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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. (A.10) Obviously, ijkijk = 3!=6. A.1.3. Let a, b, c be three arbitrary vectors. With these vectors one can define the following two types of product: 1. The mixed product a · (b × c) = ijkai b j ck , (A.11) in particular, ui · (u j × uk ) = ijk. (A.12) 2. The double cross product a × (b × c) = (a · c)b − (a · b)c. (A.13) A.2 Orthogonal Coordinate Transformations The transformation from a three-dimensional orthogonal Euclidean coordinate sys- tem S(Oxyz) to another system S(Ox yz) can be accomplished by three main procedures: i) translation of axes; ii) rotation of axes; iii) mirror reflection. Appendix A: Vectors and Vector Analysis 593 The first two types of transformations do not change the orientation of the axes of S with respect to S (proper transformations), while under mirror reflection (i.e. x =−x, y = y, z = z) a right-handed coordinate system transforms into a left-handed coordinate system (improper transformation). There are also possible combinations of these transformations, with the observation that a succession of two proper/improper transformations gives a proper transformation, while a proper transformation followed by an improper transformation leads to an improper trans- formation. , , = , , Suppose that S and S have the same origin, and let ui ui i 1 2 3bethe orthonormal associated bases. Since = = , r xi ui xi u i (A.14) we have = , = · , xi aikxk aik u i uk = , = · , , = , , , xi aki xk aki u k ui i k 1 2 3 (A.15) as well as u i = aikuk , ui = akiu k . (A.16) The coefficients aik form the matrix of the orthogonal transformation (A.15)1, and aki –thematrix of the inverse transformation (A.15)2. The invariance of the distance between two points, 2 = = , r xi xi xk xk yields the orthogonality condition aikaim = δkm, i, k, m = 1, 2, 3. (A.17) Using the rule of the product of determinants, we find det(aik) =±1. (A.18) On the other hand, det(aik) = ijka1i a2 j a3k , i, j, k = 1, 2, 3, (A.19) that is ijka1i a2 j a3k =±1, (A.20) 594 Appendix A: Vectors and Vector Analysis or ijkaliamjank =±lmn = lmn det(aik), l, m, n = 1, 2, 3. (A.21) Here the sign “+” corresponds to the case when the frames have the same orientation, and “−” to the case when the orientations are different. Under a change of orthonormal basis, we have p + q = pi ui + qi ui = + = + = + , pi amiu m qi amiu m pm u m qm u m p q · = = ( )( ) = = = · , p q pi qi ami pm asiqs δms pmqs psqs p q × = = = ( × ) ( ), p q ijk p j qk ui ijkamjaskali pmqs u l p q det aik · ( × ) = = p q r ijk pi q j rk ijkaliamjank pl qmrn = p · (q × r ) det(aik). (A.22) Consequently, the addition (subtraction) and dot product of two vectors do not change when changing the basis, while the cross product and the mixed product (of polar vectors) change sign when the bases have different orientations. We call scalars of the first kind or scalar invariants those quantities whose sign does not depend on the basis orientation (e.g., temperature, mass, mechanical work, electric charge, etc.), and scalars of the second kind or pseudoscalars those quantities which change sign when the basis changes its orientation (e.g., magnetic flux, dΦ = B · dS, the moment of a force F with respect to an axis Δ, MΔ = uΔ · (r × F), etc.). We call polar vector or proper vector a vector which transforms to its negative under the inversion of its coordinate axes (electric field intensity, velocity of a particle, gradient of a scalar, etc.), and axial vector or pseudovector a vector which is invariant under inversion of coordinate axes (magnetic induction, angular velocity, etc). A.3 Elements of Vector Analysis A.3.1 Scalar and Vector Fields If to each point P of a domain D of the Euclidean space E3 one can associate a value of a scalar ϕ(P),inD is defined a scalar field. If to each point P one can associate a vector quantity A(P),inD is defined a vector field. A scalar (or vector) field is called stationary if ϕ (or A) do not explicitly depend on time. In the opposite case, the field is non-stationary. Appendix A: Vectors and Vector Analysis 595 A.3.2 Frequently Occurring Integrals In classical/phenomenological electrodynamics one encounters three types of inte- grals: 1. Line integral P2 a · ds P1 along a curve C, taken between the points P1 and P2, where a is a vector with its origin on the curve, and ds is a vector element of C. This integral is called circulation of the vector a along the curve C, between the points P1 and P2.If the curve C is closed, the circulation is denoted by a · ds. C 2. Double integral a · dS = a · dS = a · n dS, S S where a is a vector with its origin on the surface S, dS is a vector surface element, and n is the unit vector of the external normal to dS. This integral is called the flux of the vector a through the surface S.IfS is a closed surface, the integral is denoted by a · dS. S 3. Triple integral a dτ = a dτ = a dr = a dV, V V V where V is the volume of the three-dimensional domain D ⊂ E3, and a has its origin at some point of D. A.3.3 First-Order Vector Differential Operators The nabla or del operator is defined as ∂ ∂ ∂ ∇= u1 + u2 + u3. (A.23) ∂x1 ∂x2 ∂x3 596 Appendix A: Vectors and Vector Analysis By applying it in different ways to scalars and vectors, we obtain the gradient, diver- gence, and curl. 1. Gradient.1 Let ϕ(r) be a scalar field, with ϕ(r) a continuous function with continuous derivative in D ⊂ E3. The vector field ∂ϕ A = grad ϕ =∇ϕ = ui , i = 1, 2, 3(A.24) ∂xi is the gradient of the scalar field ϕ(r). The vector field A defined by (A.24)is called conservative. Equipotential surfaces. Consider the fixed surface ϕ(x, y, z) = C(= const.). (A.25) Then dϕ =∇ϕ · dr = 0 shows that at every point of the surface (A.25) the vector ∇ϕ is oriented along the normal to the surface. Giving values to C, one obtains a family of surfaces called equipotential surfaces,orlevel surfaces. Field lines.LetA(r) be a stationary vector field, and C – a curve given by its parametric equations xi = xi (s), i = 1, 2, 3. If at every point the field A is tan- gent to the curve C, then the curve is a line of the vector field A. The differential equations of the field lines are obtained by projecting on axes the vector relation A × dr = 0, where dr is a vector element of the field line. Directional derivative. Let us project the vector ∇ϕ unto the direction defined by the unit vector u. Since dr = u|dr|=u ds, where ds is a curve element in the direction u,wehave dϕ (∇ϕ) · u = (∇ϕ) = . (A.26) u ds If, in particular, u is the unit vector n of the external normal to the surface (A.25), then dϕ ∇ϕ = ≥ 0, (A.27) n dn meaning that the gradient is oriented along the normal to the equipotential sur- faces, and points in the direction of the greatest rate of increase of the scalar field. 1As a matter of fact, gradient, divergence,andcurl are not bona fide first-order vector differential operators, but the results of the nabla operator (Hamilton’s operator) action, in different ways, upon scalar and/or vector fields. Note that the Laplacian =∇·∇=∇2 and the d’Alembertian = 1 ∂2 − v2 ∂t2 are true differential operators, but of the second order.
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