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In mathematics, a **time dependent vector field** is a construction in vector calculus which generalizes the concept of vector fields. It can be thought of as a vector field which moves as time passes. For every instant of time, it associates a vector to every point in a Euclidean space or in a manifold.

A **time dependent vector field** on a manifold *M* is a map from an open subset on

such that for every , is an element of .

For every such that the set

is nonempty, is a vector field in the usual sense defined on the open set .

Given a time dependent vector field *X* on a manifold *M*, we can associate to it the following differential equation:

which is called nonautonomous by definition.

An integral curve of the equation above (also called an integral curve of *X*) is a map

such that , is an element of the domain of definition of *X* and

- .

A time dependent vector field on can be thought of as a vector field on where does not depend on

Conversely, associated with a time-dependent vector field on is a time-independent one

on In coordinates,

The system of autonomous differential equations for is equivalent to that of non-autonomous ones for and is a bijection between the sets of integral curves of and respectively.

The flow of a time dependent vector field *X*, is the unique differentiable map

such that for every ,

is the integral curve of *X* that satisfies .

We define as

- If and then
- , is a diffeomorphism with inverse .

Let *X* and *Y* be smooth time dependent vector fields and the flow of *X*. The following identity can be proved:

Also, we can define time dependent tensor fields in an analogous way, and prove this similar identity, assuming that is a smooth time dependent tensor field:

This last identity is useful to prove the Darboux theorem.

In vector calculus and differential geometry the **generalized Stokes theorem**, also called the **Stokes–Cartan theorem**, is a statement about the integration of differential forms on manifolds, which both simplifies and generalizes several theorems from vector calculus. In particular, the fundamental theorem of calculus is the special case where the manifold is a line segment, Green’s theorem and Stokes' theorem are the cases of a surface in or and the divergence theorem is the case of a volume in Hence, the theorem is sometimes referred to as the **Fundamental Theorem of Multivariate Calculus**.

In differential geometry, the **tangent bundle** of a differentiable manifold is a manifold which assembles all the tangent vectors in . As a set, it is given by the disjoint union of the tangent spaces of . That is,

In mathematics, a **linear form** is a linear map from a vector space to its field of scalars.

In mathematics, **differential forms** provide a unified approach to define integrands over curves, surfaces, solids, and higher-dimensional manifolds. The modern notion of differential forms was pioneered by Élie Cartan. It has many applications, especially in geometry, topology and physics.

In mathematics and classical mechanics, the **Poisson bracket** is an important binary operation in Hamiltonian mechanics, playing a central role in Hamilton's equations of motion, which govern the time evolution of a Hamiltonian dynamical system. The Poisson bracket also distinguishes a certain class of coordinate transformations, called *canonical transformations*, which map canonical coordinate systems into canonical coordinate systems. A "canonical coordinate system" consists of canonical position and momentum variables that satisfy canonical Poisson bracket relations. The set of possible canonical transformations is always very rich. For instance, it is often possible to choose the Hamiltonian itself as one of the new canonical momentum coordinates.

In differential geometry, the **Lie derivative**, named after Sophus Lie by Władysław Ślebodziński, evaluates the change of a tensor field, along the flow defined by another vector field. This change is coordinate invariant and therefore the Lie derivative is defined on any differentiable manifold.

In mathematics, the **Hodge star operator** or **Hodge star** is a linear map defined on the exterior algebra of a finite-dimensional oriented vector space endowed with a nondegenerate symmetric bilinear form. Applying the operator to an element of the algebra produces the **Hodge dual** of the element. This map was introduced by W. V. D. Hodge.

In mathematics, a Lie algebroid is a vector bundle together with a Lie bracket on its space of sections and a vector bundle morphism , satisfying a Leibniz rule. A Lie algebroid can thus be thought of as a "many-object generalisation" of a Lie algebra.

