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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.
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

{\displaystyle M}
is a smooth mapping of the total space of the tangent bundle of

M

{\displaystyle M}
to

R

{\displaystyle \mathbb {R} }
whose restriction to each fibre is a linear functional on the tangent space. Symbolically,

where

α

x

{\displaystyle \alpha _{x}}
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

f

i

{\displaystyle f_{i}}
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.

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