Everything about One-form totally explained
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. Explicitly, a one-form on a manifold
M is a smooth mapping of the
total space of the
tangent bundle of
M to
R whose restriction to each fibre is a linear functional on the tangent space. Symbolically,
» , 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's 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.
In terms of the
de Rham complex, one has an assignment from
zero-forms (scalar functions) to one-forms for example
.
A one-form is said to be a
closed one-form if it's differentiable and its
exterior derivative is everywhere equal to zero.
Further Information
Get more info on 'One-form'.
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