In differential geometry, a one-form (or covector field) on a differentiable manifold is a differential form of degree one, that 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. Let U {\displaystyle U} be an open subset of M {\displaystyle M} and p ∈ U {\displaystyle p\in U} . Then
ω : U → ⋃ p ∈ U T p ∗ ( M ) p ↦ ω p ∈ T p ∗ ( M ) {\displaystyle {\begin{aligned}\omega :U&\rightarrow \bigcup _{p\in U}T_{p}^{*}(M)\\p&\mapsto \omega _{p}\in T_{p}^{*}(M)\end{aligned}}}
defines a one-form ω {\displaystyle \omega } . ω p {\displaystyle \omega _{p}} is a covector. 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 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.
Examples The most basic non-trivial differential one-form is the "change in angle" form d θ . {\displaystyle d\theta .} This is defined as the derivative of the angle "function" θ ( x , y ) {\displaystyle \theta (x,y)} (which is only defined up to an additive constant), which can be explicitly defined in terms of the atan2 function. Taking the derivative yields the following formula for the total derivative:
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