I’m very grateful for this series. You’re an amazing teacher, proven by the fact that you bring these high level concepts in a very understandable way and always with a good motivation behind them, which is often skimmed over.
At 33:45, is it possible to explain the origin of the (-1)^k factor from this geometric perspective? I understand its origin in an algebraic context (the antisymmetric property of the wedge product), but I'm struggling to form an equivalent explanation in this geometric picture P.S. this series has to be the best resource on differential geometry I have ever come across. Most courses I've been taught just hammer you with the algebraic rules and properties, with little regard for the equivalent geometric picture. You'd think the geometric intuition for differential GEOMETRY would be more widely taught! Anyways, thank you so much for this; it has filled in numerous intuitive blanks I've had in this field (and physics too!)
Thanks for the Videos Prof! As someone studying visual computing for my master's I have followed your series on Computer Graphics and now, I plan to binge on some differential geometry once I'm finished with my exams. I also want to thank you for really piquing my interest in geometry in general and what you can do with geometries. And this is why I plan to apply to MIT's summer geometry processing program, of which you're also a part. Hope I get accepted :)
Basically, the antisymmetry of the wedge product comes from computing determinants, from which the -1 comes. For more detailed derivations, check out Michael Penn's videos titled "The product rule for the exterior derivative" and "Algebraic Properties of the Wedge Product".
Good description of the directional derivative. "Walk away from p with velocity X." That's a very good way to think about it in the case where X is not a unit vector.
Extraordinary!I am just totally speechless about such beautiful Visual lecture series (beyond the traditional bookish equations)! Kindly make some videos about differential equations with such visual explanations if possible. Thank you very much for all your videos and efforts.💟
I really appreciate your videos -- probably some of my favorite on RUclips! In this lecture we essentially attempt to directly generalise the otherwise ad-hoc-ly defined grad, div and curl. For instance when justifying d^2 = 0 you use the fact that curl(div) = 0 as a motivation. In my opinion this kind of misses why d is so important. It's not important because it generalises/unifies grad, div and curl but rather because it's the map making Stokes' theorem true in general. I think Terrance Tao uses this perspective (that the boundary and exterior derivative are adjoint) -- taking Stokes' theorem kind of as the "definition" for d. However, Spivak introduces d similar to you, providing an axiomatic definition and proving its existence, then later establishing the relationship with the boundary operator. But.. the way he sets out his book is he first defines d (proving its existence and uniqueness), then he defines chains and the boundary operator, then he proves stokes theorem from this. [but it's also clear that to an extent, Spivak's goal with his book is to make it quite terse] I personally like Tao's approach since it lets you sort of follow a "path of discovery" to invent d on your own.
One more thing is that what about Cartan geometry by rolling balls?(does it have any significance in this relevant course/field?)i.e. Instead of investigating manifold by tangent planes,it(beautifully) is done by rolling(without slipping) a tangent ball over the surface which encodes more geometric informations Here is a picture about rolling hamster ball.www.researchgate.net/figure/Hamster-ball-rolling-over-a-manifold-thanks-to-D-Wise-for-the-picture-concept_fig1_230899575
Why does div(X) = ⋆d(⋆X♭) not have a sharp operator? I think it's because the result is a scalar... but does this mean all 0-forms are exactly identical to 0-vectors?
Yes. 0-forms and scalars are completely indistinguishable, since conversion between the two does not involve a choice of inner product. Not true with, say, 1-forms vs. vectors. A fancier way of saying this is: the only structure you need to put on your manifold to talk about scalar functions / differential 0-forms is a smooth structure. But to talk about vector fields vs. 1-forms you need a Riemannian metric.
Excelent lecture Prof. I just have a question. At 36:49 you set to zero a term of the form ddx since "d of d is zero" due to the exactness of the exterior derivative. However we were asked to think about "dx" as basis 1-form, unrelated to derivatives. Hence I would say that ddx=0 since dx is a constant 1-form. Is this a notational coincidence? Is dx really the exterior derivative of x in some way? (I guess so, since in a previous video you depicted dx and dy with "gradient fields" for x and y) thanks again!
