World Library  
Flag as Inappropriate
Email this Article

One-form

Article Id: WHEBN0000314692
Reproduction Date:

Title: One-form  
Author: World Heritage Encyclopedia
Language: English
Subject: Dual space, Hodge dual, Integral, Linear form, Coframe
Collection: 1 (Number), Differential Forms
Publisher: World Heritage Encyclopedia
Publication
Date:
 

One-form

Linear functionals (1-forms) α, β and their sum σ and vectors u, v, w, in 3d Euclidean space. The number of (1-form) hyperplanes intersected by a vector equals the inner product.[1]

In linear algebra, a one-form on a vector space is the same as a linear functional on the space. The usage of one-form in this context usually distinguishes the one-forms from higher-degree multilinear functionals on the space. For details, see linear functional.

In differential geometry, a one-form on a differentiable manifold is a smooth section of the cotangent bundle. Equivalently, a one-form on a manifold M is a smooth mapping of the total space of the tangent bundle of M to \mathbb{R} whose restriction to each fibre is a linear functional on the tangent space. Symbolically,

\alpha : TM \rightarrow {\mathbb{R}},\quad \alpha_x = \alpha|_{T_xM}: T_xM\rightarrow {\mathbb{R}}

where αx is linear.

Often one-forms are described locally, particularly in local coordinates. In a local coordinate system, a one-form is a linear combination of the differentials of the coordinates:

\alpha_x = f_1(x) \, dx_1 + f_2(x) \, dx_2+ \cdots +f_n(x) \, dx_n

where the fi are smooth functions. From this perspective, a one-form has a covariant transformation law on passing from one coordinate system to another. Thus a one-form is an order 1 covariant tensor field.

Contents

  • Examples 1
    • Linear 1.1
    • Differential 1.2
  • Differential of a function 2
  • See also 3
  • References 4

Examples

Linear

Many real-world concepts can be described as one-forms:

  • Indexing into a vector: The second element of a three-vector is given by the one-form [0, 1, 0]. That is, the second element of [x ,y ,z] is
[0, 1, 0] · [xyz] = y.
  • Mean: The mean element of an n-vector is given by the one-form [1/n, 1/n, ..., 1/n]. That is,
\operatorname{mean}(v) = [1/n, 1/n,\dots,1/n]\cdot v.
  • Sampling: Sampling with a kernel can be considered a one-form, where the one-form is the kernel shifted to the appropriate location.
\mathrm{NPV}(R(t)) = \langle w, R\rangle = \int_{t=0}^\infty \frac{R(t)}{(1+i)^{t}}\,dt.

Differential

The most basic non-trivial differential one-form is the "change in angle" form d\theta. This is defined as the derivative of the angle "function" \theta(x,y) (which is only defined up to a constant), which can be explicitly defined in terms of the atan2 function \operatorname{atan2}(y,x) = \operatorname{arctan}(y/x). Taking the derivative yields the following formula for the total derivative:

\begin{align} d\theta &= \partial_x\left(\operatorname{atan2}(y,x)\right) dx + \partial_y\left(\operatorname{atan2}(y,x)\right) dy \\ &= -\frac{y}{x^2 + y^2} dx + \frac{x}{x^2 + y^2} dy \end{align}

While the angle "function" cannot be continuously defined – the function atan2 is discontinuous along the negative y-axis – which reflects the fact that angle cannot be continuously defined, this derivative is continuously defined except at the origin, reflecting the fact that infinitesimal (and indeed local) changes in angle can be defined everywhere except the origin. Integrating this derivative along a path gives the total change in angle over the path, and integrating over a closed loop gives the winding number.

In the language of differential geometry, this derivative is a one-form, and it is closed (its derivative is zero) but not exact (it is not the derivative of a 0-form, i.e., a function), and in fact it generates the first de Rham cohomology of the punctured plane. This is the most basic example of such a form, and it is fundamental in differential geometry.

Differential of a function

Let U \subseteq \mathbb{R} be open (e.g., an interval (a,b) ), and consider a differentiable function f: U \to \mathbb{R} , with derivative f'. The differential df of f, at a point x_0\in U , is defined as a certain linear map of the variable dx. Specifically, df(x_0, dx): dx \mapsto f'(x_0) dx . (The meaning of the symbol dx is thus revealed: it is simply an argument, or independent variable, of the function df.) Hence the map x \mapsto df(x,dx) sends each point x to a linear functional df(x,dx). This is the simplest example of a differential (one-)form.

In terms of the de Rham complex, one has an assignment from zero-forms (scalar functions) to one-forms i.e., f\mapsto df.

See also

References

  1. ^ J.A. Wheeler, C. Misner, K.S. Thorne (1973). Gravitation. W.H. Freeman & Co. p. 57.  
This article was sourced from Creative Commons Attribution-ShareAlike License; additional terms may apply. World Heritage Encyclopedia content is assembled from numerous content providers, Open Access Publishing, and in compliance with The Fair Access to Science and Technology Research Act (FASTR), Wikimedia Foundation, Inc., Public Library of Science, The Encyclopedia of Life, Open Book Publishers (OBP), PubMed, U.S. National Library of Medicine, National Center for Biotechnology Information, U.S. National Library of Medicine, National Institutes of Health (NIH), U.S. Department of Health & Human Services, and USA.gov, which sources content from all federal, state, local, tribal, and territorial government publication portals (.gov, .mil, .edu). Funding for USA.gov and content contributors is made possible from the U.S. Congress, E-Government Act of 2002.
 
Crowd sourced content that is contributed to World Heritage Encyclopedia is peer reviewed and edited by our editorial staff to ensure quality scholarly research articles.
 
By using this site, you agree to the Terms of Use and Privacy Policy. World Heritage Encyclopedia™ is a registered trademark of the World Public Library Association, a non-profit organization.
 


Copyright © World Library Foundation. All rights reserved. eBooks from Project Gutenberg are sponsored by the World Library Foundation,
a 501c(4) Member's Support Non-Profit Organization, and is NOT affiliated with any governmental agency or department.