Introduction to 1-Forms

This video introduces the idea of a 1-Form including its definition and how it acts on vectors. It looks at tangent and co-tangent spaces and how these spaces are related to vectors and 1-forms using pictorial representations to support the ideas covered.

[@johnwarren8032]

This is a good high-level review but I would not understand it at all without background. So thanks for the review, but if you want this to be truly helpful to people who don't understand the material you will have to explain key concepts with more care. For instance, the discussion of tangent vectors and tangent spaces is a hand-wave and it is unclear what the expression in terms of the derivative-like basis really means. I think it would be helpful to go through a more complete explanation, and explain that tangent vectors are actually linear functions from the space of continuous functions in the manifold to the reals. A tangent vector is like a little machine that measures how much any given real valued function on the manifold would change for an increment with the length and direction associated with that tangent vector. Concepts like this really need to be explained thoroughly for students to really get it, in my opinion. Thanks for the opportunity to give feedback.

[@TensorCalculusRobertDavie]

Thank you for your thoughtful feedback! You make an excellent point about the need for a more thorough explanation of key concepts like tangent vectors and tangent spaces, especially for viewers who are new to the material. I’ll certainly consider creating a follow-up video or expanding the current content to delve deeper into these foundational ideas. For example, explaining tangent vectors as linear functionals that measure the rate of change of functions on the manifold is a fantastic approach that could help bridge the gap for beginners. I appreciate your suggestion and your engagement with the content-it helps me refine these explanations to be more accessible for all viewers. Stay tuned for updates!

[@BCarli1395]

Thanks. This is a good way to visualize-it “clicked” for me.

[@TensorCalculusRobertDavie]

Thanks for that feedback. Cheers.

[@lkapitan8232]

Thank you for this, I've really been struggling with the concepts.

[@TensorCalculusRobertDavie]

I'm glad to hear you found it helpful. Cheers.

[@BrotherBobby-IN]

What's the difference between tangent space and cotangent space?

[@TensorCalculusRobertDavie]

Great question! The concepts of tangent space and cotangent space are fundamental in differential geometry and general relativity. Here's a detailed explanation: Tangent Space The tangent space at a point on a manifold is the set of all possible tangent vectors at that point. Intuitively, you can think of the tangent space as a plane that "touches" the manifold at a given point, containing all the directions in which one can tangentially pass through that point. For a manifold 𝑀 at a point 𝑝, the tangent space is denoted by 𝑇_𝑝𝑀. Example: Consider a 2D surface (like a sphere) in 3D space. The tangent space at a point on the surface is the plane that just touches the sphere at that point, containing all possible directions you can move while staying on the surface. Cotangent Space The cotangent space at a point on a manifold is the dual space to the tangent space. While the tangent space consists of tangent vectors, the cotangent space consists of covectors (also known as one-forms). These are linear functionals that take a tangent vector and return a real number. For a manifold 𝑀 at a point 𝑝, the cotangent space is denoted by 𝑇_𝑝^∗𝑀. Example: If you have a function 𝑓 defined on the manifold, the gradient of 𝑓 at a point 𝑝 is an element of the cotangent space at 𝑝. This gradient can take any tangent vector at 𝑝 and return the rate of change of the function 𝑓 in the direction of that vector. Key Differences Nature: Tangent vectors are elements of the tangent space and represent directions in which one can move on the manifold. Covectors (one-forms) are elements of the cotangent space and are linear functionals acting on tangent vectors. Duality: The cotangent space is the dual space to the tangent space. This means each element of the cotangent space is a linear map that takes an element of the tangent space and returns a scalar. Dimensions: At each point 𝑝 on an 𝑛-dimensional manifold, both the tangent space 𝑇_𝑝𝑀 and the cotangent space 𝑇_𝑝^∗𝑀 are 𝑛-dimensional vector spaces. Application: Tangent vectors are used to describe velocities and directions of curves on the manifold. Covectors are used to describe gradients, differentials, and other quantities that can act on tangent vectors. Visualization Imagine standing on a hill: The tangent space at your feet represents all possible directions you can walk. The cotangent space can be thought of as representing measurements of how steep the hill is in various directions. Each co-vector in the cotangent space can tell you the slope (rate of change) of the hill in a specific direction. I hope this helps clarify the distinction between tangent and cotangent spaces! If you have any further questions, feel free to ask.