One-form
From Wikipedia, the free encyclopedia
- This article deals with differential one-forms, a global construction on differential manifolds. For one-forms on vector spaces, i.e. the corresponding local concept, see linear functional.
A one-form on a vector space is simply a linear functional. A (differential) one-form on a differential manifold is a section of the cotangent bundle--in other words, a map that sends each point of the manifold to an element of the cotangent space (i.e. a continuous linear functional on the tangent space) at the point in question. A one-form on a manifold may thus be viewed as a linear functional that varies from point to point, in the same way that a vector field (resp. tensor field) may be thought of as a vector (resp. tensor) that varies from point to point. (One then sees that the use of the term "one-form" in this context is actually an abuse of language of the same type as the use of "tensor" to mean "tensor field"--though presumably more acceptable than the latter, due to the awkwardness of a term such as "one-form-field".)
[edit] Special cases
Let 
be open (e.g. an interval (a,b)), and consider a differentiable function 
, with derivative f'. The differential df of f, at a point 
, is defined as a certain linear map of the variable dx. Specifically, 
. (The meaning of the symbol dx is thus revealed: it is simply an argument, or independent variable, of the function df.) Hence the map 
sends each point x to a linear functional df(x,dx). This is the simplest example of a differential (one-)form.
 |
This mathematics-related article is a stub. You can help Wikipedia by expanding it. |
