx,dy) to just dy. Think of dx and dy (the 1-forms) as the "elementary 1-forms" just like you think of e1 and e2 as the elementary vectors in R2. I'm guessing this leap is more natural if we understand more intimately what the dual of a vector space is.
Hi Michael, thank you for a great series on differential forms. I have a question at about 17:11 in the video. I understand the symbolic manipulation you performed to get the final expression for w1 (wedge) w2. However, in doing that, the term, say dx, went from being an element of a vector in TpR^2 space to a differential form that itself now operates on an element from the TpR^2 space. In other words, it looks like this symbolic manipulation somehow completely transformed the nature of the object 'dx'. Can you please elaborate on this a little?
For others confused by this: I think we're supposed to, in this context, think of dx as the 1-form that maps a vector (v1,v2) to the first component: v1. And think of dy as a 1-form that maps a vector(v1,v2) to the 2nd component: v2. In other words, in the language he has used, dx takes (dx,dy) to just dx (abusive notation here: using dx as the name of a function and as one of the input vectors to that function). And (again with abusive notation) dy takes (dx,dy) to just dy. Think of dx and dy (the 1-forms) as the "elementary 1-forms" just like you think of e1 and e2 as the elementary vectors in R2. I'm guessing this leap is more natural if we understand what the dual of a vector space is.
somewhat like the grassmann algebra approach i read in rudin better but I really appreciate seeing this perspective on things. Thanks so much for doing this series!
I have studied differential forms both in undergrad and more recently in my master's. I know where your determinant formula comes from. But I never thought of like, going backwards and translating the "usual formula" for the exterior product given in terms of the tensor product and just, turning it back onto a determinant to see it as a signed "hypervolume" (quotes because I'm not sure if a complex value could be consider a hypervolume, heh).
Thanks a bunch for this series! I've been pushing off learning about differential forms for a while, and it's so nice seeing these ideas worked out with rigor on a blackboard. Also, for people craving some visuals to supplement the algebraic geometry, I highly highly highly highly highly recommend this playlist from Professor Ghrist's multivariable calculus course: ua-cam.com/video/cFscZ9c0AIk/v-deo.html. He uses the ideas of differential forms to tackle vector calculus from an elegant, general standpoint.
Started this series because I'm taking a class in differential geometry, but differential forms aren't covered. Especially regarding the wedge product, forms are starting to smell a lot like tensors w/ the tensor product, or at least covectors. Will forms end up being equivalent to tensors in some way?
Why is this function defined using the cross product instead of the dot product? Would that not also give a linear function that has a geometric definition? Also, I always had a notion that cross product doesn't have a definition outside of R2 and R3, while dot products generalize to any dimensionality. Is this a faulty notion? Or is that correct but just not matter because we're already transforming things into the R2 space of the w's?
Why are the dx and dy suddenly one forms? We started off with dw being a one form and dx, dy were some arbitrary vectors of the tangent space. Why are dx and dy also one-forms, intuitively?
dx and dy aren't (primarily) considered forms here, the differential operators d/dx and d/dy are. It's because dx and dy are functions that take in vectors (d/dx and d/dy) and output scalars, that's the definition of a form. For example, dx(d/dx) = 1, while dx(d/dy) = 0. Here, we define dx(d/dx) to be equal to df/dx, where "f" is the function f(x)=x (here represented by dx) and d/dx means derivative with respect to ex.
How did you get from a 2x2 matrix determinant to the 2x1 determinant to give the 2-form dx ∧ dy geometric interpretation on the end of the video? Can you even define a 2x1 matrix determinant? Didn't get that part. Your series are awesome btw.
He wrote the vectors as the rows of the matrix at the end. Slightly confusing, but since each vector has two components, it still represents a 2x2 matrix.
I guess it would be more intuitive the usage column vectors and swapping the terms on the secondary diagonal to resemble the jacobian matrix. Does not affect the result though.
Also by root you probably mean solution. In which case ∞ technically solves it. But it feels kinda wonky to even consider ∞ as a possibility for a solution...
p chaitanya Well you can’t just plug in infinity into an equation, you can take the limit as x approaches infinity and you can see that both sides do go towards infinity but it’s not correct to say that 2*inifity= infinity.
