Just wanted to let you know how amazing your channel is. Thanks for being such a generous dude and sharing so much with the public. You've been a huge help in my self-education in math. I always come here first to introduce myself to a new a field. It's a beautiful thing to freely offer people -- especially given how financially prohibitive higher education has become. Your easy, unpretentious grasp of analysis and set theory lets you provide the most succinct, clear, and precise "operational" definitions I've encountered. Also, while your presentation style is dense it's also really accessible as a result. In short, you're simply the best at what you do for your target audience. I'm sure thousands of people are just as grateful for your work as I am. Thank you!
When people offer visualizations of what exactly are these objects generated by wedging vectors in your vector space (or in our case for this series: the dual space/cotangent space at a point on the manifold, rather than the tangent space directly), I noticed there are one of two ways people like to visualize them: 1. As flat geometric objects with direction, analogous to directed line segments for vectors, e.g. * 'directed' plane segments (wedge product between two vectors, or bivectors) * 'directed' cubes/volume segments (wedge product between three vectors, i.e. trivectors) * ... (this is one you'll find in the clifford/geometric algebra community) 2. As stacks of these same flat geometric objects, but now being pierced by the vector it would be measuring (like you'll see in Tristan Needham's 'Visual Differential Geometry and Forms', or in other differential geometry texts that don't avoid visualizing them) When I was starting to learn exterior algebra, I get that everyone will have many ways to visualize the same thing... but I would often get slightly confused as a beginner as to why there seems to be *consistently* two different ways people use to visualize the geometry going on with these objects, but I realized that it's based on which vector space you generated your exterior algebra: * If you generated it from the vector space directly i.e. Λ(V), you'll come to visualization 1. * If you generated it from its dual space i.e. Λ(V*), you'll arrive at 2.
Thank you for the great video. Correct me if I'm wrong, but I think you need to scale the sum by cardinality of S_{k+s}, which is (k+s)! instead of k! s!.
Just wanted to let you know how amazing your channel is. Thanks for being such a generous dude and sharing so much with the public. You've been a huge help in my self-education in math. I always come here first to introduce myself to a new a field. It's a beautiful thing to freely offer people -- especially given how financially prohibitive higher education has become.
Your easy, unpretentious grasp of analysis and set theory lets you provide the most succinct, clear, and precise "operational" definitions I've encountered. Also, while your presentation style is dense it's also really accessible as a result. In short, you're simply the best at what you do for your target audience. I'm sure thousands of people are just as grateful for your work as I am. Thank you!
Thank you very much :) That is really nice! I am glad that I can help!
When people offer visualizations of what exactly are these objects generated by wedging vectors in your vector space (or in our case for this series: the dual space/cotangent space at a point on the manifold, rather than the tangent space directly), I noticed there are one of two ways people like to visualize them:
1. As flat geometric objects with direction, analogous to directed line segments for vectors, e.g.
* 'directed' plane segments (wedge product between two vectors, or bivectors)
* 'directed' cubes/volume segments (wedge product between three vectors, i.e. trivectors)
* ...
(this is one you'll find in the clifford/geometric algebra community)
2. As stacks of these same flat geometric objects, but now being pierced by the vector it would be measuring (like you'll see in Tristan Needham's 'Visual Differential Geometry and Forms', or in other differential geometry texts that don't avoid visualizing them)
When I was starting to learn exterior algebra, I get that everyone will have many ways to visualize the same thing... but I would often get slightly confused as a beginner as to why there seems to be *consistently* two different ways people use to visualize the geometry going on with these objects, but I realized that it's based on which vector space you generated your exterior algebra:
* If you generated it from the vector space directly i.e. Λ(V), you'll come to visualization 1.
* If you generated it from its dual space i.e. Λ(V*), you'll arrive at 2.
Thank you for this series of videos on manifolds, they are really great :)
Good ol’ exterior algebra
Thank you for the great video. Correct me if I'm wrong, but I think you need to scale the sum by cardinality of S_{k+s}, which is (k+s)! instead of k! s!.