TheInfoList

In
vector calculus Vector calculus, or vector analysis, is concerned with differentiation Differentiation may refer to: Business * Differentiation (economics), the process of making a product different from other similar products * Product differentiation, in ...
and physics, a vector field is an assignment of a
vector Vector may refer to: Biology *Vector (epidemiology) In epidemiology Epidemiology is the study and analysis of the distribution (who, when, and where), patterns and risk factor, determinants of health and disease conditions in defined pop ...
to each point in a subset of
space Space is the boundless extent in which and events have relative and . In , physical space is often conceived in three s, although modern s usually consider it, with , to be part of a boundless known as . The concept of space is considere ...
. For instance, a vector field in the plane can be visualised as a collection of arrows with a given magnitude and direction, each attached to a point in the plane. Vector fields are often used to model, for example, the speed and direction of a moving fluid throughout space, or the strength and direction of some
force In physics, a force is an influence that can change the motion (physics), motion of an Physical object, object. A force can cause an object with mass to change its velocity (e.g. moving from a Newton's first law, state of rest), i.e., to acce ...

, such as the
magnetic Magnetism is a class of physical attributes that are mediated by s. s and the s of elementary particles give rise to a magnetic field, which acts on other currents and magnetic moments. Magnetism is one aspect of the combined phenomenon of . The ...

or
gravitational Gravity (), or gravitation, is a natural phenomenon by which all things with mass Mass is both a property Property (''latin: Res Privata'') in the Abstract and concrete, abstract is what belongs to or with something, whether as an ...

force, as it changes from one point to another point. The elements of
differential and integral calculus Calculus, originally called infinitesimal calculus or "the calculus of infinitesimal In mathematics, infinitesimals or infinitesimal numbers are quantities that are closer to zero than any standard real number, but are not zero. They do not ex ...
extend naturally to vector fields. When a vector field represents
force In physics, a force is an influence that can change the motion (physics), motion of an Physical object, object. A force can cause an object with mass to change its velocity (e.g. moving from a Newton's first law, state of rest), i.e., to acce ...

, the
line integral In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
of a vector field represents the
work Work may refer to: * Work (human activity), intentional activity people perform to support themselves, others, or the community ** Manual labour, physical work done by humans ** House work, housework, or homemaking * Work (physics), the product of ...

done by a force moving along a path, and under this interpretation
conservation of energy In physics Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates to the order of nature, or, in other words, to the regular s ...
is exhibited as a special case of the
fundamental theorem of calculus The fundamental theorem of calculus is a theorem that links the concept of derivative, differentiating a function (mathematics), function (calculating the gradient) with the concept of integral, integrating a function (calculating the area under t ...
. Vector fields can usefully be thought of as representing the velocity of a moving flow in space, and this physical intuition leads to notions such as the
divergence In vector calculus Vector calculus, or vector analysis, is concerned with derivative, differentiation and integral, integration of vector fields, primarily in 3-dimensional Euclidean space \mathbb^3. The term "vector calculus" is sometimes ...

(which represents the rate of change of volume of a flow) and
curl Curl or CURL may refer to: Science and technology * Curl (mathematics) In vector calculus Vector calculus, or vector analysis, is concerned with derivative, differentiation and integral, integration of vector fields, primarily in 3-dimension ...
(which represents the rotation of a flow). In coordinates, a vector field on a domain in ''n''-dimensional
Euclidean space Euclidean space is the fundamental space of . Originally, it was the of , but in modern there are Euclidean spaces of any nonnegative integer , including the three-dimensional space and the ''Euclidean plane'' (dimension two). It was introduce ...
can be represented as a
vector-valued function A vector-valued function, also referred to as a vector function, is a function (mathematics), mathematical function of one or more variables whose range of a function, range is a set of multidimensional Euclidean vector, vectors or infinite-dimensi ...

