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 is a smooth mapping of the total space of the tangent bundle of to whose restriction to each fibre is a linear functional on the tangent space. Symbolically,
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:
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
Many real-world concepts can be described as one-forms:
- Indexing into a vector: The second element of a three-vector is given by the one-form That is, the second element of is
- Mean: The mean element of an -vector is given by the one-form That is,
- Sampling: Sampling with a kernel can be considered a one-form, where the one-form is the kernel shifted to the appropriate location.
- Net present value of a net cash flow, is given by the one-form where is the discount rate. That is,
The most basic non-trivial differential one-form is the "change in angle" form This is defined as the derivative of the angle "function" (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:
While the angle "function" cannot be continuously defined – the function atan2 is discontinuous along the negative
-axis – which reflects the fact that angle cannot be continuously defined, this derivative is continuously defined except at the origin, reflecting the fact that infinitesimal (and indeed local) changes
in angle can be defined everywhere except the origin. Integrating this derivative along a path gives the total change in angle over the path, and integrating over a closed loop gives the winding number
In the language of differential geometry, this derivative is a one-form, and it is closed (its derivative is zero) but not exact (it is not the derivative of a 0-form, that is, a function), and in fact it generates the first de Rham cohomology of the punctured plane. This is the most basic example of such a form, and it is fundamental in differential geometry.
Differential of a function
Let be open (for example, an interval ), and consider a differentiable function with derivative The differential of at a point is defined as a certain linear map of the variable Specifically, (The meaning of the symbol is thus revealed: it is simply an argument, or independent variable, of the linear function ) Hence the map sends each point to a linear functional This is the simplest example of a differential (one-)form.
In terms of the de Rham cochain complex, one has an assignment from zero-forms (scalar functions) to one-forms; that is,