electrodynamics / Electromagnetic Field Theory - Bo Thide

F.4 Vector Relations

Let x be the radius vector (coordinate vector) from the origin to the point (x1;x2;x3) (x;y;z) and let jxj denote the magnitude (“length”) of x. Let further (x); (x);::: be arbitrary scalar fields and a(x);b(x);c(x);d(x);::: arbitrary vector fields.

The differential vector operator r is in Cartesian coordinates given by

where xˆi, i = 1;2;3 is the ith unit vector and xˆ1 xˆ, xˆ2 yˆ, and xˆ3 zˆ. In component (tensor) notation r can be written

Special relations

In the following k is an arbitrary constant vector.

Integral relations

Let V(S ) be the volume bounded by the closed surface S (V). Denote the 3- dimensional volume element by d3x( dV) and the surface element, directed along the outward pointing surface normal unit vector nˆ, by dS( d2x nˆ).

If S (C) is an open surface bounded by the contour C(S ), whose line element is dl, then

C S

C S

Bibliography

[1]George B. Arfken and Hans J. Weber. Mathematical Methods for Physicists. Academic Press, Inc., San Diego, CA . . . , fourth, international edition, 1995. ISBN 0-12-059816-7.

[2]Philip M. Morse and Herman Feshbach. Methods of Theoretical Physics. Part I. McGraw-Hill Book Company, Inc., New York, NY . . . , 1953. ISBN 07-043316-8.

[3]Wolfgang K. H. Panofsky and Melba Phillips. Classical Electricity and Magnetism. Addison-Wesley Publishing Company, Inc., Reading, MA . . . , second edition, 1962. ISBN 0-201-05702-6.

M

Mathematical

Methods

M.1 Scalars, Vectors and Tensors

Every physical observable can be described by a geometric object. We will describe the observables in classical electrodynamics mathematically in terms of scalars, pseudoscalars, vectors, pseudovectors, tensors or pseudotensors and will not exploit differential forms to any significant degree.

A scalar describes a scalar quantity which may or may not be constant in time and/or space. A vector describes some kind of physical motion due to vection and a tensor describes the motion or deformation due to some form of tension. However, generalisations to more abstract notions of these quantities are commonplace. The difference between a scalar, vector and tensor and a pseudoscalar, pseudovector and a pseudotensor is that the latter behave differently under such coordinate transformations which cannot be reduced to pure rotations.

Throughout we adopt the convention that Latin indices i; j;k;l;::: run over the range 1;2;3 to denote vector or tensor components in the real Euclidean three-dimensional (3D) configuration space 3 , and Greek indices ; ; ; ;:::, which are used in four-dimensional (4D) space, run over the range 0;1;2;3.

M.1.1 Vectors

Radius vector

Any vector can be represented mathematically in several different ways. One suitable representation is in terms of an ordered N-tuple, or row vector, of the coordinates xN where N is the dimensionality of the space under consideration. The most basic vector is radius vector which is the vector from the origin to the point of interest. Its N-tuple representation simply enumerates the coordinates

159

which describe this point. In this sense, the radius vector from the origin to a point is synonymous with the coordinates of the point itself.

In the 3D space 3 , we have N =3 and the radius vector can be represented by the triplet (x1;x2;x3) of coordinates xi, i = 1;2;3. The coordinates xi are scalar quantities which describe the position along the unit base vectors xˆi which span 3 . Therefore a representation of the radius vector in 3 is

i=1

where we have introduced Einstein's summation convention (E ) which states that a repeated index in a term implies summation over the range of the index in question. Whenever possible and convenient we shall in the following always assume E and suppress explicit summation in our formulae. Typographically, we represent a 3D vector by a boldface letter or symbol in a Roman font.

Alternatively, we may describe the radius vector in component notation as follows:

This component notation is particularly useful in 4D space where we can represent the radius vector either in its contravariant component form

The relation between the covariant and contravariant forms is determined by the metric tensor (also known as the fundamental tensor) whose actual form is dictated by the physics. The dual representation of vectors in contravariant and covariant forms is most convenient when we work in a non-Euclidean vector space with an indefinite metric. An example is Lorentz space 94 which is a 4D Riemannian space. 94 is often utilised to formulate the special theory of relativity.

We note that for a change of coordinates x ! x0 = x0 (x0;x1;x2;x3), due to a transformation from a system to another system 0, the differential radius vector dx transforms as

M.1.2 Fields

A field is a physical entity which depends on one or more continuous parameters. Such a parameter can be viewed as a “continuous index” which enumerates the “coordinates” of the field. In particular, in a field which depends on the usual radius vector x of 3 , each point in this space can be considered as one degree of freedom so that a field is a representation of a physical entity which has an infinite number of degrees of freedom.

Scalar fields

We denote an arbitrary scalar field in 3 by

This field describes how the scalar quantity varies continuously in 3D 3 space.

In 4D, a four-scalar field is denoted

which indicates that the four-scalar depends on all four coordinates spanning this space. Since a four-scalar has the same value at a given point regardless of coordinate system, it is also called an invariant.

Analogous to the transformation rule, Equation (M.5) on the facing page, for the differential dx , the transformation rule for the differential operator @=@x under a transformation x ! x0 becomes

which, again, follows trivially from the rules of differentiation.

Vector fields

We can represent an arbitrary vector field a(x) in 3 as follows:

In component notation this same vector can be represented as

In 4D, an arbitrary four-vector field in contravariant component form can be represented as

where x is the radius four-vector. Again, the relation between a and a is determined by the metric of the physical 4D system under consideration.

Whether an arbitrary N-tuple fulfils the requirement of being an (N-dimen- sional) contravariant vector or not, depends on its transformation properties during a change of coordinates. For instance, in 4D an assemblage y = (y0;y1;y2;y3) constitutes a contravariant four-vector (or the contravariant components of a four-vector) if and only if, during a transformation from a systemwith coordinates x to a system 0 with coordinates x0 , it transforms to the new system according to the rule

i.e., in the same way as the differential coordinate element dx transforms according to Equation (M.5) on page 160.

The analogous requirement for a covariant four-vector is that it transforms, during the change from to 0, according to the rule

i.e., in the same way as the differential operator @=@x transforms according to Equation (M.8) on the preceding page.

Tensor fields

We denote an arbitrary tensor field in 3 by A(x). This tensor field can be represented in a number of ways, for instance in the following matrix form:

where, in the last member, we have again used the more compact component notation. Strictly speaking, the tensor field described here is a tensor of rank two.