that associates an ''n''-tuple of real numbers to each point of the domain. This representation of a vector field depends on the coordinate system, and there is a well-defined transformation law in passing from one coordinate system to the other. Vector fields are often discussed on open subsets of Euclidean space, but also make sense on other subsets such as
surface File:Water droplet lying on a damask.jpg, Water droplet lying on a damask. Surface tension is high enough to prevent floating below the textile. A surface, as the term is most generally used, is the outermost or uppermost layer of a physical obje ...
s, where they associate an arrow tangent to the surface at each point (a
tangent vector :''For a more general — but much more technical — treatment of tangent vectors, see tangent space.'' In mathematics, a tangent vector is a Vector (geometry), vector that is tangent to a curve or Surface (mathematics), surface at a given point. T ...
). More generally, vector fields are defined on
differentiable manifold In mathematics, a differentiable manifold (also differential manifold) is a type of manifold The real projective plane is a two-dimensional manifold that cannot be realized in three dimensions without self-intersection, shown here as Boy's surfa ...
s, which are spaces that look like Euclidean space on small scales, but may have more complicated structure on larger scales. In this setting, a vector field gives a tangent vector at each point of the manifold (that is, a
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 sign ...
of the
tangent bundle Image:Tangent bundle.svg, Informally, the tangent bundle of a manifold (which in this case is a circle) is obtained by considering all the tangent spaces (top), and joining them together in a smooth and non-overlapping manner (bottom). In differen ...

to the manifold). Vector fields are one kind of
tensor field In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...
.

Definition

Vector fields on subsets of Euclidean space

Given a subset in , a vector field is represented by a
vector-valued function A vector-valued function, also referred to as a vector function, is a function (mathematics), mathematical function of one or more variables whose range of a function, range is a set of multidimensional Euclidean vector, vectors or infinite-dimensi ...

in standard
Cartesian coordinates A Cartesian coordinate system (, ) in a plane Plane or planes may refer to: * Airplane or aeroplane or informally plane, a powered, fixed-wing aircraft Arts, entertainment and media *Plane (Dungeons & Dragons), Plane (''Dungeons & Dragons'') ...

. If each component of is continuous, then is a continuous vector field, and more generally is a vector field if each component of is times
continuously differentiable In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gene ...
. A vector field can be visualized as assigning a vector to individual points within an ''n''-dimensional space. Given two -vector fields , defined on and a real-valued -function defined on , the two operations scalar multiplication and vector addition :$\left(fV\right)\left(p\right) := f\left(p\right)V\left(p\right)$ :$\left(V+W\right)\left(p\right) := V\left(p\right) + W\left(p\right)$ define the
module Module, modular and modularity may refer to the concept of modularity. They may also refer to: Computing and engineering * Modular design, the engineering discipline of designing complex devices using separately designed sub-components * Modula ...
of -vector fields over the ring of -functions where the multiplication of the functions is defined pointwise (therefore, it is commutative with the multiplicative identity being ).

Coordinate transformation law

In physics, a
vector Vector may refer to: Biology *Vector (epidemiology) In epidemiology Epidemiology is the study and analysis of the distribution (who, when, and where), patterns and risk factor, determinants of health and disease conditions in defined pop ...
is additionally distinguished by how its coordinates change when one measures the same vector with respect to a different background coordinate system. The transformation properties of vectors distinguish a vector as a geometrically distinct entity from a simple list of scalars, or from a
covector In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gene ...
. Thus, suppose that (''x''1,...,''x''''n'') is a choice of Cartesian coordinates, in terms of which the components of the vector ''V'' are :$V_x = \left(V_,\dots,V_\right)$ and suppose that (''y''1,...,''y''''n'') are ''n'' functions of the ''x''''i'' defining a different coordinate system. Then the components of the vector ''V'' in the new coordinates are required to satisfy the transformation law Such a transformation law is called contravariant. A similar transformation law characterizes vector fields in physics: specifically, a vector field is a specification of ''n'' functions in each coordinate system subject to the transformation law () relating the different coordinate systems. Vector fields are thus contrasted with
scalar field In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...

s, which associate a number or ''scalar'' to every point in space, and are also contrasted with simple lists of scalar fields, which do not transform under coordinate changes.

Vector fields on manifolds

Given a
differentiable manifold In mathematics, a differentiable manifold (also differential manifold) is a type of manifold The real projective plane is a two-dimensional manifold that cannot be realized in three dimensions without self-intersection, shown here as Boy's surfa ...
$M$, a vector field on $M$ is an assignment of a
tangent vector :''For a more general — but much more technical — treatment of tangent vectors, see tangent space.'' In mathematics, a tangent vector is a Vector (geometry), vector that is tangent to a curve or Surface (mathematics), surface at a given point. T ...
to each point in $M$. More precisely, a vector field $F$ is a
mapping Mapping may refer to: * Mapping (cartography), the process of making a map * Mapping (mathematics), a synonym for a mathematical function and its generalizations ** Mapping (logic), a synonym for functional predicate Types of mapping * Animated ...
from $M$ into the
tangent bundle Image:Tangent bundle.svg, Informally, the tangent bundle of a manifold (which in this case is a circle) is obtained by considering all the tangent spaces (top), and joining them together in a smooth and non-overlapping manner (bottom). In differen ...

