Professor Penn, the rate at which you produce top-notch content is just mind boggling! I hope you have some time over for other things like, idk, eating, sleeping, breathing... Lol!
It just gets better and better. A few weeks down the road we will hear about Poincare duality for de Rham cohomologies, CW-complexes or even Atiyah-Signer theorem.
Nice video series but it looks like there is a big mistake on the board at 15:25 (for instance). The summation sign is missing in the approximation to the integral...
Great video, as always. I've heard there are some issues with applying the Riemann integral to higher dimensions due to limit processes. Is it possible to cast an integral of a differential 2-form in the form of a Lebesgue integral?
Very nice explanation. I have the following question: In this video, we are integrating arbitrary differential 2-forms. In particular, the 2-form on the tangent space is arbitrary. I understand how given such an arbitrary 2-form, the integration is well-defined, and this video nicely demonstrates that. However, I am wondering what 2-form concretely one should select so that the integration will result in the usual surface-area-based integration of scalar functions on surfaces. So, I am interesting in knowing how we could use the integration definition of this lecture to integrate a scalar function over a surface (say, embedded into R^3, or into any R^n for n >= 3), in the usual sense where we need to multiply the function value with a differential of the surface area, and integrate over the entire surface. This doesn't seem possible in the framework presented in this lecture. The issue is that we would need to find a 2-form on R^n such that, given two vectors v1 = dphi/du and v2 = dphi/dv, the form returns the signed area of the parallelogram spanned by v1 and v2. I know this is possible when n=2, but such a 2-form doesn't seem to exist when n >= 3. Namely, the area is given by the sqrt of the determinant of the 2x2 Gramm matrix, i.e., sqrt(det( {v_i dot v_j}_{i=1,2; j=1,2})). In other words, we need the sqrt. Also, even the mapping {v1, v2} -> det( {v_i dot v_j}_{i=1,2; j=1,2} ) itself is not even bilinear, so this cannot be the sought-after 2-form. For example, when n=3, this Gramm matrix is exactly the first fundamental form of the surface, and we know that the surface area is given as sqrt( det( firstFundamentalForm) ) du dv. But -- in this lecture, we never pre-prend sqrt to the output of the 2-form, and I don't see how sqrt(det( {v_i dot v_j}_{i=1,2; j=1,2})) could be somehow linked to a 2-form, or otherwise incorporated into the integration framework given in this lecture. And so, this framework seems fundamentally unable to integrate scalar functions over surfaces using the usual Euclidean-based notion of a surface area. This seems to me to be the most common usage case, and so I am perplexed why it doesn't fit into the framework. What I am missing? How is this issue addressed? Thank you!!
At 12:10, why is there no Jacobian in this sum? The vectors after the transformation are not necessarily orthogonal or normalized. It seems like the integral as defined has no physical significance...
I am a little confused by the step around 14:19 where you multiply by delta_u and delta_v while only dividing by delta_u for the first component and delta_v for the second component. I don't see how you are able to ignore the delta_v for the first component and delta_u for the second component.
I have very little idea what's going on in this, but maybe it has something to do with the fact that delta_u and delta_v are perpendicular vectors? Maybe their multiplication on the other tangent vector cancels out in some way? Edit: delta_u and delta_v aren't vectors, but the tangent vectors are perpendicular, so maybe that's it?
The reason is that the definition of bilinear function is separate for each component, not simultaneous; so phi(ax,y)=phi(x,ay)=a*phi(x,y). Think of this as a quadratic function, like x*y. We do not have (ax)*(ay)=a*(xy), we have (ax)*(ay)=(a*a)*(x*y), the constant being separately pulled out of each factor
Thanks for the video! It's not at all obvious to me that the definition of the integral given is independent of the parameterization of the surface. Could you do a video where you give some idea why this is the case?
I can`t believe this is free, this is 100 times better than my Calc4/Calc3 proffessor
This is so good. I cannot believe something that's free with just 10,000 views is this good.
Really clear. This is exactly how a top quality maths lecture should be. Great work. Thanks.
Professor Penn, the rate at which you produce top-notch content is just mind boggling! I hope you have some time over for other things like, idk, eating, sleeping, breathing... Lol!
It just gets better and better. A few weeks down the road we will hear about Poincare duality for de Rham cohomologies, CW-complexes or even Atiyah-Signer theorem.
You saved me with These Videos, i have an exam tomorrow in manifolds and m-forms, thank you very much!!!
Marvellous! Do you have plans on covering Lie Groups & Algebras afterwards?
From 12:23 on there is a sum over i,j missing
It is understood explicitly have u heard of Einstein's summation convention?
