One-form

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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.

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

where 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:

where the 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.

ExamplesEdit

ApplicationsEdit

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   That is, the second element of   is
     
  • Mean: The mean element of an  -vector is given by the one-form   That is,
     
  • Sampling: Sampling with a kernel can be considered a one-form, where the one-form is the kernel shifted to the appropriate location.
  • Net present value of a net cash flow,   is given by the one-form   where   is the discount rate. That is,
     

DifferentialEdit

The most basic non-trivial differential one-form is the "change in angle" form   This is defined as the derivative of the angle "function"   (which is only defined up to an additive constant), which can be explicitly defined in terms of the atan2 function   Taking the derivative yields the following formula for the total derivative:

 
While the angle "function" cannot be continuously defined – the function atan2 is discontinuous along the negative  -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 times  

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, that is, 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 functionEdit

Let   be open (for example, an interval  ), and consider a differentiable function   with derivative   The differential   of   at a point   is defined as a certain linear map of the variable   Specifically,   (The meaning of the symbol   is thus revealed: it is simply an argument, or independent variable, of the linear function  ) Hence the map   sends each point   to a linear functional   This is the simplest example of a differential (one-)form.

In terms of the de Rham cochain complex, one has an assignment from zero-forms (scalar functions) to one-forms; that is,  

See alsoEdit

ReferencesEdit

  1. ^ J.A. Wheeler; C. Misner; K.S. Thorne (1973). Gravitation. W.H. Freeman & Co. p. 57. ISBN 0-7167-0344-0.