Thank you again. Your way of using diagrams is very helpful because these are basically visual and spacial concepts. People need to 'see' it in their mind's eye to really understand. Just processing it all verbally with the left side of my brain isn't enough for me and it is likely the same for most people . I made a comment on a previous video about the notation in the diagrams, though. It seems to me that the pullback mapping (purple arrow) should be indicated as f*, not f* \omega_f(p) . f*\omega_f(p) is actually the image of the pullback mapping, so it is a form in the space T*M (image space of the pullback map). A similar issue applies in the pullback mapping associated with the inverse of f. This could possibly cause some misunderstanding, I believe. Thanks.
Thank you for your question! Yes, these topics are generally covered in courses related to differential geometry or advanced calculus on manifolds, often at the advanced undergraduate level. Courses Where You Might Encounter These Topics: 1. Differential Geometry: Many universities offer an undergraduate course in differential geometry, which typically covers the basics of manifolds, tangent spaces, vector fields, and curvature. You’ll also likely encounter topics like the geometry of curves and surfaces, the concept of geodesics, and sometimes an introduction to differential forms. Differential geometry is foundational in physics, particularly in general relativity, where the geometry of spacetime is modeled using similar tools. 2. Advanced Calculus on Manifolds / Multivariable Analysis: Some programs offer a course under names like "Advanced Calculus on Manifolds" or "Multivariable Analysis." These courses often delve into differential forms, the exterior derivative, integration on manifolds, and Stokes’ theorem, which are essential tools in more advanced differential geometry. 3. Courses in Theoretical Physics or General Relativity: If you’re studying physics, courses on general relativity or advanced classical mechanics will often touch on these topics, especially if they introduce tensors, differential forms, or the geometry of spacetime. Where to Start? If you’re interested in these topics but don’t have access to a full differential geometry course, you might start by looking for textbooks or online resources on differential geometry for physicists or for advanced calculus. These resources often cover the basics in a way that’s approachable at the undergraduate level. Feel free to ask more questions as you explore these subjects! Thanks for engaging with the video.
Fascinating! 🎉😊
Thank you!
great videos, thank you
You're welcome!
Thank you again. Your way of using diagrams is very helpful because these are basically visual and spacial concepts. People need to 'see' it in their mind's eye to really understand. Just processing it all verbally with the left side of my brain isn't enough for me and it is likely the same for most people .
I made a comment on a previous video about the notation in the diagrams, though. It seems to me that the pullback mapping (purple arrow) should be indicated as f*, not f* \omega_f(p) . f*\omega_f(p) is actually the image of the pullback mapping, so it is a form in the space T*M (image space of the pullback map). A similar issue applies in the pullback mapping associated with the inverse of f. This could possibly cause some misunderstanding, I believe.
Thanks.
this is a lot like classical vector calculus
Not surprising when you consider that R^n is a manifold. Lucky for us only a single coordinate chart is needed to cover it.
is there an undergraduate course where these topics would be explored? would this fall under differential geometry?
Thank you for your question! Yes, these topics are generally covered in courses related to differential geometry or advanced calculus on manifolds, often at the advanced undergraduate level.
Courses Where You Might Encounter These Topics:
1. Differential Geometry:
Many universities offer an undergraduate course in differential geometry, which typically covers the basics of manifolds, tangent spaces, vector fields, and curvature. You’ll also likely encounter topics like the geometry of curves and surfaces, the concept of geodesics, and sometimes an introduction to differential forms.
Differential geometry is foundational in physics, particularly in general relativity, where the geometry of spacetime is modeled using similar tools.
2. Advanced Calculus on Manifolds / Multivariable Analysis:
Some programs offer a course under names like "Advanced Calculus on Manifolds" or "Multivariable Analysis." These courses often delve into differential forms, the exterior derivative, integration on manifolds, and Stokes’ theorem, which are essential tools in more advanced differential geometry.
3. Courses in Theoretical Physics or General Relativity:
If you’re studying physics, courses on general relativity or advanced classical mechanics will often touch on these topics, especially if they introduce tensors, differential forms, or the geometry of spacetime.
Where to Start?
If you’re interested in these topics but don’t have access to a full differential geometry course, you might start by looking for textbooks or online resources on differential geometry for physicists or for advanced calculus. These resources often cover the basics in a way that’s approachable at the undergraduate level.
Feel free to ask more questions as you explore these subjects! Thanks for engaging with the video.