1-forms. Line Integrals. Degree

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In this video, we introduce the simplest type of integration on manifolds, the analogues of line integrals. We define 1-forms, which are the objects to be integrated in this case. These are dual to the notion of vector fields. We finish off the video by looking at an interesting 1-form on the circle which is intimately related to the notion of degree in topology. This is the first of many examples of the interesting connection between differential forms and topology.
  1. DanielChanMaths
  2. Daniel Chan
  3. mathematics
  4. differential forms
  5. line integrals
  6. degree
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All I can say is, "wow." Great video my friend, thank you.

gamcdermgamcderm
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In your notation: w is a 1-form and r' is a vector. The inner product <w, r'> does not make sense. Do you mean w(r')?

mochen
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@oldtom541
1 second ago
Like everyone else I find, you did not justify the partial / /partial x_i notation for basis vectors of T_p. What does this get to differentiate if there cannot be a position vector for the manifold?

Otherwise super lecture, thanks!

oldtom
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How i wish you will talk on Riemannian Manifold, Its curvature and its optimazations

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