Interesting question. Once infinity comes up, it's fairly intuitive to start thinking that limiting expressions equate to algebraic truths. However, the idea of infinity comes from limits. Take the functions "x" and "2x". For all x>0, we know that 2x>x. Now it's time to play a game: move x to the other side and consider the expression 2x - x >0. What x do I need such that 2x is 10 + x? Well, x = 10 right. What about 100; 1,000; 1,000,000; etc? It's clear that we can construct a value of x such that the difference between "x" and "2x" is arbitrarily large. For that reason, we realize that claiming "infinity" as a solution is silly - if anything, the expressions grow arbitrarily far apart. The same kind of reasoning holds for x
15:44 I did it as follows: Take (aω_1+bω_2) ^ ω_3 (v_1,v_2) with a,b real numbers. Evaluating the determinant from the definition of the wedge product, it is obtained ( aω_1(v_1) + bω_2(v_1) ) ω_3(v_2) - ω_3(v_1) ( aω_1(v_2) + bω_2(v_2) ) Which can be rewritten as a ( ω_1(v_1) ω_3(v_2) - ω_3(v_1) ω_1(v_2) ) + b ( ω_2(v_1) ω_3(v_2) - ω_3(v_1) ω_2(v_2) ) Which in turn is identical to aω_1 ^ ω_3 + bω_2 ^ ω_3 I dropped the (v_1,v_2) for simplicity. The wedge product is evidently right-distributive. A very similar procedure shows that it is also left-distributive. The wedge product is thus bilinear.
"Begs us to ask" maybe? I know it's a losing battle, but I'd like to keep "begging the question" reserved for its technical use as a logical fallacy. D: Ah well, so it goes.
Not hearing "That's a good place to stop" at the end of the video feels like a proof without QED or a black square
I studied mathematics and had various courses on differential geometry and aspects of lie-theory.
Feel like i am now getting a real understanding of some of this stuff for the first time - great content, keep going!
17:00 mind blown. It's so satisfying seeing it all come together.
You are natural to explain that the mathematical definition is so natural. :) Thanks for the great video!!
You’re putting out some of the best mathematical content on UA-cam right now!
I really struggled with my first manifolds course since we never received any intuitive explanations.. so this series is amazing.
Yeah agree, wish I had these videos available when I was in grad school. I did have some use of these sets of lecture notes arxiv.org/abs/0908.1395
Glenn Wouda im trying to go into theoretical particle physics so thanks for the useful link!
Cosmic Dissonance Cosmic Dissonance Well everyone is at different levels of education and I think it’s fine that some people may overestimate their expertise on a subject. It’s an easy trap to fall into. If they’re really serious about pursuing maths they will find out sooner or later that their knowledge is lacking. I agree that their are a lot of channels who do present more challenging topics but their presentation is just awful! Michael’s clarity in these videos is awesome. Personally, I wouldn’t say maths is taught extremely poorly but then everyone has had different experiences. I’ve had many professors who have been able to convey their content well. On the flip side though, you will get professors who are just bad teachers. I usually find that they will write down and prove many definitions and theorems that are second nature to them, but then this leaves the whole class not being able to remember anything, or not being able to connect anything, especially without imagery. I think visuals really help internalise an idea which most academics have already done and perhaps they don’t realise that students haven’t.
@Cosmic Dissonance This is one of the most pretentious comments I've ever had the pleasure of reading lol
I am glad you like those videos. It inspires me to charge less when I will do mine.
I'm sure I've commented before, consider this additional engagement! Truly a wonderful gift these lectures you've shared with us, thank you!
This is lovely to study after finishing vector calc and linear algebra. It brings everything together
Oh my! After so much time spent on form in order to study general relativity and electromagnetism, I finally get the intuitive mental imagery of form contrast to the algebraic expression which I didn't understand, thank you, signed area explanation was brilliant
hands down the best mind clearing explanation, thanks you!
Please continue making this series! This really are great videos
10:34 It's obvious, but just in case anyone is confused, he's not correcting himself to say it's not 2 but 3 vectors (like it sounds). He's just calling those vectors "3-vectors."
