Vectors and Vector Analysis
Appendix A Vectors and Vector Analysis A.1 Vector Algebra A.1.1. Let ai , i = 1, 2, 3 be the components of a vector a in the orthonormal basis ui of an Euclidean three-dimensional space. Using Einstein’s summation convention, the analytical expression of a is a = ai ui . (A.1) The analytical expression of the radius-vector is then r = xi ui . (A.2) A.1.2. Let a and b be two arbitrary vectors. Skipping addition and subtraction, one can define 1. The scalar product or dot product of the two vectors: a · b = (ai ui ) · (bk uk ) = ai bi = ab cos(a, b), (A.3) since ui · uk = δik. (A.4) 2. The vector product,orcross product of the two vectors: a × b =−b × a = ijka j bk ui , |a × b|=ab sin(a, b), (A.5) as well as 1 u × u = u , u = u × u , (A.6) i j ijk k s 2 sij i j © Springer-Verlag Berlin Heidelberg 2016 591 M. Chaichian et al., Electrodynamics, DOI 10.1007/978-3-642-17381-3 592 Appendix A: Vectors and Vector Analysis where ⎧ ⎨ +1, if i, j, k are an even permutation of 1, 2, 3, = − , , , , , , ijk ⎩ 1 if i j k are an odd permutation of 1 2 3 (A.7) 0, if any two indices are equal, is the Levi-Civita permutation symbol (see Sect. A.5). One can easily verify the property δil δim δin ijklmn = δ jl δ jm δ jn . (A.8) δkl δkm δkn For i = l,(A.8) becomes ijkimn = δ jmδkn − δ jnδkm, (A.9) while for i = l, j = m it becomes ijkijn = 2δkn.
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