Vector field

From Citizendium, the Citizens' Compendium

Jump to: navigation, search


This article is developing and not approved.
Main Article
Talk
Related Articles  [?]
Bibliography  [?]
External Links  [?]
 
http://en.citizendium.org/wiki?title=Vector_field&printable=yes
This is a draft article, under development and not meant to be cited but you can help to improve it. These unapproved articles are subject to a disclaimer.

In physics, a vector field is a vector function on a space. That is, there is a vector associated with every point in the space. The term field in this context should not be confused with the algebraic concept of field, which is a set of numbers (most often real or complex numbers).

In non-relativistic physics, the space in question is usually the three-dimensional Euclidean space \scriptstyle \mathbb{E}^3—the infinite (Newtonian) world that we live in. A vector field may be pictured as a set of arrows, one arrow for each point of space. In this picture an arrow represents a vector, which is defined by its length and by its direction.

A vector field is called homogeneous if all vectors are parallel and of the same length. If the vectors vary from point to point in length or direction, the field is called non-homogeneous.

The vectors may be time-dependent, i.e., the length and direction of the vectors may change as a function of time; in that case the field is—non-surprisingly—said to be time-dependent.

Contents

Mathematical description

A vector field is a function of position. When we choose a Cartesian coordinate system for \scriptstyle \mathbb{E}^3 with origin O, then the vector \scriptstyle \overrightarrow{OP} pointing from O to a point P of \scriptstyle \mathbb{E}^3 has coordinates x, y and z, and the vector field is a vector function v(x,y,z),

\mathbf{v}:\quad \mathbb{E}^3 \rightarrow \mathbb{R}^3

that is,

\mathbf{v}(x,y,z) = \begin{pmatrix} v_x(x,y,z)\\ v_y(x,y,z)\\ v_z(x,y,z)\end{pmatrix}.

The length of the vectors is

|\mathbf{v}(x,y,z)| \equiv \sqrt{v^2_x+v^2_y+v^2_z},

where |v(x,y,z)| is the strength (also known as intensity) of the field at (x,y,z).

Indicating unit vectors along the Cartesian coordinate axes by ex, ey, ez, we have

\overrightarrow{OP} = \mathbf{e}_x x + \mathbf{e}_y y + \mathbf{e}_z z \equiv \big(\mathbf{e}_x,\; \mathbf{e}_y,\; \mathbf{e}_z \big) \begin{pmatrix}x\\y\\z\end{pmatrix} \equiv \big(\mathbf{e}_x,\; \mathbf{e}_y,\; \mathbf{e}_z \big) \mathbf{r}.

Equivalently,

\vec{v}(\mathbf{r}) = \big(\mathbf{e}_x,\; \mathbf{e}_y,\; \mathbf{e}_z \big) \begin{pmatrix}v_x(\mathbf{r})\\v_y(\mathbf{r})\\v_z(\mathbf{r})\end{pmatrix}\equiv \big(\mathbf{e}_x,\; \mathbf{e}_y,\; \mathbf{e}_z \big)\mathbf{v}(\mathbf{r}).

Rotation

Rotation of the coordinate system affects r and v(r), and both vectors obey the same rotation rule. Indeed, if A is a rotation matrix, a real 3×3 matrix obeying ATA = E (the identity matrix), then rotation of the coordinate axes by A gives

\mathbf{r}' = \mathbf{A} \mathbf{r} \quad\hbox{and}\quad \mathbf{v}'(\mathbf{A}\mathbf{r}) = \mathbf{A} \mathbf{v}(\mathbf{r}).

To show this we require the following invariance under rotation,

\vec{v}(\mathbf{r}) = \big(\mathbf{e}_x, \, \mathbf{e}_y, \, \mathbf{e}_z \big) \begin{pmatrix}v_x(\mathbf{r}) \\ v_y(\mathbf{r}) \\ v_z(\mathbf{r})\\ \end{pmatrix} = \big(\mathbf{e}'_x, \, \mathbf{e}'_y, \, \mathbf{e}'_z \big) \begin{pmatrix}v'_x(\mathbf{r}') \\ v'_y(\mathbf{r}') \\ v'_z(\mathbf{r}')\\ \end{pmatrix}

with

\big(\mathbf{e}'_x, \, \mathbf{e}'_y, \, \mathbf{e}'_z \big) \equiv \big(\mathbf{e}_x, \, \mathbf{e}_y, \, \mathbf{e}_z \big) \mathbf{A}^\textrm{T} \quad\hbox{and}\quad \begin{pmatrix}x' \\ y' \\ z'\\ \end{pmatrix} \equiv \mathbf{A} \begin{pmatrix}x \\ y \\ z\\ \end{pmatrix} \quad\hbox{or}\quad \mathbf{r}' = \mathbf{A}\mathbf{r}.

This expression states in fact that the vector field exists, independent of choice of coordinate system.

Hence

\mathbf{A}^\textrm{T} \begin{pmatrix}v'_x(\mathbf{r}') \\ v'_y(\mathbf{r}') \\ v'_z(\mathbf{r}')\\ \end{pmatrix} = \begin{pmatrix}v_x(\mathbf{r}) \\ v_y(\mathbf{r}) \\ v_z(\mathbf{r})\\ \end{pmatrix} \quad \Longrightarrow \mathbf{A}^\textrm{T}\mathbf{v}'(\mathbf{A}\mathbf{r}) = \mathbf{v}(\mathbf{r}) \Longrightarrow \mathbf{v}'(\mathbf{A}\mathbf{r}) = \mathbf{A} \mathbf{v}(\mathbf{r}),

where we used A AT = E. The last equation can be written as[1]

v'_\alpha(\mathbf{r}') = \sum_\beta A_{\alpha \beta} v_\beta(\mathbf{r}), \qquad \alpha,\beta = x,y,z.

Decomposition

It can be shown (see Helmholtz decomposition) that any vector field A(r) can be decomposed in two terms, an irrotational (also known as curl-free or longitudinal) field and a solenoidal (also known as divergence-free, or transverse) field,

\mathbf{A}(\mathbf{r}) = \mathbf{A}_\parallel(\mathbf{r}) + \mathbf{A}_\perp(\mathbf{r})

with

\boldsymbol{\nabla}\times \mathbf{A}_\parallel(\mathbf{r})=0, \qquad \boldsymbol{\nabla}\cdot \mathbf{A}_\perp(\mathbf{r})=0.

Reference

  1. ↑ J. D. Jackson, Classical Electrodynamics, 2nd ed. Wiley (1975), p. 247, Eq. (6.145)
Retrieved from "http://en.citizendium.org/wiki/Vector_field"
Views
Personal tools