Where do Differential forms come from? Answer: One day Élie Cartan was having a lot of Absinthe and suddenly he had a strange idea: "What if I take the change of variable formula and represent it without that annoying Jacobian determinant?" First Mr. Cartan expressed, as usual, the integrand and the region with new coordinate function X' = ϕ(X). Then he formally expressed the differentials with new differentials, dx'= ∑∂(x')/∂x*dx. Now he could extract the Jacobian determinant if he treated differentials in a very *bizarre* way, dx*dy = -dy*dx, just like with matrix columns in determinant. So he could get rid of the Jacobian determinant Det[J] and encode that information into the differentials: Det[J]*d(x_1)∧...∧d(x_n)=Det[J]Det[x_1,..., x_n]=Det[J*X]=Det[X']=Det[x'_1,..., x'_n]=d(x'_1)∧...∧d(x'_n). ---> ∫f(X')*d(x'_1)∧...∧d(x'_n)= ∫f((ϕ(X))*Det[J_ϕ]*d(x_1)∧...∧d(x_n) As a true Frenchman Cartan quickly axiomatized and generalized his powerful invention, and left future generations wondering what the Hell's going on...
23:50 is the clearest proof of the linear independence, and therefore spanning set and basis, of the basis of the dual space I've ever seen in any textbook, and I've read the big names prior to this. They aren't as clear as this course. This course doesn't cease to amaze me; great Job.
I can't believe this course is covering the dual space. This truly is awesome, it might even ultimately be better than Hubbard and Hubbard's textbook, and after reading both, I think it's also at least on par with Calculus on Manifolds by Spivak, which says a lot -- perhaps even better than Spivak with its treatment of limits in general. I think the one and only drawback of this class is that it doesn't cover the Lebesgue integral. But that is the only flaw I can find in it, except perhaps for avoiding the relationship between eigenstuff and optimization (lagrange multipliers and definiteness) and rushing through the theory of eigenstuff in general. I can only guess that eigenstuff is more elaborately explained in an ODE course that comes after this sequence.
The elements w of the dual space are ‘linear functionals’ taking the elements of the vector space v to R, thus is in R. In the case of matrices [w ]’ [v] = r in R. Because of isomorphism why don’t we ever say v is a linear functional eg , or conversely that w is just a covector? Why is it always only presented w is a functional and v a vector?
thumbs up for this easy 2 understand lecture, this is truely amazing, cool!!! Please go on with differential form leactures! :-)) Thanks a lot for this terrific work!! No comparison to the usual Show off lectures ;-)
After weeks of reading and watching, it was you who finally made me understand the dual vector space. Thank you so so much for this.
Where do Differential forms come from?
Answer: One day Élie Cartan was having a lot of Absinthe and suddenly he had a strange idea: "What if I take the change of variable formula and represent it without that annoying Jacobian determinant?" First Mr. Cartan expressed, as usual, the integrand and the region with new coordinate function X' = ϕ(X). Then he formally expressed the differentials with new differentials, dx'= ∑∂(x')/∂x*dx. Now he could extract the Jacobian determinant if he treated differentials in a very *bizarre* way, dx*dy = -dy*dx, just like with matrix columns in determinant. So he could get rid of the Jacobian determinant Det[J] and encode that information into the differentials: Det[J]*d(x_1)∧...∧d(x_n)=Det[J]Det[x_1,..., x_n]=Det[J*X]=Det[X']=Det[x'_1,..., x'_n]=d(x'_1)∧...∧d(x'_n).
---> ∫f(X')*d(x'_1)∧...∧d(x'_n)= ∫f((ϕ(X))*Det[J_ϕ]*d(x_1)∧...∧d(x_n)
As a true Frenchman Cartan quickly axiomatized and generalized his powerful invention, and left future generations wondering what the Hell's going on...
23:50 is the clearest proof of the linear independence, and therefore spanning set and basis, of the basis of the dual space I've ever seen in any textbook, and I've read the big names prior to this. They aren't as clear as this course. This course doesn't cease to amaze me; great Job.
Very great lectures, and what a beatiful handwriting!
I can't believe this course is covering the dual space. This truly is awesome, it might even ultimately be better than Hubbard and Hubbard's textbook, and after reading both, I think it's also at least on par with Calculus on Manifolds by Spivak, which says a lot -- perhaps even better than Spivak with its treatment of limits in general. I think the one and only drawback of this class is that it doesn't cover the Lebesgue integral. But that is the only flaw I can find in it, except perhaps for avoiding the relationship between eigenstuff and optimization (lagrange multipliers and definiteness) and rushing through the theory of eigenstuff in general. I can only guess that eigenstuff is more elaborately explained in an ODE course that comes after this sequence.
Hey viewers! This lecture has been refilmed and a complete version can be found in the link in the comments!
where...?
The elements w of the dual space are ‘linear functionals’ taking the elements of the vector space v to R, thus is in R. In the case of matrices [w ]’ [v] = r in R. Because of isomorphism why don’t we ever say v is a linear functional eg , or conversely that w is just a covector? Why is it always only presented w is a functional and v a vector?
Many thx for the cool stuff there. but the course seems incomplete?🙄
"your furry friend, the determinant"
sadly The video is incomplete... what are we missing? thank you
Excellent talk, but seems to finish before the end of the lecture!
Yes, sadly we noticed this when it was too late to find the missing bits. We will try to rerecord approximately this lecture next spring.
What was said at he end? I got really excited!
Good teacher, great course.
Dr. Shifrin is the student of the legendary Chern. No wonder...
thumbs up for this easy 2 understand lecture, this is truely amazing, cool!!!
Please go on with differential form leactures! :-)) Thanks a lot for this terrific work!! No comparison to the usual Show off lectures ;-)
excellent!
Gee, why do i got the feeling that it is a sheriff teaching math there? 🤔