In
linear algebraLinear algebra is a branch of mathematics that studies vector spaces, also called linear spaces, along with linear functions that input one vector and output another. Such functions are called linear maps and can be represented by matrices if a basis is given. Thus matrix theory is often...
, a oneform on a
vector spaceA vector space is a mathematical structure formed by a collection of vectors: objects that may be added together and multiplied by numbers, called scalars in this context. Scalars are often taken to be real numbers, but one may also consider vector spaces with scalar multiplication by complex...
is the same as a
linear functionalIn linear algebra, a linear functional or linear form is a linear map from a vector space to its field of scalars. In Rn, if vectors are represented as column vectors, then linear functionals are represented as row vectors, and their action on vectors is given by the dot product, or the...
on the space. The usage of oneform in this context usually distinguishes the oneforms from higherdegree
multilinear functionals on the space. For details, see
linear functionalIn linear algebra, a linear functional or linear form is a linear map from a vector space to its field of scalars. In Rn, if vectors are represented as column vectors, then linear functionals are represented as row vectors, and their action on vectors is given by the dot product, or the...
.
In differential geometry, a oneform on a
differentiable manifoldA differentiable manifold is a type of manifold that is locally similar enough to a linear space to allow one to do calculus. Any manifold can be described by a collection of charts, also known as an atlas. One may then apply ideas from calculus while working within the individual charts, since...
is a
smoothIn mathematical analysis, a differentiability class is a classification of functions according to the properties of their derivatives. Higher order differentiability classes correspond to the existence of more derivatives. Functions that have derivatives of all orders are called smooth.Most of...
sectionIn the mathematical field of topology, a section of a fiber bundle π is a continuous right inverse of the function π...
of the
cotangent bundleIn mathematics, especially differential geometry, the cotangent bundle of a smooth manifold is the vector bundle of all the cotangent spaces at every point in the manifold...
. Equivalently, a oneform on a manifold M is a smooth mapping of the total space of the
tangent bundleIn differential geometry, the tangent bundle of a differentiable manifold M is the disjoint unionThe disjoint union assures that for any two points x1 and x2 of manifold M the tangent spaces T1 and T2 have no common vector...
of M to R whose restriction to each fibre is a linear functional on the tangent space. Symbolically,
where α
_{x} is linear.
Often oneforms are described
locallyIn mathematics, a phenomenon is sometimes said to occur locally if, roughly speaking, it occurs on sufficiently small or arbitrarily small neighborhoods of points.Properties of a single space:...
, particularly in
local coordinatesLocal coordinates are measurement indices into a local coordinate system or a local coordinate space. A simple example is using house numbers to locate a house on a street; the street is a local coordinate system within a larger system composed of city townships, states, countries, etc.Local...
. In a local coordinate system, a oneform is a linear combination of the
differentialsIn differential geometry, the exterior derivative extends the concept of the differential of a function, which is a 1form, to differential forms of higher degree. Its current form was invented by Élie Cartan....
of the coordinates:
where the f
_{i} are smooth functions. Note the use of upper numerical indices, not to be confused with powers. From this perspective, a oneform has a covariant transformation law on passing from one coordinate system to another. Thus a oneform is an order 1 covariant
tensor fieldIn mathematics, physics and engineering, a tensor field assigns a tensor to each point of a mathematical space . Tensor fields are used in differential geometry, algebraic geometry, general relativity, in the analysis of stress and strain in materials, and in numerous applications in the physical...
.
Examples
Many realworld concepts can be described as oneforms:
 Indexing into a vector: The second element of a threevector is given by the oneform [0, 1, 0]. That is, the second element of [x ,y ,z] is

 [0, 1, 0] · [x, y, z] = y.
 Mean
In statistics, mean has two related meanings:* the arithmetic mean .* the expected value of a random variable, which is also called the population mean....
: The mean element of an nvector is given by the oneform [1/n, 1/n, ..., 1/n]. That is,


 Sampling
In signal processing, sampling is the reduction of a continuous signal to a discrete signal. A common example is the conversion of a sound wave to a sequence of samples ....
: Sampling with a kernel can be considered a oneform, where the oneform is the kernel shifted to the appropriate location.
 Net present value
In finance, the net present value or net present worth of a time series of cash flows, both incoming and outgoing, is defined as the sum of the present values of the individual cash flows of the same entity...
of a net cash flowCash flow is the movement of money into or out of a business, project, or financial product. It is usually measured during a specified, finite period of time. Measurement of cash flow can be used for calculating other parameters that give information on a company's value and situation.Cash flow...
, R(t), is given by the oneform w(t) := (1 + i)^{−t} where i is the discount rateThe discount rate can mean*an interest rate a central bank charges depository institutions that borrow reserves from it, for example for the use of the Federal Reserve's discount window....
. That is,


Differential of a function
Let
be
openThe concept of an open set is fundamental to many areas of mathematics, especially pointset topology and metric topology. Intuitively speaking, a set U is open if any point x in U can be "moved" a small amount in any direction and still be in the set U...
(e.g., an interval
), and consider a differentiable
functionIn mathematics, a function associates one quantity, the argument of the function, also known as the input, with another quantity, the value of the function, also known as the output. A function assigns exactly one output to each input. The argument and the value may be real numbers, but they can...
, with
derivativeIn calculus, a branch of mathematics, the derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much one quantity is changing in response to changes in some other quantity; for example, the derivative of the position of a...
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 function df.) Hence the map
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 zeroforms (scalar functions) to oneforms i.e.,
.
See also
 Twoform
In linear algebra, a twoform is another term for a bilinear form, typically used in informal discussions, or sometimes to indicate that the bilinear form is skewsymmetric....
 Reciprocal lattice
In physics, the reciprocal lattice of a lattice is the lattice in which the Fourier transform of the spatial function of the original lattice is represented. This space is also known as momentum space or less commonly kspace, due to the relationship between the Pontryagin duals momentum and...
 Intermediate treatment of tensors
 Inner product