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One-form
  CATEGORIES ABOUT ONE-FORM
   
differential forms
   
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INTRODUCTION

A one-form is a Tensor of type \begin{pmatrix} 0 \ 1 \end{pmatrix} . It is the simplest non-scalar tensor.

Let ilde{f} represent a one-form which acts on vectors of space ''V'', including vectors ec u and ec v. Then the Linearity properties of ilde{f} are
: ilde{f} ( ec u + ec v) = ilde{f} ( ec u) + ilde{f} ( ec v)
: ilde{f} (\alpha ec v) = \alpha ilde{f} ( ec v)
where ''α'' is a Scalar .

The set of all one-forms definable on the vector space ''V'' can also itself be a vector space if one-forms can be added to each other or be multiplied by scalars in a pointwise linear manner. That is, if the vectors of the space ''V'' are position vectors of points, then for every point ec v in the space ''V'', the following should hold true:
: ( ilde{f} + ilde{g}) ( ec v) = ilde{f}( ec v) + ilde{g}( ec v)
: (\alpha ilde{f}) ( ec v) = \alpha ilde{f}( ec v).
If these last two conditions are true for every ec v \isin V then the one-forms constitute a vector space.

If ''V'' is an Inner-product Space with Inner Product 〈 , 〉 then every vector ec v can be mapped to a dual one-form ilde{v} defined by
: ilde{v} := \langle ec v, \ angle
(i.e. ilde{v} := \lambda x. \langle ec v, x angle in Lambda notation)
so that the one-form ilde{v} applied to a vector ec u yields
: ilde{v} ( ec u) = \langle ec v, ec u angle.
Thus the inner product provides a Bijection of each vector in ''V'' to a one-form of its dual vector space ilde{V}.


VISUALIZING ONE-FORMS

A vector is usually visualized as an arrow extending from the origin to a point in space. A one-form can be visualized as a set of equally spaced parallel planes which partition the entire space. The magnitude of a one-form is directly proportional to the density of parallel planes and inversely proportional to the spacing between pairs of neighboring planes. To find the result of applying a one-form to a vector, basically count the number of planes which a vector cuts through. (Note: this visualization is discrete whereas one-forms and vectors have magnitudes which range continuously over the real numbers. The visualization can be interpolated linearly, as it were, to increase the precision.)


BASIS OF THE DUAL SPACE

Let the vector space ''V'' have a basis { ec e}_1,\ { ec e}_2, … , { ec e}_n, not necessarily Orthonormal nor even Orthogonal . Then the Dual Space ilde{V} has a basis ilde{\omega}^1, \ ilde{\omega}^2, … , \ ilde{\omega}^n which in the three-dimensional case (''n'' = 3) can be defined by
: ilde{\omega}^i = {1 \over 2} \, \left\langle { \epsilon^{ijk} \, ( ec e_j imes ec e_k) \over ec e_1 \cdot ec e_2 imes ec e_3} , \qquad ight angle
where \epsilon\,\! is the Levi-Civita Symbol . This definition has the special property that
: ilde{\omega}^i ( ec e_j) = \delta^i {}_j
where δ is the Kronecker Delta . Thus, these two dual bases are mutually orthonormal even if each basis is not self-orthonormal.

''N.B.'' The superscripts of the basis one-forms are not exponents but are instead Contravariant indices.

A one-form ilde{u} belonging to the dual space ilde{V} can be expressed as a Linear Combination of basis one-forms, with coefficients ("components") ''ui'' ,
: ilde{u} = u_i \, ilde{\omega}^i
Then, applying one-form ilde{u} to a basis vector ''ej'' yields
: ilde{u}( ec e_j) = (u_i \, ilde{\omega}^i) ec e_j = u_i ( ilde{\omega}^i ( ec e_j))
due to linearity of scalar multiples of one-forms and pointwise linearity of sums of one-forms. Then
: ilde{u}({ ec e}_j) = u_i ( ilde{\omega}^i ({ ec e}_j)) = u_i \delta^i {}_j = u_j
that is
: ilde{u} ( ec e_j) = u_j.
This last equation shows that an individual component of a one-form can be extracted by applying the one-form to a corresponding basis vector.


DIFFERENTIAL ONE-FORMS

A differential one-form is a one-form the components of which are all Differential . It is the simplest non-scalar Differential Form .


SEE ALSO



REFERENCE

  • Bernard F. Schutz (1985, 2002). ''A first course in general relativity.'' Cambridge University Press: Cambridge, UK. Chapter 3. ISBN 0-521-27703-5 .