Thank you and Merry Christmas to you and your family. Thank you for your earlier comments on one of the videos from push forward of vectors on manifolds part five. I replaced that video with two that were vastly better. Your comments made a real difference.
Thank you for your thoughtful question! In the last line I left the summation symbol out in order to save space and keep everything on the one line. This seemed a reasonable choice given that ω was defined above. Also, I left the wedge product out for the same reason when it should really be present when describing an integral over a manifold.
14:40 Integral of a 2-Form on a Surface: Why is the z coordinate ignored in the Jacobian matrix? I think the Jacobian matrix, in this case, is not square so there is no determinant.
Hello Daniel and thank you for spotting that. You are quite right. The basis 1-form dz is not not supposed to be there as the question was dealing with the integral of a 2-form over a surface. It should have read: ω=f(x,y,z)dx∧dy. Thanks again for spotting that.
Thank you again. I want to bring in a concept that seems like it could be part of this discussion, the tensor properties of differential forms. One of the main applications of the pullback here is change of coordinates for purposes of integration of a form. That could be a change of cartesian coordinates to polar, say. True, the change of coordinates can be described as a pullback from the manifold in the old coordinates to (a copy of) the same manifold with the new coordinates. By the properties of the pullback you have given us, the integral will be unchanged so we can interpret this operation as a change of coordinates in the integral of a form over a manifold. But the form itself is a (covariant) tensor, no? Tensors are defined through their transformation properties under changes of coordinates. So shouldn't it be possible to approach this problem directly using the tensor properties of differential forms under a change of coordinate systems? Shouldn't we get the same answer as using a pullback? Can this be demonstrated (or not)?. Many thanks if you can a connection for us.
I think your playlist (Differential Geometry) is upside down! You should order the playlist according to the order in which you want people to watch it.
I have to say, this channel is gold. Thank you so much for your videos!
Thank you for that.
Thank you. Happy Holidays.
Thank you and Merry Christmas to you and your family.
Thank you for your earlier comments on one of the videos from push forward of vectors on manifolds part five. I replaced that video with two that were vastly better. Your comments made a real difference.
Es de admirar su trabajo profesor
Thank you! If you mean professor as teacher, then yes to that.
23:14 Don't you need a summation sign over the i indices for the last integral on the bottom line?
Thank you for your thoughtful question! In the last line I left the summation symbol out in order to save space and keep everything on the one line. This seemed a reasonable choice given that ω was defined above. Also, I left the wedge product out for the same reason when it should really be present when describing an integral over a manifold.
14:40 Integral of a 2-Form on a Surface: Why is the z coordinate ignored in the Jacobian matrix? I think the Jacobian matrix, in this case, is not square so there is no determinant.
Hello Daniel and thank you for spotting that. You are quite right. The basis 1-form dz is not not supposed to be there as the question was dealing with the integral of a 2-form over a surface. It should have read: ω=f(x,y,z)dx∧dy.
Thanks again for spotting that.
@@TensorCalculusRobertDavie Thank you for this wonderful channel that has taught me so much!
Thanks for your lectures! What software do you use to make your classes? Please.
You're welcome. I use Paint 2D, PPT and Mathematica.
Thank you again. I want to bring in a concept that seems like it could be part of this discussion, the tensor properties of differential forms.
One of the main applications of the pullback here is change of coordinates for purposes of integration of a form. That could be a change of cartesian coordinates to polar, say. True, the change of coordinates can be described as a pullback from the manifold in the old coordinates to (a copy of) the same manifold with the new coordinates. By the properties of the pullback you have given us, the integral will be unchanged so we can interpret this operation as a change of coordinates in the integral of a form over a manifold.
But the form itself is a (covariant) tensor, no? Tensors are defined through their transformation properties under changes of coordinates. So shouldn't it be possible to approach this problem directly using the tensor properties of differential forms under a change of coordinate systems? Shouldn't we get the same answer as using a pullback? Can this be demonstrated (or not)?.
Many thanks if you can a connection for us.
I think your playlist (Differential Geometry) is upside down! You should order the playlist according to the order in which you want people to watch it.
Hello Daniel and thank you for that! It should be fixed now.