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

$\alpha : TM \rightarrow {\mathbf R},\quad \alpha_x = \alpha|_{T_xM}: T_xM\rightarrow {\mathbf R}$

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:

$\alpha_x = f_1(x) \, dx^1 + f_2(x) \, dx^2+ \cdots +f_n(x) \, dx^n$

where the f_i 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 an order 1 covariant tensor field.

Examples

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.

• Mean: The mean element of an n-vector is given by the one-form [1/n, 1/n, ..., 1/n]. That is,

$\operatorname{mean}(v) = [1/n, 1/n,\dots,1/n]\cdot v.$

• 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,

$\mathrm{NPV}(R(t)) = \langle w, R\rangle = \int_{t=0}^\infty \frac{R(t)}{(1+i)^{t}}\,dt.$

Differential of a function

Main article: Differential of a function

Let $U \subseteq \mathbb{R}$ be open (e.g., an interval (a,b)), and consider a differentiable function $f: U \to \mathbb{R}$, with derivative f'. The differential df of f, at a point $x_0\in U$, is defined as a certain linear map of the variable dx. Specifically, $df(x_0, dx): dx \mapsto f'(x_0) dx$. (The meaning of the symbol dx is thus revealed: it is simply an argument, or independent variable, of the function df.) Hence the map $x \mapsto df(x,dx)$ 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., $f\mapsto df$.

See also

• Two-form
• Reciprocal lattice
• Intermediate treatment of tensors
• Inner product

References