Appendix A. The Dirac Delta Distribution The Dirac delta is an element of the set of mathematical objects called distri­ butions. These are a generalization of functions. Some of them are particularly suitable for describing physical entities in an approximated way. For instance, the Dirac delta can be used to represent highly concentrated distributions of mass or charge. The theory of distributions was developed in the first half of the twentieth century by the French mathematician L. Schwartz, and its name is due precisely to its connection with mass and charge distributions. Under appropriate conditions, the Dirac delta distribution 6(x) can be treated, cum grano salis, as a function of the variable x E (-00,00), defined by the conditions 6(x) = 0 for x -I- 0, { (A.I) J~oo 6(x) dx = 1. This is an even distribution, 6(x) = 6( -x), and it is infinitely concentrated at x = O. The Dirac delta distribution can be approximated by functions, called approximant kernels of 6 (x), in appropriate limits. Some of these approximant kernels are: f(x) = (27ra)-1/2 exp( _x2 /2a) for a -+ 0 (Gaussian), for a -+ 0 (Lorentzian), (A.2) f(x) = '\exp( -2'\lxl) for ,\ -+ 00, f(x) = sin(ax)/7rx for a -+ 00. The main property of the Dirac delta of interest for applications is that for any function f (x) b f(xo) = 1 f(x) 6(x - Xo) dx, (A.3) where a < Xo < b, if f(x) is continuous at x = Xo. Integrating by parts, the "derivatives" of 6(x) can be introduced. They are also distributions, defined by the property 274 A. The Dirac Delta Distribution (A A) where the labels (n) indicate nth-order differentiation. When the argument of the delta distribution is a function ¢( x), a variable change makes it possible to prove that (A.5) where Xi are the zeros of ¢(x) in the interval (a, b). In particular, lb f(x) 8[Q(x - xo)] dx = IQI- 1 f(xo), (A.6) or, formally, 8(Qx) = 8(x)/IQI. The product of Dirac deltas allows us to define the multidimensional version of this distribution. For instance, in three dimensions, 8(r) = 8(x)8(y)8(z), (A.7) where (x, y, z) are the Cartesian components of r. In spherical coordinates (r, (), '1') we find 1 1 8(r) = 2--=-() 8(r)8(())8(cp) = 2 8(r)8(cos())8(cp). (A.8) r sm r The multidimensional delta satisfies i f(r) 8(r - ro) d3r = f(ro), (A.9) if ro is in the integration volume V. Other important properties of the Dirac delta are the following: d 8(x) = dx ()(x), (A.lO) where ()(x) is the Heaviside function ()(x) = {O ~fx < 0, (A.ll) 1 If x> 0, also called the step function. Moreover, 8(r) = _~\72~, (A.12) 411" r which, according to Gauss's law, implies that 8(r) is the charge density to be associated with a unit point charge at the origin. Finally, 00 8(x) = -1 1 exp(ikx) dk, (A.13) 211" -00 which relates the Dirac delta to the Fourier tranform of a constant function. B. Legendre Polynomials and Spherical Harmonics From Laplace's equation, \7 2 4> = 0, (B.1) the separation of variables in spherical coordinates, 4>( r, (), 'P) = R( r) Y ((), 'P), produces for Y ((), 'P) the differential equation sin() ae(sin() aeY) + a!ipY = -AY, (B.2) where A is a separation constant. Under conditions of azimuthal symmetry, the solution of Laplace's equa­ tion does not depend on 'P. Requiring regularity conditions at () = °and () = Jr, the solutions of the angular part of the equation are the Leg­ endre polynomials P1(x), 1= 0,1,2, ... , evaluated in terms of x = cos(): Y((),'P) = Y(()) = P1(cos()). The lth-degree Legendre polynomial P1(x) is defined by 1 d1 2 I P1(x) = ii' -dI (x - 1) . (B.3) 2. x Legendre polynomials satisfy the orthogonality relation l 2 J Pdx) P1(x) dx = -1- 511', (B.4) -1 2 + 1 and form a set of orthogonal functions in the interval (-1, 1). Any well­ behaved function can thus be expanded in a series of Legendre polynomials within that interval, and such an expansion is unique. The generating function of PI (x), (Xl I Ir - r /I -1 = ""'~ ---r:t=lP1(cos/,),r < (B.5) 1=0 r> where /' is the angle determined by rand r/, and r < = min( r, r/), r> = max(r, r/), turns out to be particularly important in applications to electro­ statics (see Chap. 4). 