$TM$ so that $p\circ F$ is the identity mapping where $p$ denotes the projection from $TM$ to $M$. In other words, a vector field is a
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 sign ...
of the
tangent bundle Image:Tangent bundle.svg, Informally, the tangent bundle of a manifold (which in this case is a circle) is obtained by considering all the tangent spaces (top), and joining them together in a smooth and non-overlapping manner (bottom). In differen ...

. An alternative definition: A smooth vector field $X$ on a manifold $M$ is a
linear map In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...

$X: C^\infty\left(M\right) \rightarrow C^\infty\left(M\right)$ such that $X$ is a
derivation Derivation may refer to: * Derivation (differential algebra), a unary function satisfying the Leibniz product law * Derivation (linguistics) * Formal proof or derivation, a sequence of sentences each of which is an axiom or follows from the precedi ...
: $X\left(fg\right)=fX\left(g\right)+X\left(f\right)g$ for all $f,g \in C^\infty\left(M\right)$. If the manifold $M$ is smooth or analytic—that is, the change of coordinates is smooth (analytic)—then one can make sense of the notion of smooth (analytic) vector fields. The collection of all smooth vector fields on a smooth manifold $M$ is often denoted by $\Gamma \left(TM\right)$ or $C^\infty \left(M,TM\right)$ (especially when thinking of vector fields as
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 sign ...
s); the collection of all smooth vector fields is also denoted by $\textstyle \mathfrak \left(M\right)$ (a
fraktur Fraktur () is a calligraphic hand of the Latin alphabet and any of several blackletter typefaces derived from this hand. The blackletter lines are broken up; that is, their forms contain many angles when compared to the curves of the Antiqua ...
"X").

Examples

* A vector field for the movement of air on Earth will associate for every point on the surface of the Earth a vector with the wind speed and direction for that point. This can be drawn using arrows to represent the wind; the length (
magnitude Magnitude may refer to: Mathematics *Euclidean vector, a quantity defined by both its magnitude and its direction *Magnitude (mathematics), the relative size of an object *Norm (mathematics), a term for the size or length of a vector *Order of ...
) of the arrow will be an indication of the wind speed. A "high" on the usual
barometric pressure Atmospheric pressure, also known as barometric pressure (after the barometer A barometer is a scientific instrument that is used to measure air pressure Atmospheric pressure, also known as barometric pressure (after the barometer A barom ...
map would then act as a source (arrows pointing away), and a "low" would be a sink (arrows pointing towards), since air tends to move from high pressure areas to low pressure areas. *
Velocity The velocity of an object is the Time derivative, rate of change of its Position (vector), position with respect to a frame of reference, and is a function of time. Velocity is equivalent to a specification of an object's speed and direction ...

field of a moving
fluid In physics Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates to the order of nature, or, in other words, to the regular s ...
. In this case, a
velocity The velocity of an object is the Time derivative, rate of change of its Position (vector), position with respect to a frame of reference, and is a function of time. Velocity is equivalent to a specification of an object's speed and direction ...

vector is associated to each point in the fluid. *
Streamlines, streaklines and pathlines.'') Streamlines, streaklines and pathlines are field lines in a fluid flow In physics Physics (from grc, φυσική (ἐπιστήμη), physikḗ (epistḗmē), knowledge of nature, from ''phýsis'' 'nature'), , is the natural s ...
are 3 types of lines that can be made from (time-dependent) vector fields. They are: *:streaklines: the line produced by particles passing through a specific fixed point over various times *:pathlines: showing the path that a given particle (of zero mass) would follow. *:streamlines (or fieldlines): the path of a particle influenced by the instantaneous field (i.e., the path of a particle if the field is held fixed). *
Magnetic field A magnetic field is a vector field In vector calculus and physics, a vector field is an assignment of a vector to each point in a subset of space. For instance, a vector field in the plane can be visualised as a collection of arrows with ...

s. The fieldlines can be revealed using small
iron Iron () is a with Fe (from la, ) and 26. It is a that belongs to the and of the . It is, on , right in front of (32.1% and 30.1%, respectively), forming much of Earth's and . It is the fourth most common . In its metallic state, iron ...