18:12 "recall" is referring to lecture 1, ua-cam.com/video/PaWj0WxUxGg/v-deo.html
Excellent series of videos!!
I loved to study this back at uni. I like this forms much more because they helped me understand what a gradient, rotor amd divergence are in higher dimensions, Also Green's theorem and Stokes' become much simpler to remember. Awsome video, keep it up.☺
Your videos are amazing. Very clear presentation. Thank you very much!
I'm watching these for a better understanding of "n-forms", because they keep coming up, but I can't help but think that a bivector(/trivector/quadvector/etc.) formulation might be conceptually simpler (at least to me). I guess I've just never had much appreciation for this distinction between vector spaces and their duals.
For others confused near 16:30, as to the nature of dx and dy (are they no longer components of vectors? He's taking the wedge product of them?):
Here's what I think is going on.
EDIT: 18:42 describes exactly what I was saying here... oops.
I think we're supposed to, in this context, think of dx as the 1-form that maps a vector (v1,v2) to the first component: v1.
And think of dy as a 1-form that maps a vector(v1,v2) to the 2nd component: v2.
In other words, in the language he has used, dx takes (dx,dy) to just dx (abusive notation here: using dx as the name of a function and as one of the input vectors to that function).
And (again with abusive notation) dy takes (dx,dy) to just dy. Think of dx and dy (the 1-forms) as the "elementary 1-forms" just like you think of e1 and e2 as the elementary vectors in R2.
I'm guessing this leap is more natural if we understand more intimately what the dual of a vector space is.
this is brilliant, and it pains my that this was not included at the start of my GR course
Amazing lectures , thank you Michael!
Hi Michael, thank you for a great series on differential forms.
I have a question at about 17:11 in the video. I understand the symbolic manipulation you performed to get the final expression for w1 (wedge) w2. However, in doing that, the term, say dx, went from being an element of a vector in TpR^2 space to a differential form that itself now operates on an element from the TpR^2 space. In other words, it looks like this symbolic manipulation somehow completely transformed the nature of the object 'dx'. Can you please elaborate on this a little?
For others confused by this:
I think we're supposed to, in this context, think of dx as the 1-form that maps a vector (v1,v2) to the first component: v1.
And think of dy as a 1-form that maps a vector(v1,v2) to the 2nd component: v2.
In other words, in the language he has used, dx takes (dx,dy) to just dx (abusive notation here: using dx as the name of a function and as one of the input vectors to that function).
And (again with abusive notation) dy takes (dx,dy) to just dy. Think of dx and dy (the 1-forms) as the "elementary 1-forms" just like you think of e1 and e2 as the elementary vectors in R2.
I'm guessing this leap is more natural if we understand what the dual of a vector space is.
somewhat like the grassmann algebra approach i read in rudin better but I really appreciate seeing this perspective on things.
Thanks so much for doing this series!
At 5:39 we're looking for at least one number that will satisfy the definition ? So we use the area of a parallelogram?
I have studied differential forms both in undergrad and more recently in my master's. I know where your determinant formula comes from. But I never thought of like, going backwards and translating the "usual formula" for the exterior product given in terms of the tensor product and just, turning it back onto a determinant to see it as a signed "hypervolume" (quotes because I'm not sure if a complex value could be consider a hypervolume, heh).
I wish you were my college teacher... you know, officially.
Thank you 👍👍👍👍👍
Thanks a bunch for this series! I've been pushing off learning about differential forms for a while, and it's so nice seeing these ideas worked out with rigor on a blackboard. Also, for people craving some visuals to supplement the algebraic geometry, I highly highly highly highly highly recommend this playlist from Professor Ghrist's multivariable calculus course: ua-cam.com/video/cFscZ9c0AIk/v-deo.html. He uses the ideas of differential forms to tackle vector calculus from an elegant, general standpoint.
Couldn't agree more. His course is so amazing and yet somehow so unknown.
sei il numero uno !!
