B.Thide - Electromagnetic Field Theory

which is the d'Alembert operator, sometimes denoted , and sometimes defined with an opposite sign convention.

END OF EXAMPLE M.6C

With the help of the del operator we can define the gradient, divergence and curl of a tensor (in the generalised sense).

The gradient

The gradient of an R3 scalar field (x), denoted r (x), is an R3 vector field a(x):

From this we see that the boldface notation for the nabla and del operators is very handy as it elucidates the 3D vectorial property of the gradient.

In 4D, the four-gradient is a covariant vector, formed as a derivative of a four-scalar field (x ), with the following component form:

i

EXAMPLE M.7 BGRADIENTS OF SCALAR FUNCTIONS OF RELATIVE DISTANCES IN 3D

Very often electrodynamic quantities are dependent on the relative distance in R3 between two vectors x and x0, i.e., on jx x0j. In analogy with Equation (M.68) on page 182, we can define the “primed” del operator in the following way:

Using this, the “unprimed” version, Equation (M.68) on page 182, and elementary rules of differentiation, we obtain the following two very useful results:

END OF EXAMPLE M.7C

The divergence

We define the 3D divergence of a vector field in R3 as

which, as indicated by the notation (x), is a scalar field in R3. We may think of the divergence as a scalar product between a vectorial operator and a vector. As is the case for any scalar product, the result of a divergence operation is a scalar. Again we see that the boldface notation for the 3D del operator is very convenient.

The four-divergence of a four-vector a is the following four-scalar:

i

which demonstrates how the “primed” divergence, defined in terms of the “primed” del operator in Equation (M.79) on the facing page, works.

END OF EXAMPLE M.8C

The Laplacian

The 3D Laplace operator or Laplacian can be described as the divergence of the gradient operator:

The symbol r2 is sometimes read del squared. If, for a scalar field (x), r2 < 0 at some point in 3D space, it is a sign of concentration of at that point.

END OF EXAMPLE M.9C

The curl

In R3 the curl of a vector field a(x), denoted r a(x), is another R3 vector field b(x) which can be defined in the following way:

where use was made of the Levi-Civita tensor, introduced in Equation (M.18) on page 173.

The covariant 4D generalisation of the curl of a four-vector field a (x ) is the antisymmetric four-tensor field

A vector with vanishing curl is said to be irrotational.

i

EXAMPLE M.10 BTHE CURL OF A GRADIENT

Using the definition of the R3 curl, Equation (M.87) on the preceding page, and the gradient, Equation (M.77) on page 183, we see that

so that the four-curl of a four-gradient vanishes just as does a curl of a gradient in R3. Hence, a gradient is always irrotational.

END OF EXAMPLE M.10C

EXAMPLE M.11 BTHE DIVERGENCE OF A CURL

With the use of the definitions of the divergence (M.82) and the curl, Equation (M.87) on the preceding page, we find that

Using the definition for the Levi-Civita symbol, defined by Equation (M.18) on

i

END OF EXAMPLE M.11C

Numerous vector algebra and vector analysis formulae are given in Chapter F. Those which are not found there can often be easily derived by using the component forms of the vectors and tensors, together with the Kronecker and Levi-Civita tensors and their generalisations to higher ranks. A short but very useful reference in this respect is the article by A. Evett [3].

M.2 Analytical Mechanics

M.2.1 Lagrange's equations

As is well known from elementary analytical mechanics, the Lagrange function or Lagrangian L is given by

where qi is the generalised coordinate, T the kinetic energy and V the potential energy of a mechanical system, The Lagrangian satisfies the Lagrange equations

i

To the generalised coordinate qi one defines a canonically conjugate momentum pi according to

M.2.2 Hamilton's equations

From L, the Hamiltonian (Hamilton function) H can be defined via the Legendre transformation

After differentiating the left and right hand sides of this definition and setting them equal we obtain

According to the definition of pi, Equation (M.99) above, the second and fourth terms on the right hand side cancel. Furthermore, noting that according to Equation (M.100) the third term on the right hand side of Equation (M.102) above is equal to p˙idqi and identifying terms, we obtain the Hamilton equations:

Bibliography

[1]G. B. ARFKEN AND H. J. WEBER, Mathematical Methods for Physicists, fourth, international ed., Academic Press, Inc., San Diego, CA . . . , 1995, ISBN 0-12- 059816-7.

[2]R. A. DEAN, Elements of Abstract Algebra, John Wiley & Sons, Inc., New York, NY . . . , 1967, ISBN 0-471-20452-8.

i

[3]A. A. EVETT, Permutation symbol approach to elementary vector analysis,

American Journal of Physics, 34 (1965), pp. 503–507.

[4]P. M. MORSE AND H. FESHBACH, Methods of Theoretical Physics, Part I. McGraw-Hill Book Company, Inc., New York, NY . . . , 1953, ISBN 07-043316-8.

[5]B. SPAIN, Tensor Calculus, third ed., Oliver and Boyd, Ltd., Edinburgh and London, 1965, ISBN 05-001331-9.

[6]W. E. THIRRING, Classical Mathematical Physics, Springer-Verlag, New York, Vienna, 1997, ISBN 0-387-94843-0.

i

i

Index

i

i