Manifolds #9 - Vector Fields (Example)

Автор: WHYB maths

Загружено: 2020-03-29

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Notes are on my GitHub! github.com/rorg314/WHYBmaths

In this video I give an example of a vector field on R^2, showing how it appears in the manifold (and consequently the x, y coordinate chart) and in the polar coordinate r, theta chart. I use this to emphasise how the vector can appear differently in various coordinate systems, and how the vector should be understood as an independent quantity on the manifold, that has different components in various coordinate bases. I then use the chain rule to show how the basis vectors can be easily transformed, and from the definitions of the coordinates (transition functions) I show how you can easily obtain the vector components in the new coordinate system.

Manifolds #9 - Vector Fields (Example)

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