Started this series because I'm taking a class in differential geometry, but differential forms aren't covered. Especially regarding the wedge product, forms are starting to smell a lot like tensors w/ the tensor product, or at least covectors. Will forms end up being equivalent to tensors in some way?
great video
Thanck you, could you show some examples of application, as that would be very inlighting, P,S, graghs and pictours , my specilty ions.Bye.
This was great!
Why is this function defined using the cross product instead of the dot product? Would that not also give a linear function that has a geometric definition?
Also, I always had a notion that cross product doesn't have a definition outside of R2 and R3, while dot products generalize to any dimensionality. Is this a faulty notion? Or is that correct but just not matter because we're already transforming things into the R2 space of the w's?
Why are the dx and dy suddenly one forms?
We started off with dw being a one form and dx, dy were some arbitrary vectors of the tangent space. Why are dx and dy also one-forms, intuitively?
dx and dy aren't (primarily) considered forms here, the differential operators d/dx and d/dy are. It's because dx and dy are functions that take in vectors (d/dx and d/dy) and output scalars, that's the definition of a form. For example, dx(d/dx) = 1, while dx(d/dy) = 0. Here, we define dx(d/dx) to be equal to df/dx, where "f" is the function f(x)=x (here represented by dx) and d/dx means derivative with respect to ex.
How did you get from a 2x2 matrix determinant to the 2x1 determinant to give the 2-form dx ∧ dy geometric interpretation on the end of the video? Can you even define a 2x1 matrix determinant? Didn't get that part. Your series are awesome btw.
He wrote the vectors as the rows of the matrix at the end. Slightly confusing, but since each vector has two components, it still represents a 2x2 matrix.
I guess it would be more intuitive the usage column vectors and swapping the terms on the secondary diagonal to resemble the jacobian matrix. Does not affect the result though.
Why dy = a2 ???
If dx= then dy =
Is +/-infinity a root of the equation X=2X
That's not usually defined tho
The equation is just X=0, introducing Infinity won't work if you're not very clear about what ∞-2∞ even means
Also by root you probably mean solution. In which case ∞ technically solves it. But it feels kinda wonky to even consider ∞ as a possibility for a solution...
@@ilonachan yes but infinity +infinity is just written as infinity. So 2×infinity=infinity
p chaitanya Well you can’t just plug in infinity into an equation, you can take the limit as x approaches infinity and you can see that both sides do go towards infinity but it’s not correct to say that 2*inifity= infinity.
Interesting question. Once infinity comes up, it's fairly intuitive to start thinking that limiting expressions equate to algebraic truths. However, the idea of infinity comes from limits. Take the functions "x" and "2x". For all x>0, we know that 2x>x. Now it's time to play a game: move x to the other side and consider the expression 2x - x >0. What x do I need such that 2x is 10 + x? Well, x = 10 right. What about 100; 1,000; 1,000,000; etc? It's clear that we can construct a value of x such that the difference between "x" and "2x" is arbitrarily large. For that reason, we realize that claiming "infinity" as a solution is silly - if anything, the expressions grow arbitrarily far apart. The same kind of reasoning holds for x
great ,if you give some intuition to why such calculation relate to the are of parallelogram ,that will much better
Amazing
15:44 I did it as follows:
Take
(aω_1+bω_2) ^ ω_3 (v_1,v_2)
with a,b real numbers. Evaluating the determinant from the definition of the wedge product, it is obtained
( aω_1(v_1) + bω_2(v_1) ) ω_3(v_2) - ω_3(v_1) ( aω_1(v_2) + bω_2(v_2) )
Which can be rewritten as
a ( ω_1(v_1) ω_3(v_2) - ω_3(v_1) ω_1(v_2) ) + b ( ω_2(v_1) ω_3(v_2) - ω_3(v_1) ω_2(v_2) )
Which in turn is identical to
aω_1 ^ ω_3 + bω_2 ^ ω_3
I dropped the (v_1,v_2) for simplicity. The wedge product is evidently right-distributive. A very similar procedure shows that it is also left-distributive. The wedge product is thus bilinear.
"Begs us to ask" maybe? I know it's a losing battle, but I'd like to keep "begging the question" reserved for its technical use as a logical fallacy. D: Ah well, so it goes.