Forms
==================

Forms are used in many contexts of relativity. It might be difficult to visualize a general n-form, 1-form, on the other hand, carries a simple geometrical meaning even to physicists.

1-form can be viewed as the dual space of vectors. In many textbooks, vectors are named as contravariant vectors. In any case, **vectors are visualized using arrows**.

By definition, contraction of 1-form :math:`\tilde \omega` and a vector :math:`v^a` should result in a number. In the field of relativity, we talk about real fields, so

.. math::
   \tilde \omega v^a \in \mathscr{R}.

A 1-form maps a vector to a real number. From this point of view, 1-form is a set of contour lines. Given this set of contour lines, it maps an arrow to a number.

.. figure:: assets/forms/1-form-contour-lines.png
   :align: center

   1-forms as contour lines. Figure (a) shows a 1-form using contour lines in a neighbourhood of a point. Figure (b) shows how a 1-form (contour lines) maps a vector (arrow) to a real number. In this case, we could assign the result real number as the number of contour lines that the arrow crossed. Different 1-forms (contour lines) take the same vector (arrow) to different real number, 4 and 2 using our definition for (b) and (c). Taken from [Schutz]_.




Refs & Notes
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.. [Schutz] Schutz, A First Course in General Relativity(Second Edition).