filings. *
Maxwell's equations Maxwell's equations are a set of coupled partial differential equation In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), ...
allow us to use a given set of initial and boundary conditions to deduce, for every point in
Euclidean space Euclidean space is the fundamental space of . Originally, it was the of , but in modern there are Euclidean spaces of any nonnegative integer , including the three-dimensional space and the ''Euclidean plane'' (dimension two). It was introduce ...
, a magnitude and direction for the
force In physics, a force is an influence that can change the motion (physics), motion of an Physical object, object. A force can cause an object with mass to change its velocity (e.g. moving from a Newton's first law, state of rest), i.e., to acce ...

experienced by a charged test particle at that point; the resulting vector field is the
electromagnetic field An electromagnetic field (also EM field or EMF) is a classical (i.e. non-quantum) field Field may refer to: Expanses of open ground * Field (agriculture), an area of land used for agricultural purposes * Airfield, an aerodrome that lacks the in ...
. *A
gravitational field In physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion and behavior through Spacetime, space and time, and the related entities of energy and force. "P ...

generated by any massive object is also a vector field. For example, the gravitational field vectors for a spherically symmetric body would all point towards the sphere's center with the magnitude of the vectors reducing as radial distance from the body increases.

Vector fields can be constructed out of
scalar field In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...

s using the
gradient In vector calculus Vector calculus, or vector analysis, is concerned with differentiation Differentiation may refer to: Business * Differentiation (economics), the process of making a product different from other similar products * Prod ...

operator (denoted by the
del Del, or nabla, is an operator used in mathematics (particularly in vector calculus Vector calculus, or vector analysis, is concerned with derivative, differentiation and integral, integration of vector fields, primarily in 3-dimensional ...

: ∇). A vector field ''V'' defined on an open set ''S'' is called a gradient field or a conservative field if there exists a real-valued function (a scalar field) ''f'' on ''S'' such that :$V = \nabla f = \bigg\left(\frac, \frac, \frac, \dots ,\frac\bigg\right).$ The associated
flow Flow may refer to: Science and technology * Flow (fluid) or fluid dynamics, the motion of a gas or liquid * Flow (geomorphology), a type of mass wasting or slope movement in geomorphology * Flow (mathematics), a group action of the real numbers on ...
is called the , and is used in the method of
gradient descent Gradient descent is a first-order iterative Mathematical optimization, optimization algorithm for finding a local minimum of a differentiable function. The idea is to take repeated steps in the opposite direction of the gradient (or approximate ...

. The path integral along any
closed curve In mathematics, a curve (also called a curved line in older texts) is an object similar to a line (geometry), line, but that does not have to be Linearity, straight. Intuitively, a curve may be thought of as the trace left by a moving point (geo ...
''γ'' (''γ''(0) = ''γ''(1)) in a conservative field is zero: :$\oint_\gamma V\left(\boldsymbol \right)\cdot \mathrm\boldsymbol = \oint_\gamma \nabla f\left(\boldsymbol \right)\cdot \mathrm\boldsymbol = f\left(\gamma\left(1\right)\right) - f\left(\gamma\left(0\right)\right).$

Central field in euclidean spaces

A ''C''-vector field over R''n'' \ is called a central field if :$V\left(T\left(p\right)\right) = T\left(V\left(p\right)\right) \qquad \left(T \in \mathrm\left(n, \mathbf\right)\right)$ where O(''n'', R) is the
orthogonal group In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
. We say central fields are invariant under orthogonal transformations around 0. The point 0 is called the center of the field. Since orthogonal transformations are actually rotations and reflections, the invariance conditions mean that vectors of a central field are always directed towards, or away from, 0; this is an alternate (and simpler) definition. A central field is always a gradient field, since defining it on one semiaxis and integrating gives an antigradient.

