One-forms

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 is a smooth mapping of the total space of the tangent bundle of M to




R



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




α
:
T
M
→


R


,


α

x


=
α


|



T

x


M


:

T

x


M
→


R


,


{\displaystyle \alpha :TM\rightarrow {\mathbb {R} },\quad \alpha _{x}=\alpha |_{T_{x}M}:T_{x}M\rightarrow {\mathbb {R} },}
where αx 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:





α

x


=

f

1


(
x
)

d

x

1


+

f

2


(
x
)

d

x

2


+
⋯
+

f

n


(
x
)

d

x

n


,


{\displaystyle \alpha _{x}=f_{1}(x)\,dx_{1}+f_{2}(x)\,dx_{2}+\cdots +f_{n}(x)\,dx_{n},}
where the fi 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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