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.

**Physics Forums - The Fusion of Science and Community**

Dismiss Notice

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# How are 1-forms related to infinitesimal displacements?

Loading...

Similar Threads - forms related infinitesimal | Date |
---|---|

I Some Question on Differential Forms and Their Meaningfulness | Feb 19, 2018 |

A Two cones connected at their vertices do not form a manifold | Jan 10, 2017 |

A Notation in Ricci form | Oct 17, 2016 |

How to relate the Ehresmann connection to connection 1-form? | Aug 24, 2015 |

How to relate a one form to a vector. | Apr 8, 2007 |

**Physics Forums - The Fusion of Science and Community**