Operations on vector fields

Line integral

A common technique in physics is to integrate a vector field along a
curve In mathematics, a curve (also called a curved line in older texts) is an object similar to a line (geometry), line, but that does not have to be Linearity, straight. Intuitively, a curve may be thought of as the trace left by a moving point (geo ...
, also called determining its
line integral In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
. Intuitively this is summing up all vector components in line with the tangents to the curve, expressed as their scalar products. For example, given a particle in a force field (e.g. gravitation), where each vector at some point in space represents the force acting there on the particle, the line integral along a certain path is the work done on the particle, when it travels along this path. Intuitively, it is the sum of the scalar products of the force vector and the small tangent vector in each point along the curve. The line integral is constructed analogously to the
Riemann integral The partition does not need to be regular, as shown here. The approximation works as long as the width of each subdivision tends to zero. In the branch of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such to ...
and it exists if the curve is rectifiable (has finite length) and the vector field is continuous. Given a vector field and a curve ,
parametrized In mathematics, a parametric equation defines a group of quantities as Function (mathematics), functions of one or more independent variables called parameters. Parametric equations are commonly used to express the coordinates of the points that m ...
by in (where and are
real number In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no g ...
s), the line integral is defined as :$\int_\gamma V\left(\boldsymbol \right)\cdot \mathrm\boldsymbol = \int_a^b V\left(\gamma\left(t\right)\right)\cdot\dot \gamma\left(t\right)\; \mathrmt.$

Divergence

The
divergence In vector calculus Vector calculus, or vector analysis, is concerned with derivative, differentiation and integral, integration of vector fields, primarily in 3-dimensional Euclidean space \mathbb^3. The term "vector calculus" is sometimes ...

of a vector field on Euclidean space is a function (or scalar field). In three-dimensions, the divergence is defined by :$\operatorname \mathbf = \nabla \cdot \mathbf = \frac + \frac+\frac,$ with the obvious generalization to arbitrary dimensions. The divergence at a point represents the degree to which a small volume around the point is a source or a sink for the vector flow, a result which is made precise by the
divergence theorem In vector calculus, the divergence theorem, also known as Gauss's theorem or Ostrogradsky's theorem, reprinted in is a theorem which relates the flux of a vector field through a closed surface (mathematics), surface to the divergence of the fie ...

. The divergence can also be defined on a
Riemannian manifold In differential geometry Differential geometry is a mathematical Mathematics (from Greek Greek may refer to: Greece Anything of, from, or related to Greece Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a c ...
, that is, a manifold with a
Riemannian metric In differential geometry, a Riemannian manifold or Riemannian space is a real manifold, real, smooth manifold ''M'' equipped with a positive-definite Inner product space, inner product ''g'p'' on the tangent space ''T'p'M'' at each poin ...
that measures the length of vectors.

Curl in three dimensions

The
curl Curl or CURL may refer to: Science and technology * Curl (mathematics) In vector calculus Vector calculus, or vector analysis, is concerned with derivative, differentiation and integral, integration of vector fields, primarily in 3-dimension ...
is an operation which takes a vector field and produces another vector field. The curl is defined only in three dimensions, but some properties of the curl can be captured in higher dimensions with the
exterior derivative On a differentiable manifold In mathematics, a differentiable manifold (also differential manifold) is a type of manifold The real projective plane is a two-dimensional manifold that cannot be realized in three dimensions without self-interse ...
. In three dimensions, it is defined by :$\operatorname\,\mathbf = \nabla \times \mathbf = \left\left(\frac- \frac\right\right)\mathbf_1 - \left\left(\frac- \frac\right\right)\mathbf_2 + \left\left(\frac- \frac\right\right)\mathbf_3.$ The curl measures the density of the
angular momentum In , angular momentum (rarely, moment of momentum or rotational momentum) is the rotational equivalent of . It is an important quantity in physics because it is a —the total angular momentum of a closed system remains constant. In three , the ...

of the vector flow at a point, that is, the amount to which the flow circulates around a fixed axis. This intuitive description is made precise by
Stokes' theorem An illustration of Stokes' theorem, with surface , its boundary and the normal vector . Stokes' theorem, also known as Kelvin–Stokes theorem Nagayoshi Iwahori, et al.:"Bi-Bun-Seki-Bun-Gaku" :ja:裳華房, Sho-Ka-Bou(jp) 1983/12Written in Ja ...
.

