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Im trying to read some mathematical physics and have problems with the understanding of vector fields. Th questions are regarding the explanations in the book "Geometrical methods of mathematical physics"..

The author, Bernard Schutz, writes:

"

*Given a coordinate system x^i, it is often convenient to adopt {∂/∂x^i} as a basis for vector fields.*"

This seems like a good basis for the tangent space to a given point (which can be described by our coordinates) but the vector space of our vector fields on M is an infinite dimensional space.. Im not really sure what he means by the set {∂/∂x^i}?? The elements doesnt even seem to be vector fields.

Secondly:

He writes that noncoordinate basis is a basis that can not be expressed by any coordinate system.. but when he shows that two vector fields not necessary commutes he actually expands them in the coordinate system! Im not really sure what he wants to show. (page 44.)

I am really thankful for your effort!

All the best!

/Kontilera