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,
- <math>\alpha : TM \rightarrow {\mathbf R},\quad \alpha_x = \alpha|_{T_xM}: T_xM\rightarrow {\mathbf R}</math>
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:
- <math>\alpha_x = f_1(x)dx^1+f_2(x)dx^2+\dots+f_n(x)dx^n</math>
where the fi are smooth functions. Note the use of upper numerical indices, not to be confused with powers. From this perspective, a one-form has a covariant transformation law on passing from one coordinate system to another. Thus a one-form is a rank 1 covariant tensor field.
Special cases
Let <math> U \subseteq \mathbb{R} </math> be open (e.g. an interval <math> (a,b) </math>), and consider a differentiable function <math> f: U \to \mathbb{R} </math>, with derivative f'. The differential df of f, at a point <math> x_0\in U </math>, is defined as a certain linear map of the variable dx. Specifically, <math>df(x_0, dx): dx \mapsto f'(x_0) dx </math>. (The meaning of the symbol dx is thus revealed: it is simply an argument, or independent variable, of the function df.) Hence the map <math>x \mapsto df(x,dx) </math> 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 i.e. <math>f\mapsto df</math>. A one-form is said to be a closed one-form if it is differentiable and its exterior derivative is everywhere equal to zero.
