Consider a point(adsbygoogle = window.adsbygoogle || []).push({}); pin a manifold with coordinates [tex]x^\alpha[/tex] and another point nearby with coordinates [tex]x^\alpha + dx^\alpha[/tex] wheredxare infinitesimal or arbitrarily small. Suppose we have a functionfon 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 functionfto 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 atp. 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 pointpcan also be thought of as directional derivatives to curves passing throughp, visually it is tangent vectors which seem to me more like small displacements, rather than their dual the 1-forms.

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# How are 1-forms related to infinitesimal displacements?

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