# One-form

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 whose restriction to each fibre is a linear functional on the tangent space. Symbolically,

where *α*_{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} 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

### Applications

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 [0, 1, 0]. That is, the second element of [
*x*,*y*,*z*] is

- [0, 1, 0] · [
*x*,*y*,*z*] =*y*.

- [0, 1, 0] · [

- Mean: The mean element of an
*n*-vector is given by the one-form [1/*n*, 1/*n*, ..., 1/*n*]. 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,
*R*(*t*), is given by the one-form*w*(*t*) := (1 +*i*)^{−t}where*i*is the discount rate. That is,

### Differential

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 a 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 *y*-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, i.e., 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 (e.g., an interval ), and consider a differentiable function , with derivative *f'*. The differential *df* of *f*, at a point , is defined as a certain linear map of the variable *dx*. Specifically, . (The meaning of the symbol *dx* is thus revealed: it is simply an argument, or independent variable, of the linear function .) Hence the map sends each point *x* to a linear functional . 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., .

## See also

## References

- J.A. Wheeler; C. Misner; K.S. Thorne (1973).
*Gravitation*. W.H. Freeman & Co. p. 57. ISBN 0-7167-0344-0.