I'm confused about your description of the gradient. The way you write it in the equation with the nabla operator is as if abla \phi is an element of the tangent space, whose basis are the partial differential operators \partial x^i . But I thought that the gradient is actually an element of the co-tangent space: To compute the directional derivative with respect to a vector X in the tangent space: X = X^i e_i, where the e_i are the basis of the tangent space, that is, the partial derivative operators. If you apply this vector X to a function f you get X(f) = X^i e_i(f) = X^i (\partial f / \partial x^i),
I hope that sharp and flat can be applied to scalars. If so, then all three of div grad and curl could be written with a flat on the inside and a sharp on the outside.
You say that the gradient depends on the choice of the inner product and the differential doesn't, but then give an equation that lets them be defined in terms of each other using the musical isomorphisms. How does that flexibility get preserved? Do the musical isomorphisms depend on the space you're working in (and thus the choice of inner product)? Something else?
43:34 One can show that a curl of a gradient is 0 using Stokes' theorem without referring to any coordinates, hint: closed curve integral of a gradient is 0.
Thank you for the fantastic lectures! Just to be sure, is the exponent k in the expression of d(a /\ b) the "degree" of a, b or a /\ b ? The example calculations you go through seem to indicate that a is the k-form, but in that case, I think there should be a minus sign in the expression at 33:30 , since alpha is a 1-form. But then that would break the geometric intuition. Perhaps I'm missing something ?
I'm a little confused by the product rule at T = 13:14. are we to understand that (a V b) has order k, or does (-1)^k refer to the order of a only? a -> alpha b -> beta V -> Wedge symbol From looking around a bit, it appears that the latter is true, and by analogy (probably dangerous) to the curl of the product of two vector functions in 3-space. Ah. This is clarified at T = 31:45
The lectures are extremely well organized. And thanks for your amazing course in this subject. I'm studying it just for fun, and I've already gained a lot just from six lectures.
This video was helpful in understanding this material on a more intuitive level. I am learning from Lee's introduction to smooth manifolds currently, and sometimes it's hard to actually visualize and understand conceptually what is going on in all the calculations/proofs. Glad I found your channel.
The divergence formule here ruclips.net/video/jeiDXhCiF44/видео.html has a typo: \frac {\partial v}{\partial \mu} should be \frac {\partial v}{\partial y}
Quick question: if exactness is a necessary condition for the exterior derivative, then using inexact differentials in thermodynamics means they must be using a different operator, right? Is that case considered in this theoretical structure? Thank you for the lectures! :)
Oh I think now I understand it: in the notes of thermodynamics formalism of Danny Calegar, the "inexact" differential are presented as conventional exact 1-forms but that occur in an equilibrium manifold with the same initial and final state as the irreversible process (the fact that it is done in an irreversible manner is what makes it inexact).
I’m very grateful for this series. You’re an amazing teacher, proven by the fact that you bring these high level concepts in a very understandable way and always with a good motivation behind them, which is often skimmed over.
00:30 Exterior Calculus-Overview
01:19 Integration and Differentiation
02:46 Motivation for Exterior Calculus
Exterior Derivative
04:59 Derivative-Many Interpretations…
06:31 Review: Vector Derivatives
09:13 Review: Vector Derivatives in Coordinates
11:21 Exterior Derivative
Exterior Derivative-Differential
14:29 Review: Directional Derivative
17:09 Review: Gradient
20:44 Differential of a Function
22:51 Gradient vs. Differential
Exterior Derivative-Product Rule
26:43 Review: Product of Numbers
28:04 Product Rule-Derivative
31:21 Product Rule-Exterior Derivative
34:29 Product Rule-“Recursive Evaluation”
37:36 Exterior Derivative-Examples
Exterior Derivative-Exactness
42:25 Review: Curl of Gradient
43:58 What Happens if d ∘ d = 0?