Index of a vector field

The index of a vector field is an integer that helps to describe the behaviour of a vector field around an isolated zero (i.e., an isolated singularity of the field). In the plane, the index takes the value -1 at a saddle singularity but +1 at a source or sink singularity. Let the dimension of the manifold on which the vector field is defined be n. Take a small sphere S around the zero so that no other zeros lie in the interior of S. A map from this sphere to a unit sphere of dimensions ''n'' − 1 can be constructed by dividing each vector on this sphere by its length to form a unit length vector, which is a point on the unit sphere Sn-1. This defines a continuous map from S to Sn-1. The index of the vector field at the point is the
degree Degree may refer to: As a unit of measurement * Degree symbol (°), a notation used in science, engineering, and mathematics * Degree (angle), a unit of angle measurement * Degree (temperature), any of various units of temperature measurement ...
of this map. It can be shown that this integer does not depend on the choice of S, and therefore depends only on the vector field itself. The index of the vector field as a whole is defined when it has just a finite number of zeroes. In this case, all zeroes are isolated, and the index of the vector field is defined to be the sum of the indices at all zeroes. The index is not defined at any non-singular point (i.e., a point where the vector is non-zero). it is equal to +1 around a source, and more generally equal to (−1)k around a saddle that has k contracting dimensions and n-k expanding dimensions. For an ordinary (2-dimensional) sphere in three-dimensional space, it can be shown that the index of any vector field on the sphere must be 2. This shows that every such vector field must have a zero. This implies the
hairy ball theorem A hair whorl The hairy ball theorem of algebraic topology 250px, A torus, one of the most frequently studied objects in algebraic topology Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topologi ...
, which states that if a vector in R3 is assigned to each point of the unit sphere S2 in a continuous manner, then it is impossible to "comb the hairs flat", i.e., to choose the vectors in a continuous way such that they are all non-zero and tangent to S2. For a vector field on a compact manifold with a finite number of zeroes, the Poincaré-Hopf theorem states that the index of the vector field is equal to the
Euler characteristic#REDIRECT Euler characteristic In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathema ...
of the manifold.

Physical intuition

Michael Faraday Michael Faraday (; 22 September 1791 – 25 August 1867) was an English scientist A scientist is a person who conducts scientific research The scientific method is an Empirical evidence, empirical method of acquiring knowledge ...

, in his concept of ''
lines of force A line of force in Faraday Michael Faraday (; 22 September 1791 – 25 August 1867) was an English scientist A scientist is a person who conducts Scientific method, scientific research to advance knowledge in an Branches of science, ...
,'' emphasized that the field ''itself'' should be an object of study, which it has become throughout physics in the form of field theory. In addition to the magnetic field, other phenomena that were modeled by Faraday include the electrical field and
light field The light field is a vector function that describes the amount of light Light or visible light is electromagnetic radiation within the portion of the electromagnetic spectrum that is visual perception, perceived by the human eye. Visible lig ...
.

Flow curves

Consider the flow of a fluid through a region of space. At any given time, any point of the fluid has a particular velocity associated with it; thus there is a vector field associated to any flow. The converse is also true: it is possible to associate a flow to a vector field having that vector field as its velocity. Given a vector field ''V'' defined on ''S'', one defines curves γ(''t'') on ''S'' such that for each ''t'' in an interval ''I'' :$\gamma\text{'}\left(t\right) = V\left(\gamma\left(t\right)\right)\,.$ By the
Picard–Lindelöf theorem In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
, if ''V'' is
Lipschitz continuous In mathematical analysis, Lipschitz continuity, named after Rudolf Lipschitz, is a strong form of uniform continuity for function (mathematics), functions. Intuitively, a Lipschitz continuous function is limited in how fast it can change: there ex ...

there is a ''unique'' ''C''1-curve γ''x'' for each point ''x'' in ''S'' so that, for some ε > 0, :$\gamma_x\left(0\right) = x\,$ :$\gamma\text{'}_x\left(t\right) = V\left(\gamma_x\left(t\right)\right) \qquad \forall t \in \left(-\varepsilon, +\varepsilon\right) \subset \mathbf.$ The curves γ''x'' are called integral curves or trajectories (or less commonly, flow lines) of the vector field ''V'' and partition ''S'' into
equivalence class In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
es. It is not always possible to extend the interval (−ε, +ε) to the whole
real number line Real may refer to: Currencies * Brazilian real The Brazilian real ( pt, real, plural, pl. '; currency symbol, sign: R\$; ISO 4217, code: BRL) is the official currency of Brazil. It is subdivided into 100 centavos. The Central Bank of Brazil i ...

