Encyclopedia
In
mathematics a
vector field is a construction in vector calculus which associates a vector to every point in a Euclidean space.
Vector fields are often used in
physics to model for example the speed and direction of a moving fluid throughout space, or the strength and direction of some force, such as the
magnetic or
gravitational force, as it changes from point to point.
In the rigorous mathematical treatment, vector fields are defined on
manifolds as
sections of the manifold's tangent bundle.
Definition
Given a subset
S in
Rn a
vector field is represented by a
vector-valued functionin standard Euclidean coordinates . If there is another coordinate system
y, then
is the expression for the same vector field in the new coordinates. In particular a vector field is
not many scalar fields.
We say
V is a C
k vector field if
V is
k times continuously
differentiable. A point
p in
S is called
stationary if the vector at that point is zero
.
A vector field can be visualized as a
n-dimensional space with a
n-dimensional vector attached to each point.
Given two C
k-vector fields
V,
W defined on
S and a real valued C
k-function
f defined on
S, the two operations scalar multiplication and vector addition
define the module of C
k-vector fields over the ring of C
k-functions.
Notes
Vector fields should be compared to scalar fields, which associate a number or
scalar to every point in space . Vector fields similarily associate a length or
magnitude, as well as a
direction to every point in space. For example, in the common three-space, every point in the manifold can be associated parametrically with magnitudes of x, y and z components.
The divergence and curl are two operations on a vector field which result in a scalar field and another vector field respectively. The first of these operations is defined in any number of dimensions . The curl however, is defined only for
n=3, but it can be generalized to an arbitrary dimension using the
exterior product and
exterior derivative.
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 of the arrow will be an indication of the wind speed. A "high" on the usual barometric pressure map would then act as a source , and a "low" would be a sink , since air tends to move from high pressure areas to low pressure areas.
- Velocity field of a moving fluid. In this case, a velocity vector is associated to each point in the fluid.
- Streamlines, Streaklines and Pathlines are 3 types of lines that can be made from vector fields. They are :
streaklines — as revealed in
wind tunnels using smoke.
streamlines — as a line depicting the instantaneous field at a given time.
pathlines — showing the path that a given particle would follow.
- Magnetic fields. The fieldlines can be revealed using small iron filings.
- Maxwell's equations allow us to use a given set of initial conditions to deduce, for every point in Euclidean space, a magnitude and direction for the force experienced by a charged test particle at that point; the resulting vector field is the electromagnetic field.
Gradient field
Vector fields can be constructed out of scalar fields using the vector operator
gradient which gives rise to the following definition.
A vector field
V over
S is called a
gradient field or a
conservative field if there exists a real valued function
f on
S such that
.
The path integral along any closed curve γ in a gradient field is zero:
.
Central field
A
C∞-vector field over
Rn \ is called a
central field if
where O is the orthogonal group. 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 definition.
A central field is always a gradient field, since defining it on one semiaxis and integrating gives an antigradient.
Curve integral
A common technique in physics is to integrate a vector field along a curve: a
line integral. Given a particle in a gravitational vector field, where each vector represents the force acting on the particle at this point in space, the curve integral is the work done on the particle when it travels along a certain path.
The curve integral is constructed analogously to the
Riemann integral and it exists if the curve is rectifiable and the vector field is continuous.
Given a vector field
V and a curve γ parametrized by [0, 1] the curve integral is defined as
Flow curves
Vector fields have a nice interpretation in terms of autonomous, first order
ordinary differential equations.
Given a vector field
V defined on
S, we can try to define curves γ on
S such that for each
t in an interval
IIf
V is Lipschitz continuous we can find a
unique C1-curve γ
x for each point
x in
S so that
The curves γ
x are called
flow curves of the vector field
V and partition
S into equivalence classes. It is not always possible to extend the interval to the whole real number line. 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 given rise to a flow 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 then the particle will remain at
p.
Typical applications are streamline in fluid, geodesic flow, and one-parameter subgroups and the
exponential map in Lie groups.
Difference between scalar and vector field
The difference between a scalar and vector field is not that a scalar is just one number while a vector is several numbers. The difference is in how their coordinates respond to coordinate transformations. A scalar
is a coordinate whereas a vector
can be described by coordinates, but it
is not the collection of its coordinates.
Example 1
This example is about 2-dimensional Euclidean space where we examine Euclidean and
polar coordinates . Thus
x =
r cos θ and
y =
r sin θ and also
r2 =
x2 +
y2, cos θ =
x/ and sin θ =
y/. Suppose we have a scalar field which is given by the constant function 1, and a vector field which attaches a vector in the
r-direction with length 1 to each point. More precisely, they are given by the functions
Let us convert these fields to Euclidean coordinates. The vector of length 1 in the
r-direction has the
x coordinate cos θ and the
y coordinate sin θ. Thus in Euclidean coordinates the same fields are described by the functions
We see that while the scalar field remains the same, the vector field now looks different. The same holds even in the 1-dimensional case, as illustrated by the next example.
Example 2
Consider the 1-dimensional Euclidean space
R with its standard Euclidean coordinate
x. Also consider the coordinate ξ := 2
x. Suppose we have a scalar field and a vector field which are both given in the ξ coordinates by the constant function 1,
Thus, we have a scalar field which has the value 1 everywhere and a vector field which attaches a vector in the ξ-direction with magnitude 1 unit of ξ to each point. But if ξ changes one unit then
x changes 2 units. Then, this vector field has a magnitude of 2 in units of
x. Therefore, in the
x coordinate the scalar field and the vector field are described by the functions
which are different.
Example 3
In 1D, an example of a scalar field is the electric potential
V, which is e.g. 20 volt at a particular point. This is a scalar, not depending on the coordinate system. An electric field at that point of 5 volt/metre in some coordinate system is −5 volt/metre in an inverse coordinate system. Since a physical quantity is not just a number, but a number times a unit, there is no change of coordinate system that gives any other than one of these two values for the electric field at the point.
See also
- scalar field
- tensor field
- vector calculus
- Lie derivative
- differential geometry of curves
- Time-dependent vector field
- Vector fields in cylindrical and spherical coordinates
External links
- -- Mathworld
- -- PlanetMath
-
- Java applet illustrating vectors fields
-
- An interactive application to show the effects of vector fields