@@pythoncure6755 No he just missed it and didn't intend to employ Einstein summation convention
At 13:01, is there a summation sign missing after the Limit symbol?
Most probably
Very interesting question! Enjoyed it so much
Thanks for making these videos, well done as always.
Michael Penn my beloved
Hi from Kazakhstan
Nice video series but it looks like there is a big mistake on the board at 15:25 (for instance). The summation sign is missing in the approximation to the integral...
I think in the "so far" at 12:33, there is a missing Sum
Great video, as always. I've heard there are some issues with applying the Riemann integral to higher dimensions due to limit processes. Is it possible to cast an integral of a differential 2-form in the form of a Lebesgue integral?
Why would you want to? Manifolds are smooth, Riemann integral will suffice.
@VeryEvilPettingZoo Thanks for the thoughtful answer.
Very nice explanation. I have the following question:
In this video, we are integrating arbitrary differential 2-forms. In particular, the 2-form on the tangent space is arbitrary. I understand how given such an arbitrary 2-form, the integration is well-defined, and this video nicely demonstrates that. However, I am wondering what 2-form concretely one should select so that the integration will result in the usual surface-area-based integration of scalar functions on surfaces.
So, I am interesting in knowing how we could use the integration definition of this lecture to integrate a scalar function over a surface (say, embedded into R^3, or into any R^n for n >= 3), in the usual sense where we need to multiply the function value with a differential of the surface area, and integrate over the entire surface. This doesn't seem possible in the framework presented in this lecture.
The issue is that we would need to find a 2-form on R^n such that, given two vectors v1 = dphi/du and v2 = dphi/dv, the form returns the signed area of the parallelogram spanned by v1 and v2. I know this is possible when n=2, but such a 2-form doesn't seem to exist when n >= 3. Namely, the area is given by the sqrt of the determinant of the 2x2 Gramm matrix, i.e., sqrt(det( {v_i dot v_j}_{i=1,2; j=1,2})). In other words, we need the sqrt. Also, even the mapping {v1, v2} -> det( {v_i dot v_j}_{i=1,2; j=1,2} ) itself is not even bilinear, so this cannot be the sought-after 2-form.
For example, when n=3, this Gramm matrix is exactly the first fundamental form of the surface, and we know that the surface area is given as sqrt( det( firstFundamentalForm) ) du dv. But -- in this lecture, we never pre-prend sqrt to the output of the 2-form, and I don't see how sqrt(det( {v_i dot v_j}_{i=1,2; j=1,2})) could be somehow linked to a 2-form, or otherwise incorporated into the integration framework given in this lecture.
And so, this framework seems fundamentally unable to integrate scalar functions over surfaces using the usual Euclidean-based notion of a surface area. This seems to me to be the most common usage case, and so I am perplexed why it doesn't fit into the framework. What I am missing? How is this issue addressed? Thank you!!
You need a Riemannian metric to do the kinds of integrals you're asking about.
At 12:10, why is there no Jacobian in this sum? The vectors after the transformation are not necessarily orthogonal or normalized. It seems like the integral as defined has no physical significance...
Well explained!
I am a little confused by the step around 14:19 where you multiply by delta_u and delta_v while only dividing by delta_u for the first component and delta_v for the second component. I don't see how you are able to ignore the delta_v for the first component and delta_u for the second component.
I have very little idea what's going on in this, but maybe it has something to do with the fact that delta_u and delta_v are perpendicular vectors? Maybe their multiplication on the other tangent vector cancels out in some way?
Edit: delta_u and delta_v aren't vectors, but the tangent vectors are perpendicular, so maybe that's it?
Bilinearity, you should watch the previous videos on 2 forms to understand this property
The reason is that the definition of bilinear function is separate for each component, not simultaneous; so phi(ax,y)=phi(x,ay)=a*phi(x,y). Think of this as a quadratic function, like x*y. We do not have (ax)*(ay)=a*(xy), we have (ax)*(ay)=(a*a)*(x*y), the constant being separately pulled out of each factor
Hello Michael, I was watching your real analysis playlist and there are some videos you put in private, can you help me with that?
Really good lecture!! I hope see your math guide more!
Thanks for the video! It's not at all obvious to me that the definition of the integral given is independent of the parameterization of the surface. Could you do a video where you give some idea why this is the case?
Why is it the determinant of the transpose of the matrix
Why do we have phi((u_i,v_{j+1}) - phi(u_i,v_j) as the tangent vector?
It seems like we are limited to surfaces that have no holes and to maps that are single-valued on the surface.
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