. The flow may for example reach the edge of ''S'' in a finite time. In two or three dimensions one can visualize the vector field as giving rise to a
flow Flow may refer to: Science and technology * Flow (fluid) or fluid dynamics, the motion of a gas or liquid * Flow (geomorphology), a type of mass wasting or slope movement in geomorphology * Flow (mathematics), a group action of the real numbers on ...
on ''S''. If we drop a particle into this flow at a point ''p'' it will move along the curve γ''p'' in the flow depending on the initial point ''p''. If ''p'' is a stationary point of ''V'' (i.e., the vector field is equal to the zero vector at the point ''p''), then the particle will remain at ''p''. Typical applications are
pathline.'') Streamlines, streaklines and pathlines are field line A field line is a graphical visual aid for visualizing vector field In vector calculus and physics, a vector field is an assignment of a vector to each point in a subset of space. F ...
in
fluid In physics Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates to the order of nature, or, in other words, to the regular s ...
, geodesic flow, and
one-parameter subgroup In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
s and the exponential map in
Lie group In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...
s.

By definition, a vector field is called complete if every one of its flow curves exist for all time. In particular, compactly supported vector fields on a manifold are complete. If $X$ is a complete vector field on $M$, then the
one-parameter group In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
of
diffeomorphism In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
s generated by the flow along $X$ exists for all time. On a compact manifold without boundary, every smooth vector field is complete. An example of an incomplete vector field $V$ on the real line $\mathbb R$ is given by $V\left(x\right) = x^2$. For, the differential equation $\frac = x^2$, with initial condition $x\left(0\right) = x_0$, has as its unique solution $x\left(t\right) = \frac$ if $x_0 \neq 0$ (and $x\left(t\right) = 0$ for all $t \in \mathbb R$ if $x_0 = 0$). Hence for $x_0 \neq 0$, $x\left(t\right)$ is undefined at $t = \frac$ so cannot be defined for all values of $t$.

f-relatedness

Given a
smooth function In mathematical analysis Analysis is the branch of mathematics dealing with Limit (mathematics), limits and related theories, such as Derivative, differentiation, Integral, integration, Measure (mathematics), measure, sequences, Series (mat ...

between manifolds, ''f'' : ''M'' → ''N'', the
derivative In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities ...

is an induced map on
tangent bundle Image:Tangent bundle.svg, Informally, the tangent bundle of a manifold (which in this case is a circle) is obtained by considering all the tangent spaces (top), and joining them together in a smooth and non-overlapping manner (bottom). In differen ...

s, ''f''* : ''TM'' → ''TN''. Given vector fields ''V'' : ''M'' → ''TM'' and ''W'' : ''N'' → ''TN'', we say that ''W'' is ''f''-related to ''V'' if the equation ''W'' ∘ ''f'' = ''f'' ∘ ''V'' holds. If ''V''i is ''f''-related to ''W''i, ''i'' = 1, 2, then the
Lie bracket In mathematics, a Lie algebra (pronounced "Lee") is a vector space \mathfrak g together with an Binary operation, operation called the Lie bracket, an Alternating multilinear map, alternating bilinear map \mathfrak g \times \mathfrak g \rightar ...
'V''1, ''V''2is ''f''-related to 'W''1, ''W''2

Generalizations

Replacing vectors by ''p''-vectors (''p''th exterior power of vectors) yields ''p''-vector fields; taking the
dual space In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gen ...
and exterior powers yields differential ''k''-forms, and combining these yields general
tensor field In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...
s. Algebraically, vector fields can be characterized as derivations of the algebra of smooth functions on the manifold, which leads to defining a vector field on a commutative algebra as a derivation on the algebra, which is developed in the theory of differential calculus over commutative algebras.

* Eisenbud–Levine–Khimshiashvili signature formula *
Field line A field line is a graphical for visualizing s. It consists of a directed line which is to the field at each point along its length. A diagram showing a representative set of neighboring field lines is a common way of depicting a vector field ...
*
Field strength In physics Physics (from grc, φυσική (ἐπιστήμη), physikḗ (epistḗmē), knowledge of nature, from ''phýsis'' 'nature'), , is the natural science that studies matter, its Motion (physics), motion and behavior through S ...
* ''Gradient flow'' and ''balanced flow'' in ''atmospheric dynamics'' * Lie derivative * Scalar field * Time-dependent vector field * Vector fields in cylindrical and spherical coordinates * Tensor fields

Bibliography

* * *

Online Vector Field Editor
*

— Mathworld
Vector field
— PlanetMath
3D Magnetic field viewer

An interactive application to show the effects of vector fields {{DEFAULTSORT:Vector Field Differential topology Vector calculus