48:05 Exterior Derivative and Curl
50:27 d ∘ d = 0
51:07 Exterior Derivative in 3D (1-forms)
53:37 Exterior Derivative and Divergence
55:12 Exterior Derivative - Divergence
55:38 Exterior vs. Vector Derivatives-Summary
Exterior Derivative-Summary
58:32 Exterior Calculus-Diagram View
01:01:36 Laplacian
01:03:15 Exterior Derivative - Summary
At 33:45, is it possible to explain the origin of the (-1)^k factor from this geometric perspective? I understand its origin in an algebraic context (the antisymmetric property of the wedge product), but I'm struggling to form an equivalent explanation in this geometric picture
P.S. this series has to be the best resource on differential geometry I have ever come across. Most courses I've been taught just hammer you with the algebraic rules and properties, with little regard for the equivalent geometric picture. You'd think the geometric intuition for differential GEOMETRY would be more widely taught! Anyways, thank you so much for this; it has filled in numerous intuitive blanks I've had in this field (and physics too!)
Thanks for the Videos Prof! As someone studying visual computing for my master's I have followed your series on Computer Graphics and now, I plan to binge on some differential geometry once I'm finished with my exams. I also want to thank you for really piquing my interest in geometry in general and what you can do with geometries. And this is why I plan to apply to MIT's summer geometry processing program, of which you're also a part. Hope I get accepted :)
Great! Please do apply.
Why does the product rule on wedges have this (-1)^k term? That is, why do we sometimes need to change the sign?
Basically, the antisymmetry of the wedge product comes from computing determinants, from which the -1 comes.
For more detailed derivations, check out Michael Penn's videos titled
"The product rule for the exterior derivative" and
"Algebraic Properties of the Wedge Product".
Good description of the directional derivative. "Walk away from p with velocity X." That's a very good way to think about it in the case where X is not a unit vector.
Extraordinary!I am just totally speechless about such beautiful Visual lecture series (beyond the traditional bookish equations)!
Kindly make some videos about differential equations with such visual explanations if possible.
Thank you very much for all your videos and efforts.💟
Impressive lectures! Thanks!
I really appreciate your videos -- probably some of my favorite on RUclips!
In this lecture we essentially attempt to directly generalise the otherwise ad-hoc-ly defined grad, div and curl. For instance when justifying d^2 = 0 you use the fact that curl(div) = 0 as a motivation. In my opinion this kind of misses why d is so important. It's not important because it generalises/unifies grad, div and curl but rather because it's the map making Stokes' theorem true in general.
I think Terrance Tao uses this perspective (that the boundary and exterior derivative are adjoint) -- taking Stokes' theorem kind of as the "definition" for d.
However, Spivak introduces d similar to you, providing an axiomatic definition and proving its existence, then later establishing the relationship with the boundary operator. But.. the way he sets out his book is he first defines d (proving its existence and uniqueness), then he defines chains and the boundary operator, then he proves stokes theorem from this. [but it's also clear that to an extent, Spivak's goal with his book is to make it quite terse]
I personally like Tao's approach since it lets you sort of follow a "path of discovery" to invent d on your own.
do you have a link to Terence's view on d?
One more thing is that what about Cartan geometry by rolling balls?(does it have any significance in this relevant course/field?)i.e.
Instead of investigating manifold by tangent planes,it(beautifully) is done by rolling(without slipping) a tangent ball over the surface which encodes more geometric informations
Here is a picture about rolling hamster ball.www.researchgate.net/figure/Hamster-ball-rolling-over-a-manifold-thanks-to-D-Wise-for-the-picture-concept_fig1_230899575
Nice series ... thanks ...
I guess Keenan is God of Geometry. Rather I am sure
Why does div(X) = ⋆d(⋆X♭) not have a sharp operator?
I think it's because the result is a scalar... but does this mean all 0-forms are exactly identical to 0-vectors?
Yes. 0-forms and scalars are completely indistinguishable, since conversion between the two does not involve a choice of inner product. Not true with, say, 1-forms vs. vectors.
A fancier way of saying this is: the only structure you need to put on your manifold to talk about scalar functions / differential 0-forms is a smooth structure. But to talk about vector fields vs. 1-forms you need a Riemannian metric.