276 B. Legendre Polynomials and Spherical Harmonics Some relevant properties of the Legendre polynomials are the following: P2n+1 (0) = 0, (B.6) (x2 -l)d~Pl(X) = lxP1(x) -lP1- 1(X). The first few Legendre polynomials are Po(x) = 1, Pdx) = x, (B.7) P2(x) = ~(3x2 - 1), P3(x) = ~(5x3 - 3x). The solutions of the angular part of Laplace's equation when there is no azimuthal symmetry are the spherical harmonic functions of order (l, m), Yim(O, <p), with l = 0,1,2, ... and m = -l, -l + 1, ... , l- 1, l. For m 2': 0, 2l + 1 (l - m)! m --(l )' PI (cosO) exp(im<p), (B.8) 47r + m . and Yi,-m = (-l)myt;'", where * indicates complex conjugation. The func­ tions Pzm (x) are the associated Legendre junctions, defined by 1 d1+1ml pm(x) = _ (1 - x2)lml/2 (x2 - 1)1. (B.9) I 21l! dx 1+1ml Note that Pz°(x) = P1(x). Spherical harmonics satisfy the following orthogonality and completeness relations: (B.IO) 00 I z= z= yt;" (0', <p')Yim (0, <p) = 8( cos () - cos O')8( <p - <p'), (B.ll) 1=0 m=-l so that they constitute a complete set of functions in °-:; 0 -:; 7r, °-:; <p -:; 27r. The following addition theorem, which is useful for several applications, holds: I PI° (cos 1') = -l-47r z= yt;" (()" <p') Yim ((), <p), (B.12) 2 + 1 m=-l B. Legendre Polynomials and Spherical Harmonics 277 where 'Y is the angle between the directions defined by the angular coordinates (0,'1') and (()', '1"). This theorem can be used to write (B.5) as 00 1 1 1 r < * (' ') ( ) 1r - r '1-1 = 471" '"~ '"~ 2l + 1 rl +1 Ylm () ,'I' Yim (), 'I' , (B.13) 1=0 m=-I > where (0, '1') and (()', '1") are the angular coordinates of the vectors rand r', respectively, and r> and r < are defined as in (B.5). The first few spherical harmonics are Yoo = (471")-1/2, YlO = (3/471")1/2 cos(), Y1,±1 = =r=(3/871") 1/2 sinO exp(±icp), (B.14) Y20 = (5/1671")1/2(3cos2 () -1), Y2,±1 = =r=(15/871") 1/2 sinO cos () exp(±icp) , Y2,±2 = =r=(15/3271")1/2sen2() exp(±2icp). C. Covariant Notation and Tensor Calculus Tensor calculus is the formalism used in special and general relativity the­ ory for operating in the four-dimensional space with spatial coordinates (x, y, z) and time coordinate ct. Generally, four-dimensional tensor calculus involves mathematical objects defined in a space with generalized coordinates (xO, xl, x2, x3). Such objects, called tensors, are characterized by transform­ ing in a well-defined way under a coordinate change to the new variables (xlO , xll, x,2, x,3), which are related to the original coordinates through cer­ tain functions x lO = x,O(xO, Xl , x2, x3) , (C.l) X,3 = x'3(xO, xl, x2, x3). A zeroth-rank tensor or scalar, s, is a quantity whose value does not change under the coordinate transformation, s' = s. (C.2) A first-rank contravariant tensor or, contravariant four-vector, is a set of four quantities (vO,vl,v2 ,V3 ), whose elements are generically denoted by vI-' and which transform according to 3 a 'I-' a 'I-' v-'I-'-L~a v-ao_~ v.0 (C.3) X O X O 0=0 We have introduced here the Einstein convention of summation over repeated indices. According to this convention, when an index appears twice in an expression it is implicitly assumed that the expression is summed over the repeated index, from 0 to 3. The summation over repeated indices is called contraction with respect to those indices. A first-rank covariant tensor, or covariant four-vector, Ul-" is a set of four quantities (UO,Ul,U2,U3) which transform according to (C.4) 280 C. Covariant Notation and Tensor Calculus Four-vector physical quantities have, in general, both a contravariant and a covariant representation. Higher rth-rank tensors are sets of 4T quantities labeled by r indices, which transform as products of r four-vector components. Each index can be of contravariant or covariant character, according to the corresponding transformation rule. For instance, any of the sixteen components of a twice­ contravariant second-rank tensor tJ.LV transforms according to 8 'J.L 8 'v 'J.LV _ X x 0.(3 (C ) t - 8xo. 8x(3 t . .5 In general, we have 8 'J.L 8 'v n 8 8 ' T'J.Lv"·=~~ ... ~~ ... To.(3 .. · (C.6) pu... 8xo. 8x(3 8x'p 8x'u 8, .. · . It is worthwhile stressing that the contraction operation preserves the co­ variant properties of the result. This implies that, in general, the contraction of two tensors over an arbitrary number of indices produces another tensor. For instance, given a four-vector vJ.L its modulus vJ.LvJ.L is a scalar, i.e. is in­ variant under coordinate transformations. The transformed coordinates of a four-vector also form a four-vector, as they are obtained from the contraction of the Jacobian tensor associated with the transformation, JJ.Lv = 8x'J.L /8xv, and the original four-vector, v'J.L = JJ.Lvvv.