In mathematics, and especially differential geometry and gauge theory, a **connection** is a device that defines a notion of parallel transport on the bundle; that is, a way to "connect" or identify fibers over nearby points. A **principal G-connection** on a principal G-bundle

In mathematics, the **Poincaré lemma** gives a sufficient condition for a closed differential form to be exact. Precisely, it states that every closed *p*-form on an open ball in **R**^{n} is exact for *p* with 1 ≤ *p* ≤ *n*. The lemma was introduced by Henri Poincaré in 1886.

In mathematics, a **Sobolev space** is a vector space of functions equipped with a norm that is a combination of *L ^{p}*-norms of the function together with its derivatives up to a given order. The derivatives are understood in a suitable weak sense to make the space complete, i.e. a Banach space. Intuitively, a Sobolev space is a space of functions possessing sufficiently many derivatives for some application domain, such as partial differential equations, and equipped with a norm that measures both the size and regularity of a function.

In mathematics, a **differentiable 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 may then apply ideas from calculus while working within the individual charts, since each chart lies within a vector space to which the usual rules of calculus apply. If the charts are suitably compatible, then computations done in one chart are valid in any other differentiable chart.

In mathematics, in particular in algebraic geometry and differential geometry, **Dolbeault cohomology** (named after Pierre Dolbeault) is an analog of de Rham cohomology for complex manifolds. Let *M* be a complex manifold. Then the Dolbeault cohomology groups depend on a pair of integers *p* and *q* and are realized as a subquotient of the space of complex differential forms of degree (*p*,*q*).

In complex analysis, functional analysis and operator theory, a **Bergman space**, named after Stefan Bergman, is a function space of holomorphic functions in a domain *D* of the complex plane that are sufficiently well-behaved at the boundary that they are absolutely integrable. Specifically, for 0 < *p* < ∞, the Bergman space *A*^{p}(*D*) is the space of all holomorphic functions in *D* for which the p-norm is finite:

In mathematics, a **holomorphic vector bundle** is a complex vector bundle over a complex manifold X such that the total space E is a complex manifold and the projection map π : *E* → *X* is holomorphic. Fundamental examples are the holomorphic tangent bundle of a complex manifold, and its dual, the holomorphic cotangent bundle. A **holomorphic line bundle** is a rank one holomorphic vector bundle.

In mathematics, the **Kodaira–Spencer map**, introduced by Kunihiko Kodaira and Donald C. Spencer, is a map associated to a deformation of a scheme or complex manifold *X*, taking a tangent space of a point of the deformation space to the first cohomology group of the sheaf of vector fields on *X*.

In physics and mathematics, and especially differential geometry and gauge theory, the **Yang–Mills equations** are a system of partial differential equations for a connection on a vector bundle or principal bundle. They arise in physics as the Euler–Lagrange equations of the **Yang–Mills action functional**. They have also found significant use in mathematics.

In differential geometry, the **integration along fibers** of a *k*-form yields a -form where *m* is the dimension of the fiber, via "integration". It is also called the **fiber integration**.

In mathematics, and especially differential geometry and mathematical physics, **gauge theory** is the general study of connections on vector bundles, principal bundles, and fibre bundles. Gauge theory in mathematics should not be confused with the closely related concept of a gauge theory in physics, which is a field theory which admits gauge symmetry. In mathematics *theory* means a mathematical theory, encapsulating the general study of a collection of concepts or phenomena, whereas in the physical sense a gauge theory is a mathematical model of some natural phenomenon.

In mathematics, **calculus on Euclidean space** is a generalization of calculus of functions in one or several variables to calculus of functions on Euclidean space as well as a finite-dimensional real vector space. This calculus is also known as **advanced calculus**, especially in the United States. It is similar to multivariable calculus but is somehow more sophisticated in that it uses linear algebra more extensively and covers some concepts from differential geometry such as differential forms and Stokes' formula in terms of differential forms. This extensive use of linear algebra also allows a natural generalization of multivariable calculus to calculus on Banach spaces or topological vector spaces.

- Lee, John M.,
*Introduction to Smooth Manifolds*, Springer-Verlag, New York (2003) ISBN 0-387-95495-3. Graduate-level textbook on smooth manifolds.

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