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Fredrik
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Oct13-04, 07:23 PM
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The tangent space [tex]T_pM[/tex] of the manifold M at point p can be defined as the vector space spanned by the basis vectors

[tex]\frac{\partial}{\partial x^\mu}\bigg\lvert_p[/tex]

where x is a coordinate system (a chart). (There is also a coordinate independent definition, but it will result in the same quantities being called tangent vectors, so it's equvalent to this one).

Let F denote the set of smooth ([tex]C^\infty[/tex]) functions from M into R (the set of real numbers). The vectors in the tangent spaces are linear functions from F into R

In the physics literature, a vector field is sometimes defined as a function that takes each point p in a subset U of M to a vector in the tangent space at p. In the mathematics literature the definition is a little more complicated, but it's still pretty close to the sloppy physicist's definition.

I'm not 100% sure that I remember the mathematical definition 100% correctly, but I think this is at least very close to it:

The tangent bundle TM of the manifold M is defined by

[tex]TM=\big\{(p,v)|p\in M, v\in \bigcup_{q\in M}T_qM\big\}[/tex]

The function [tex]\pi:TM\rightarrow M[/tex] defined by

[tex]\pi(p,v)=p[/tex]

is called the projection.

A vector field is a section of the tangent bundle. A section is a function [tex]X:M\rightarrow TM[/tex], such that

[tex]\pi(X(p))=p[/tex]