Excelent lecture Prof. I just have a question. At 36:49 you set to zero a term of the form ddx since "d of d is zero" due to the exactness of the exterior derivative. However we were asked to think about "dx" as basis 1-form, unrelated to derivatives. Hence I would say that ddx=0 since dx is a constant 1-form. Is this a notational coincidence? Is dx really the exterior derivative of x in some way? (I guess so, since in a previous video you depicted dx and dy with "gradient fields" for x and y) thanks again!
Fantastic!
I'm confused about your description of the gradient. The way you write it in the equation with the nabla operator is as if
abla \phi is an element of the tangent space, whose basis are the partial differential operators \partial x^i . But I thought that the gradient is actually an element of the co-tangent space: To compute the directional derivative with respect to a vector X in the tangent space: X = X^i e_i, where the e_i are the basis of the tangent space, that is, the partial derivative operators. If you apply this vector X to a function f you get X(f) = X^i e_i(f) = X^i (\partial f / \partial x^i),
Hello Thank you Dr for these series ,I want some books in this field Thank you...
When all I had was a book, people wanted videos. Fortunately, the book is still here: www.cs.cmu.edu/~kmcrane/Projects/DDG/paper.pdf
@@keenancrane thank you...
I hope that sharp and flat can be applied to scalars. If so, then all three of div grad and curl could be written with a flat on the inside and a sharp on the outside.
You say that the gradient depends on the choice of the inner product and the differential doesn't, but then give an equation that lets them be defined in terms of each other using the musical isomorphisms. How does that flexibility get preserved? Do the musical isomorphisms depend on the space you're working in (and thus the choice of inner product)? Something else?
43:34
One can show that a curl of a gradient is 0 using Stokes' theorem without referring to any coordinates, hint:
closed curve integral of a gradient is 0.
if your domain is simply connected, then yeah
37:16 is the answer of d(ω∧γ) a zero? it's a exterior derivative of a 3-form
Incredible teacher, best expression skills ever.
Thank you for the fantastic lectures!
Just to be sure, is the exponent k in the expression of d(a /\ b) the "degree" of a, b or a /\ b ?
The example calculations you go through seem to indicate that a is the k-form, but in that case, I think there should be a minus sign in the expression at 33:30 , since alpha is a 1-form. But then that would break the geometric intuition. Perhaps I'm missing something ?
the sum of their degrees. so a wedge b
I'm a little confused by the product rule at T = 13:14.
are we to understand that (a V b) has order k, or does (-1)^k refer to the order of a only?
a -> alpha
b -> beta
V -> Wedge symbol
From looking around a bit, it appears that the latter is true, and by analogy (probably dangerous) to the curl of the product of two vector functions in 3-space.
Ah. This is clarified at T = 31:45
Wonderful video! At 57:00, curl(Y) = (star(dY^flat))^sharp, not dX^flat (if you're saying curl(Y) instead of curl(X)) (minor notational note)
The lectures are extremely well organized. And thanks for your amazing course in this subject. I'm studying it just for fun, and I've already gained a lot just from six lectures.
This video was helpful in understanding this material on a more intuitive level. I am learning from Lee's introduction to smooth manifolds currently, and sometimes it's hard to actually visualize and understand conceptually what is going on in all the calculations/proofs. Glad I found your channel.
This is so good! I'm surprised your playlist doesn't have more views, it's absolute gold. Great for people coming from vec. Calc
The divergence formule here ruclips.net/video/jeiDXhCiF44/видео.html has a typo: \frac {\partial v}{\partial \mu} should be \frac {\partial v}{\partial y}
Thanks for the mavellous lesson. But, how can we get registered Keenan?
0:25
0:14
I enjoyed this series a lot. Thank you for this ! and for uploading the notes also.
Thank you for this amazing series .
Quick question: if exactness is a necessary condition for the exterior derivative, then using inexact differentials in thermodynamics means they must be using a different operator, right? Is that case considered in this theoretical structure? Thank you for the lectures! :)
Oh I think now I understand it: in the notes of thermodynamics formalism of Danny Calegar, the "inexact" differential are presented as conventional exact 1-forms but that occur in an equilibrium manifold with the same initial and final state as the irreversible process (the fact that it is done in an irreversible manner is what makes it inexact).
very nice
42:55