- #1

pellman

- 684

- 5

Consider a point

[tex]df=f(x^\alpha + dx^\alpha)-f(x^\alpha)=\partial_\mu f dx^\mu[/tex]

Notationally this is identical to the expression for the d operator acting on the scalar function

[tex]df=\partial_\mu f dx^\mu[/tex]

where the dx^mu here are the coordinate basis of the 1-forms at

Now I can easily repeat back to you the definition of a tangent vector as a map from functions on the manifold to real numbers, and the definition of 1-forms as the dual space to the tangent space. But I can't see the conceptual relationship to small displacements.

Furthermore, since tangent vectors at a point

*p*in a manifold with coordinates [tex]x^\alpha[/tex] and another point nearby with coordinates [tex]x^\alpha + dx^\alpha[/tex] where*dx*are infinitesimal or arbitrarily small. Suppose we have a function*f*on this manifold. Then we can write[tex]df=f(x^\alpha + dx^\alpha)-f(x^\alpha)=\partial_\mu f dx^\mu[/tex]

Notationally this is identical to the expression for the d operator acting on the scalar function

*f*to give the 1-form[tex]df=\partial_\mu f dx^\mu[/tex]

where the dx^mu here are the coordinate basis of the 1-forms at

*p*. Surely the similarity in notation is meaningful in some way. But how?Now I can easily repeat back to you the definition of a tangent vector as a map from functions on the manifold to real numbers, and the definition of 1-forms as the dual space to the tangent space. But I can't see the conceptual relationship to small displacements.

Furthermore, since tangent vectors at a point

*p*can also be thought of as directional derivatives to curves passing through*p*, visually it is tangent vectors which seem to me more like small displacements, rather than their dual the 1-forms.
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