In
differential geometry
Differential geometry is a mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of differential calculus, integral calculus, linear algebra and mult ...
, a one-form on a
differentiable manifold
In mathematics, a differentiable manifold (also differential manifold) is a type of manifold that is locally similar enough to a vector space to allow one to apply calculus. Any manifold can be described by a collection of charts (atlas). One ma ...
is a
smooth section
Section, Sectioning or Sectioned may refer to:
Arts, entertainment and media
* Section (music), a complete, but not independent, musical idea
* Section (typography), a subdivision, especially of a chapter, in books and documents
** Section sig ...
of the
cotangent bundle
In 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. It may be described also as the dual bundle to the tangent bundle. Th ...
. Equivalently, a one-form on a manifold
is a smooth mapping of the
total space
In mathematics, and particularly topology, a fiber bundle (or, in Commonwealth English: fibre bundle) is a space that is a product space, but may have a different topological structure. Specifically, the similarity between a space E an ...
of the
tangent bundle of
to
whose restriction to each fibre is a linear functional on the tangent space.
Symbolically,
where
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
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
In mathematics and physics, a tensor field assigns a tensor to each point of a mathematical space (typically a Euclidean space or manifold). Tensor fields are used in differential geometry, algebraic geometry, general relativity, in the analysis ...
.
Examples
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
In computing and mathematics, the function atan2 is the 2-argument arctangent. By definition, \theta = \operatorname(y, x) is the angle measure (in radians, with -\pi < \theta \leq \pi) between the positive 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) 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 mathematics, the winding number or winding index of a closed curve in the plane around a given point is an integer representing the total number of times that curve travels counterclockwise around the point, i.e., the curve's number of t ...
times
In the language of
differential geometry
Differential geometry is a mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of differential calculus, integral calculus, linear algebra and mult ...
, this derivative is a one-form, and it is
closed
Closed may refer to:
Mathematics
* Closure (mathematics), a set, along with operations, for which applying those operations on members always results in a member of the set
* Closed set, a set which contains all its limit points
* Closed interval, ...
(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
In mathematics, de Rham cohomology (named after Georges de Rham) is a tool belonging both to algebraic topology and to differential topology, capable of expressing basic topological information about smooth manifolds in a form particularly adap ...
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
Open or OPEN may refer to:
Music
* Open (band), Australian pop/rock band
* The Open (band), English indie rock band
* Open (Blues Image album), ''Open'' (Blues Image album), 1969
* Open (Gotthard album), ''Open'' (Gotthard album), 1999
* Open (C ...
(for example, an interval
), and consider a
differentiable function
In mathematics, a differentiable function of one real variable is a function whose derivative exists at each point in its domain. In other words, the graph of a differentiable function has a non- vertical tangent line at each interior point in ...
with
derivative
In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value). Derivatives are a fundamental tool of calculus. ...
The differential
of
at a point
is defined as a certain
linear map
In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping V \to W between two vector spaces that ...
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,
See also
*
*
*
*
References
{{Manifolds
Differential forms
1 (number)